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.
RBPEATBD MEASURBS DATA ANALYSIS
WITH NONNORMAL OUTCOMBS
by
ALULA HADGU
Department of Biostatistics
University of North Carolina
•
Institute of Statistics
Mimeo Series No. 2120T
September 1993
REPEATED MEASURES DATA ANALYSIS WITH NONNORMAL OUTCOMES
by
Alula Badgu
A Dissertation submitted to the faculty of the University of North Carolina at Chapel Bill in
partial fulfillment of the requirements for the degree of Doctor of Public Bealth in the
Department of Biostatistica.
Chapel Bill, 1993
Approved by:
~.
Advisor
--~---Reader
•
ii
ABSTRACT
ALULA HADGU. Repeated Measures Data Analysis With Nonnormal Outcomes. (Under
the direction of GARY G. KOCH).
Longitudinal data present statistical problems of interest in public health and epidemiologic
studies. In this dissertat.ion, we consider longit.udinal data in which out.come measurements
are repeatedly taken on each subject., and there is primary interest in the dependence of the
outcome variable on covariatee. This work is motivat.ed by a longitudinal study of women
and their ectopic pregnancy outcomes in Lund, Sweden, and a randomized clinical trial of
three mouthrinses in inhibiting the development of dental plaque.
The common data
structure of these studies is the presence of an intra-class or serial correlation within primary
sampling units or subjects. Recently Liang and Zeger extended Generalized Linear Models by
taking correlations within primary sampling units into account using quasi-likelihood. In this
dissertation, we review and apply the Liang-Zeger methodology to the two data sets
mentioned above. We further analyze the ectopic pregnancy data using Rosner's conditional
modeling approach which is a generalization of the beta binomial models for correlated data.
The specific questions we wish to address in the ectopic pregnancy study from Lund include:
1.
Does the presence of pelvic inflammatory disease predispose a woman to a subsequent
development of ectopic pregnancy?
2.
Does the presence of mycoplasma at index laparoscopy increase a woman's risk of having
ectopic pregnancy?
3.
What are t.he other predictors of ectopic pregnancy?
The objectives of the randomized clinical trial include:
1.
2.
JJe the two experimental mouthrinses more effective than the standard mouthrinse?
Is the effect. dependent on baseline plaque measurement? If so, in what way?
iii
Although the Liang-Zeger methodology gives regression estimates and variances that are
robust against misspecification of the correlation structure, correct specification provides more
efficient estimates. Rotnitzky and Jewell (Biometrika, 1991) provide a technique to verify if
the "guessed" correlation matrix is reasonable. In this work, we modified the GEE software
by writing additional IML codes
10
that one may use the technique of Rotnitzky and Jewell
to verify if the "gueued" correlation matrix is reasonable.
Furthermore, we propose to use the GEE approach u a way to obtain variance estimates of
sensitivity and specificity estimates when the data consist of correlated binary outcomes. We
give
rust
order approximations to the variances of sensitivity, specificity, predictive value
positive, and predictive value negative •
v
TABLE OF CONTENTS
Chapter
•
Page
1.
INTRODUcrION ...•.......•••••••.•.••.•.•••.•.••••••..••.••••••••.•.••••••••..••••••••••••.•••.•.••...••••••...•••••.••• 1
2.
LITERATURE REVIEW
3.
2.1
Linear Models for Repeated Measurement Data
2.2
Approach for NODDormal Data
5
REPEATED MEASUREMENT ANALYSIS USING GENERALIZED
ESTlMATING EQUATIONS••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••.••••••••••••••13
3.1
Generalized Linear Models and Quasi-likelihood
3.2
Quasi Likelihood
3.3
Extension of GLM to Longitudinal Data
3.4
Logistic regression
4. THE EPIDEMIOLOGY OF PELVIC INFLAMMATORY DISEASE
AND ECTOPIC PREGNANCY
5.
4.1
Epidemiology and Etiology of Ectopic Pregnancy
4.2
Epidemiology and Etiology of PID
3g
ANALYSIS OF THE ECTOPIC PREGNANCY DATA USING GEE
AND CONDITIONAL MODELS
5.1
Description of the Lund Study
5.2
Descriptive Analysis
5.3
Model Fitting
5.3.1 Analysis Using Standard Logistic Regression Model
5.3.2 Analysis Using GEE Techniques
5.3.2.1 Selection Between Correlation Structuna
4:1
vi
5.3.3 Analysis Using Rosner's Beta-Binomial Mod..:d
5.3.3.1 Extension to the General Case
5.3.3.2 Comparison with Ordinary Logiatic Regreaaion
5.3.3.3 Application of Roener's Model to the Ectopic Pregnancy Data
5.3.3.4 Comparison with Ordinary Logiatic Regression and Gee Model
5.3.3.5 Comparison of the Intercepts of the Models
5.3.4 Alternative Approach
6.
ANALYSIS OF A MOUTH-WASH CLINICAL TRlAL
6.1
92
Descriptive Analysis
6.1.1
Mouthrinse Dispensation
6.1.2 Examination
6.1.3 Descriptive Analysis
6.2 Model Fitting
6.2.1 Analysis Using GEE Techniques
6.2.2 Selection Among the Correlation Structures
6.2.3 The Effects of Specifying Different Link Functions
7. SENSITMTY AND SPECIFICITY FOR CORRELATED OUTCOMES
7.1
Introduction
7.2
Sensitivity and Specificity in Clustered Data
7.3
Sensitivity and Specificity of Test Stripe
7.4
Sample Survey Approach
8. SUMMARY AND CONCLUSIONS
8.1
Summary
8.2
Future Directions
BIBLIOGRAPHY
112
121
124
vii
LIST OF TABLES
Table 3.1
Comparison of the Discriminant and Maximum Likelihood
!:;timates of the Logistic Regression Coefficienta
Table 4.1
38
Selected c:ase-c:ontrolstudies to evaluate the relation between
the use of an intrauterine device (IUD) and the occurrence of
ectopic pregnancy
42
Table 4.2 Selected cue-c:ontrolstudies oral contraception (OC) use and
the occurrence of ectopic pregnancy
Table 4.3
3
Probability estimate of acute pelvic inflammatory disease for
various combinations of symptoms based on a logistic
regression analysis with no interaction
Table 5.1
_
Number of Pregnancies Contributed by Each Woman During the
1960-1984 Time Period
Table 5.2
46
52
Univariate Analysis of Ectopic Pregnancy that Occurred During
1960-1984 Time Period at the Department of Obstetrics and
Gynecology, University Hospital, Lund, Sweden
Table 5.3
53
Code Sheet for the Variables in the Lund Ectopic Pregnancy
Data.............................................................................................................. 61
Table 5.4
Univariate Logistic Regression Models for Ectopic Pregnancy
Table 5.5
Estimated Coefficients and Standard Errors for the
_
63
Multivariate Model Containing Variables Identified
in the Univariate Analysis
Tables 5.6
Estimated Coefficients Standard Errors for the Multivariate
Model Containing Vanables Significant in Table 5.5.
_
66
viii
Table 5.7
Log-likelihood, Likelihood Ratio Test Statistic (G), Degree of
Freedom, and P-Value for Possible Interactions of Interest to
be Added to the Main Effect Model
Table 5.8
_
Estimated Coefficients and Standard Enors for the Multivariate
Model Containing Main Effects and SignifiC&Dt Interaction
Table 5.9
68
__.69
Estimated Odds ratios and 95% C.I. for variables not involving
interaction terma............................................................................................ 69
Table 5.10
Expreesion for the Logits and Logit Differences in Tenm of
the Estimated Parameters for the Possible Combinations of
MYCO and SCORE
_
Table 5.11
Values of the Estimated Logit Differences and Odds Ratioe
Table 5.12
GEE Analysis of Ectopic Pregnancy with Logit Link and
71
Binomial Variance
Table 5.13
Relative Difference Between Robuat and Naive
74
~Statistics
for
the Independent and Exchangeable Correlation Structure
Table 5.14
77
Results of the Ectopic Pregnancy Data U.ing Roener'. BetaBinomial Modeling Approach
Table 5.16
Table 6.2
_
86
Comparison of the Intercepts ofthe Standard Logistic
Regression and Itosner'. Model
Table 6.1
--84
Estimated Logistic Regression Coefficients from Alternative
Model Predicting Ectopic Pregnancy in 1497 Women
Table 5.17
76
The Eigenvalues of Matrix Q for the Independent and
Exchangeable Correlation Structures
Table 5.15
70
_.89
Summary Statistics of Demographic Characteristics by
Treatment Group
94
Descriptive Statistics for Plaque Index Data.
95
ix
Table 6.3
GEE Analysis of the Dental Clioieal Trial with Log Link and
103
G&II1JI1& Varianee
Table 6.4
GEE Analysis of the Dental Clinieal Trial with Gamma
Variance and Log Link
Table 6.5
107
Estimated Regression Coefficients and Robust-Z Statistics for
Gamma Error and Identity-Link
Table 7.1
l06
Comparison of Robust and Naive Z Statistics· (Gamma Error
Log Link)
Table 6.6
_
109
GEE Study of Sensitivity and Specificity of the Test Strip at
8 Minutes
118
x
LIST OF FIGURES
•
•
Figure 5.1
Ectopic Pregnancy Rates by Severity and Pregnancy Order
Figure 6.1
Mean Plaque Index by Type of Mouthrinae
Figure 6.2
Fitted Gamma Distribution for Dental Plaque - Month 3
---98
Figure 6.3
Fitted Gamma Distribution for Dental Plaque - Month 6
_99
96
Figure 6.4 Gamma Quantile - Quantile Plot for Dental Plaque Index - Month 3
100
Figure 6.5 Gamma Quantile - Quantile Plot for Dental Plaque Index - Month 6
1Q1
Figure 6.6 Observed and Predicted Values for Log Link
1l0
Figure 6.7 Observed and Predicted Values for Identity Link
•
_.57
__
1l1
CHAPTER 1
INTRODUCTION
Longitudinal data present statistical problems of interest in public health and
epidemiologic studies. In this dissertation, we consider longitudinal data in which outcome
measurements are repeatedly taken on each subject and the primary question is the
dependence of the outcome variable on covariates. The common data structure of repeated
measurement designs is the presence of intradass correlation within subjects (dusters) and
this intraclass correlation must be taken into account when one analyzes the data.
This work is motivated by a longitudinal study of women and their pregnancy outcomes
in
Lund,
Sweden.
The objective of this work is to develop a statistical model and an
analysis approach which :
(1) allows missing data and covariates that change over time
(2) allows for testing of hypotheses about regression parameters
(3) accommodates to response observatioDS that are not independent.
Epidemiologic studies of pregnancy outcome may focus on one of several adverse
outcomes, such as stillbirth, low birth weight, ectopic pregnancy, malformations, and
developmental abnormalities. In this dissertation we will study only one of these pregnancy
adverse outcomes, namely ectopic pregnancy.
Some of the covariates of interest are flXed
characteristics of the woman and will remain constant for all of her pregnancies.
variables will be referred to as time-independent variables.
These
The distinction between time-
independent and time-dependent variables is important in models that employ repeated
measurements.
The logistic regression model is a commonly utilized statistical model in public health and
epidemiologic studies. However, because of the correlation among pregnancies contributed by
2
the same woman, the assumption that the observations must be independent, is not satisfied.
Researchers use different approaches in analyzing pregnancy data. Some researchers
focus their analyses on only one pregnancy, usually the most recent pregnancy. The problem
with this approach is that, though the assumption of independence is satisfied, there is a
..
substantial loss of information and loss of statistical power.
Other researchers have choeen to ignore the dependent nature of the data and proceed using
logistic regression techniques.
For repeated measures data, however, the independence
assumption is usually violated. The dataset is made up of independent clusters, and within
clusters the individual observations are usu&lly correlated. This correlation can not be ignored
without damaging the inference on the regression coefficients.
Recently an extension of generalized linear models that takes the correlations among
measurements on the same individual into account has been developed by Lian,; and l-eger
(1986). The technique of Liang and Zeger is semi-parametric since the estimating equations
are derived without fully specifying the joint distribution of a subject's observations. Instead,
•
only the likelihood for the marginal distributions and a "working" covariance matrix for the
observations are specified. Liang and Zeger show that the regression estimates obtained by
this technique have optimal statistical properties such as consistency and asymptotic
normality even if the correlation matrix has been misspecified.
In this study we use the technique of Liang and Zeger, also known as the generalized
estimating equations (GEE) method, to develop a statistical model for the Swedish ectopic
pregnancy data.
Our research interest is to study the marginal dependence of the response
variable, namely the presence or absence of ectopic pregnancy on covariates.
The specific
questions we address include:
(i)
How does the prevalence of ectopic pregnancy change with age?
(ii) How does the prevalence of ectopic pregnancy depend upon a woman's history of pelvic
inflammatory disease?
3
(iii) How does the degree of PID severity and age affect the prevalence of ectopic pregnancy?
(iv) How does the prevalence of ectopic pregnancy depend upon history of adnexal surgeries?
The results of a atatistic:al analysis using the standard logistic regression model, which
assumes independence of all the observations, are compared to the results obtained using the
GEE approach.
AB mentioned before, one of the advantages of the GEE methodology is that parameter
estimates are robust against misapecification of the correlation structure.
specification results in the mOlt efficient estimates.
However, correct
Rotnitzky and Jewell (1991) provide a
technique to a.eea if the "guessed" correlation matrix is unreasonable. Thus, in this work we
modified the GEE software by writing computer programs by which one may check if the
"working" correlation matrix is reasonable.
We
further modify the GEE programs to
provide adjusted Wald test statistics that take the correlation structure of the data into
consideration as suggested by Rao and Scott (1984, 1987). These were originally introduced
88
extensions or adjustments to the usual Pearson chi-squared tests for data arising from
complex survey sampling schemes. In this dissertation the GEE backbone program was
obtained from Dr. Rezaul Karim of Johns Hopkins University. We then modified this
program to obtain both robust and naive Wald statistics and their associated }>-values for any
composite hypothesis regarding the regression coefficients.
The scientific questions which can be addressed by repeated measurement studies are
generally the marginal dependence of response on covariates or individual changes over time
as in the case of transitional models.
Marginal models describe the occasion-specific
distribution of the response variable, whereas transitional models describe the distribution of
individual changes over time. The distinction between marginal and transitional models will
be discussed in the next chapters.
General approaches for the analysis of repeated measures data are available for both
continuous and categoric:al response
variables.
However mOlt of the methods are for
continuous response variables based on parametric models assuming a multivariate normal
4
error structure (Ware, J.H. 1981; Louis, T 1988; Timm, N.H. 1980)
The first general approach to the analysis of repeated measures when the response is
categorical was described by Koch et al. (1977) and is based upon the weighted least squares
(WLS) methodology of Grizzle, Starmer and Koch (1969). Many authors have extended the
weighted least squares methodology to a variety of response functions of repeated measures
categorical data with missing data (Stanish, Gillings, and Koch, 1978; Woolson and Clarke,
1984; Landis, Miller, Davia and Koch 1988).
However, these methoda baaed on normal theory and categorical methods baaed on the
weighted least squares approach are not always appropriate for the analysis of repeated
measures data.
First, the parametric &Uumption of multivariate normality may not be
satisfied. Secondly, in situations when the response variable is dichotomous or ordinal, the
WLS technique allows only categorical predictors, and thus cannot be used when there are
continuous independent variables.
In addition, the total sample size within each sub-
population of the multi-way croea-c:lassification of response and covariates
must be
moderately large. Thus, this may impose some limitations in practice.
In Chapter 2, some of the procedures for repeated measurements in the literature will be
reviewed. In Chapter 3, the extension of generalized linear models to repeated measures data
(Zeger, 1986;
Liang 1986) will be discussed.
The Swedish pregnancy data, and the
epidemiology, etiology, and diagnosis of both pelvic inflammatory disease and ectopic
pregnancy data is discussed in Chapter 4. In Chapter 5, the pregnancy data are analyzed
using ordinary logistic regression,
GEE methodology and Rosner's conditional modeling
approach. In Chapter 6, we use the GEE techniques to study the effectiveness of three
mouthrinses in inhibiting the development of dental plaque. In Chapter 7, we use the GEE
approach to obtain estimates of sensitivity and specificity when the data consist of correlated
binary outcomes. First order Taylor series approximations to the variances of these estimates
are provided. Data from a dental clinical study are used to motivate and illustrate the
method. In Chapter 8 we give summary, conclusions and suggestions for future research.
CHAPTER 2
LITERATURE REVIEW
A number of statistical papers, involving both continuous and categorical response
variables, have been written on repeated measures analysis. A majority of the methods were
developed for continuous response variables based on parametric models assuming a
univariate or multivariate normal error structure. These parametric techniques are reviewed
by Louis (1988), Ware (1965), and Timm (1980). One advantage of these classical techniques
is the availability of existing software, such as PROC GLM and PROC LINMOD in the SAS
system.
In this chapter we will review (1) the common methods based on parametric
assumptions and (2) the techniques for categorical response variables.
2.1
Linear Models for Repeated Measures
There are two approaches to repeated measures analysis.
The first approach is the
population average model (Zeger, Liang, and Albert, 1988) where dependence of a response
variable on the independent variables is constant and observed individual difference is due to
sampling variability. In this approach, the regression coefficients have interpretations for the
population and not for any individual (Zeger, Liang, Albert). This approach is also referred
to in the statistical literature as marginal modeling.
The second approach to modeling
repeated measures is referred to as subject specific modeling, where dependence of a response
on independent variables is subject-specific. This situation arises when obeerved individual
differences are heterogeneous and this observed subject heterogeneity can be explicitly
modeled. Example of subject-specific modela include the random effects model '(Liang and
Zeger, 1990) and the mixed-eff'ects logistic model (Stiratelli, Laird and Ware, 1980; and
Anderson and Aitkin, 1985) where one assumes natural heterogeneity among subjects in
6
regression coefficients.
The ra.ndom-effects model is given by:
where Yi is a vector o( responaes
Xi is a known matrix o( covariate.
P is a vector o( regression parameters
bi - tid N(O, D)
t
2
i - tid N(O, 17 ) independent o( b i
example:
Y·=PO
1
+ PIX.1 + b01. + b 11·X.1 + t·1
where Yi = height o( child i
Xi
=age o( child i
+ P1Xi )
(Po
is considered to be the population average growth
function a.nd
(boi
+ bliX i ) is considered as deviation specific (or child i.
A. stated by Zeger, Liang, and Albert (1988), if the primary research interest is marginal
dependence o( response on covariates then the population average model (marginal model)
should be uaed.
H the primary interest is to study the individual changes over time then
subject specific models should be uaed.
The approach to repeated measures data for continuous response is based on the standard
general linear multivariate model (GLMM), which is a generalization o( the general linear
univariate model (GLUM) which includes two or more dependent variables, all o( which share
7
the same, identical design matrix.
The general linear multivariate model is given by
(2.1)
E[y]=XP
denotes a K x t matrix of observations or measurements on t dependent variables or responae
variables, Yl' Y2,·..'Yt·
The jth column of Y is Yj and is a K x 1 vector which satisfies
GLUM (Yj; XPj , C1'ij I K )
X denotes an K x q
(K
>
where
q) design matrix whose values are fixed constants which are
assumed to be known without appreciable error.
is a q x t matrix of unknown primary parameters. One of the objectives of repeated measures
analysis is to estimate these primary parameters.
An equivalent representation of model (2.1) is:
E(vec[Y])= E
= (It
=
~ X) vec(P)
( 2.2)
The constant variance-covariance matrix for each set of dependent variables is given by
E,
a txt matrix.
( 2.3)
The correlation among the dependent variables is given by
R
=[R.~ =
Diag
(Ef 1/ 2 E
Diag
(Ef 1/ 2
( 2.4)
8
When X is of less than full rank then fJ is non-estimable and not well-defined.
P=
when X is of full rank
However,
(X'Xr 1X'Y is an unbiased estimator of fJ. Moreover for each
dependent variable separately, it provides a uniformly minimum variance estimator of fJ.
Regardless of the rank of X,
t
t
the estimator of E is given by
= (Y - Xp)' (Y - XP) / (K - rank (X»
=(Y'Y - P'X'Y)/(K - rank (X»
t
( 2.5)
is an unbiased estimator of E
Bypothesis Testing:
When testing hypotheses of interest we assume that the rows of Y are independently
distributed as t-variate normal random vectors. The hypothesis of interest has the following
form:
.
Bo : 9=CfJU=O
HI: 9
= C fJ U ¢
where C is a
X
(2.6)
0
q contrast matrix consisting of any set of linearly independent row vectors
and U is t x m and is of full rank m. C and U are chosen appropriately depending on the
hypothesis of interest. If C(X'X)- C' is nonsingular then the above hypothesis is said to be
testable.
There are many test
statistics for the hypothesis above and all of' these test
statistics are functions of the following matrices:
B =
E
8'
[C(X'X)- C'r 1 9
=(K - rank (X» U· t
(2.7)
U
(2.8)
Some of these multivariate test statistics include
(1) Wilks' Lambda -
lEI / IB + EI
(2.9)
9
= Tr (H(H+Er 1]
(2)
Pillai's trace
(3)
Hotelling -Lawley Trace
(4)
Roy's Largest Root
(2.10)
= Tr [HE- 1]
(2.11)
= ..\1 = largest eigenvalue of HE"'1
(2.12)
The general linear multivariate model has been extended by attaching a post stratification
matrix T to equation (2.1).
This approach assumes that polynomial regression of the
observations on time adequately deae:ribes the dependence of the observations on time
(Potthoff, 1964). This is the
10
called polynomial regression method usually used in growth
curve problems to compare how the response changes over time among groups.
The model
is given by
E[Y]
=
(2.13)
XPT
where Y, X and fJ are as described above and T is a matrix of polynomials in time.
By
appending a post matrix T as in equation (2.13), one imposes a time structure on the mean
responses.
An example of this type of modeling approach which incorporates covariates is
given by Grizzle and Allen (1969). Kleinbaum (1969, 1973) has extended this model to allow
for missing values.
If from the estimate of the variance-covariance matrix,
t, we learn that the 'covariance
terms are approximately equal (compound symmetry) then it may be prudent to perform a
univariate analysis, such as for a randomized block design with subjects as blocks so that we
gain power and precision. Thus a vector of observations is represented as one index, such as
an average of response. Then it equality of the average response over time between groupe is
the question of interest, a univariate ANOVA can be used. This univariate approach then
tests for main effects of groupe (between subjects effect), and main effects of time (within
subject effect). The F statistic for the fixed effects of group is distributed as F as long as
measurements are independent of group membership. The necessary and sufficient condition
10
for the F statistics to be exactly as F under the null hypothesis is that the correlation matrix
of observation vector has the following form:
1
P
P
pip
p
p
1
p
p
p
p
p
p
p
p
1
This is called "uniform" correlation by Cole and Grizzle (1966), and also referred to as
intraclass correlation or exchangeable correlation. If this is not the case, F statistics for time
efl'ecta are not distributed as F (Kirk, 1982).
Adjustment factors to correct this problem have been suggested by Greenhouse 1959),
Huynh and Feldt (1970), and Schwertman (1978) .
2.2
Approach I2t Nonpormal
12Aa
The first general approach to the analysis of repeated measures when the response
variable is categorical was given by Koch et aI. (1977) and is based on the weighted least
squares methodology of Grizzle, Starmer and Koch (1969). Other authors have extended this
technique to a variety of response functioDB for complete and incomplete repeated measures
categorical data (Stanish, Woollon, Landis). In this technique the hypotheses of interest to be
addressed in a repeated measures design include:
(1)
Differences among. aubpopulatiolll regarding the average distribution of the
responses to say d conditions (e.g., treatments ).
(2)
Difference among the d conditiolll or treatments regarding the average distribution
11
across the s--subpopulations.
Test statistics for the above hypotheses as well as parameter estimation are then
undertaken within the general framework of weighted least squares by specifying the
appropriate function F which represents estimators for the parameters under analysis and the
corresponding operator matrices (e.g. operator matrices A, design matrix X, and contrast
matrix C). A detailed explanation of the construction of these operator matrices is given in
Landis et al (1988) .
The weighted least squares methodology requires that the sample size within each
subpopulation be moderately large and can not be used when there are continuous
independent variables.
Furthermore, it doea not accommodate time-dependent covariates,
thus, in some practical situations this may impose a severe limitation.
Wei and Stram (1988) describe another approach to analyzing repeated measures data
with time-dependent and/or time-independent covariates.
In this approach subjects are
observed repeatedly at a common set of observation times. At each time point, the data are
first analyzed using the technique of McCullagh and Neider (1982). The parameters of these
models are assumed to be occasion-specific and are estimated by maximizing the occasionspecific quasi-likelihood equations.
Let vec({3) = (P1'
P2, u, P t )'
denote the pt x 1 parameter vector, where
P1, P2, ... ,
P t are the parameter estimates at time point 1, time point 2,... time point t. The authors
show that the regression coefficients over different time points are asymptotically jointly
1 2
normal for large samples; i.e. K / (vec(P) - vec({3» is asymptotically normal with mean 0
-1
and covariance matrix V {3=I:~(X'X) of dimension pt x pt. Wei and Stram then provide a
consistent estimator of V{3.
The regression estimates from the time-specific univariate regressions can be combined to
give a global picture of the effects of individual covariate:s on the response variable over the
entire time period.
For example, consider the t estimated regression coefficients for the kth covariate, say, age.
12
The sequential multiple test procedures of Marcus, Peritz and Gabriel (1976) can be used to
test if there are significant changea in the t coefficients of age over time. If there are no
significant changes then one may proceed to provide a global estimate for the covariate of
interest. Wei and Stram also provide a lack of fit test for testing the adequacy of the quuilikelihood model used at each time point.
The Wei and Stram method (1988) is cloeely related to the GEE method. If the GEE
regression coefficients are allowed to vary in time and if the independence working correlation
matrix is lUed, the two methods give identie&1 results (Zeger, 1988). However, for repeated
measures, the GEE method is usually specified to give an estimate of a single regression
parameter vector which is assumed to be constant over time. The Wei and Stram method
gives separate regression estimates at each time point which then may be combined to give a
global estimate if desired.
Stram, Wei, and Ware (1988) describe an approach for analyzing repeated measures of an
ordinal categorie&1 response variable with possibly missing observations and time-dependent
covariates.
In this approach, as in the case of Wei and Stram approach mentioned above,
subjects are observed repeatedly at a common set of observation times. At each time point
the data are first analyzed using the proportional odds regression model from a class of
models for ordinal data proposed by McCullagh (1980).
The first class of non-parametric methods for repeated measures was described by Koch,
(1968).
Other more recent nonparametric procedures include the Wei and Lachin method
(1984) and the Wei and Johnson method (1985).
Apart from marginal models for binary repeated measurements, no method was available
for nODDormal outcomes with possibly missing observations and time dependent covariates
until Liang and Zeger (1986). Liang and Zeger extended Neider and Wedderburn'. (1972)
generalized linear modeb and Wedderburn'. quasi-likelihood method for repeated measures
data so that it covers a wide class of outcome distributions. This will be discussed in the next
chapter.
.
..
CHAPTER 3
REPEATED MEASUREMENT ANALYSIS USING GENERALIZED ESTIMATING EQUATIONS
The first part of this chapter is a description of generalized linear models and quasilikelihood. The second part will discuss the extension of generalized linear models and quasilikelihood to repeated meuures data. Thus mOlt of this chapter is baaed on the concepts and
expositions given by NeIder and Wedderburn (1972), Wedderburn (1974),
McCullagh and
Neider (1983), Zeger and Liang (1986); Liang and Zeger (1986), and Liang and Zeger (1991).
•
Generalized Linear Models
3.1
Generalized Linear Models (G LM) are simply an extension of classical linear models, and
were first introduced by NeIder and Wedderburn (1972).
In classical linear models set up recall that
(3.1)
where
Y denotes a K x 1 vector of observations assumed to have mean IJ and constant
variance and often a normal distribution.
Similarly
f
is a K x 1 vector of error terms
assumed to have mean 0 and a normal distribution. The fact that
f
has zero mean and a
normal distribution excludes consideration of response variables that have Poisson, binomial,
positive exponential, etc. distributions. In order to generalize the model to cover a wide range
of dependent variables, it is important to separate clearly the deterministic linear regression
component (systematic part) from the random component.
reformulating equation (3.1) into three components.
This can be done by
14
1. The systematic linear component part, the linear predictor
'1
= X/3
where X is a Kxp matrix of known covariates and /3 is a px1 vector of unknown regression
parameters.
2. The random component part Y comes from an exponential family with E(Y)
= JJ •
3. The link component part that relates or links the mean of the response variable to the
systematic linear component X /3.
'1
=
g (JJ)
where g (JJ) is the link function which may be any monotonic differentiable function. The
link function, g, changes from one generalized linear model to another as will be shown later.
Thus generalized linear models allow the random component to come from an exponential
family such as Poisson, Gamma, Binomial, etc.. It is known that if the likelihood has the
exponential family form, maximum likelihood estimates of regression parameters can be
found using the method of iteratively reweighted least squares (Bradley, 1973). The equation
maximizing a likelihood function and the equation minimizing a weighted sum of squares are
equivalent if the likelihood function arises from the exponential family of distributions.
The
link function may be any monotonic differentiable function.
Likelihood Function
~
Generalized LiDYI Models
The probability function for a scalar oheervation y from an exponential family is
fy (y; 9, ;)
=exp {[y9 - b(9)] / a(;) + C(y, ;)}
(3.2)
for some suitably choeen functions a( ), b( ), and c( ),. ; is a dispersion parameter assumed
..
15
known.
If
tP
is unknown then fy(Yi 9,
tP) may not
be an exponential family
(NeIder &l.
Wedderburn 1972). The log likelihood for the datum y in equation (3.2) is given by
y9-b(9)
L = a(tP)
+
(3.3)
C(y,4»
Differentiating (3.3) with respect to 9, we get
8L
y-b'(9)
(3.4)
89 = a(tP)
Further differentiating
(3.4)
with respect to
9, we get
(3.5)
From the well-known relationship (Kendall and Stuart)
E
(~) = 0
=> E(y) =
~
=
b'(9)
(3.6)
=>
E (y_~]2 = Yar(y) = b"(S) a(4))
(3.7)
where b'(.) and b"(.) denote the first and second derivatives of b( ) with respect to 8,
respectively. Thus note Yar(y) hu two components:
(1) b" (9) - variance function dependent upon the canonical parameter 9 (and hence on the
mean)
a( 4» which is independent of the canonical parameter and hence independent of the
(2)
mean.
The function a( 4»
is usually of the form
16
a(<p) = <p/w
where w is a known weight function.
Example 1: For the Poisson distribution
fy (YiJJ)
= JJY e- u
/ y!
= exp {[Yln(JJ) - JJ]
+
[-In(y!)]}
= exp {(y8 ~e) + [-In(y!)]},
= exp ([[eye) - b(8)]/a(<p)]
+
C (y,
<PH
where D=InJJ
b(D)
=eO
C(y, <p) = [-In (y!)]
a(<p) = 1
Example 2. Normal distribution
fy (Yi 0, <p)
=~
1211'6-
exp[~(Y-JJ)2]
26
= exp {[yO - b(D)] / a(<p) + C(y,
<PH
= JJ
b(O) = JJ2/2
where 0
a(<p) = 6 2 = <p
C(y, <p)
= - 1/2 (y2/6 2 + In (211'6 2)]
From (3.7)
Var(y) = E [y - JJ]2 = b"(8) a(<p) =lx6 2 =6 2
Example 3.
Binomial Distribution
ny -
binomial (n, JJ)
y
=0, l/n, ..., 1
17
£(y, /J) = (oCn y ) /Joy (l_/J)o-oy
= exp {ny log (/J/(1-/J))
=
b(9)
9
log (1- /J)
y log (/J/(1-/J» - log (1 - /Jr 1
l/n
exp {
= exp {( y9 - b(9»
where
+0
/ a (C;)
+ log (oC oy )}
+
log nCny}
+ C(y, C;)}
= log 1~
-/J
=log (1 - /Jr 1 = log (1 + e9)
~ b' (0)
=
e
9
~
= /J
a(c;) = l/n
C(y, ¢) = log nCny
Var (y) = b" (9) a(¢) =
e
n(l
Note that when a(¢)
/J (1-/J) / n.
= ¢/o
When ¢
>
9
+ eO
= /J (l-/J)
)2
then y is said to have an extra-binomial variation, Var (y)
1 then we have the case of overdispersion; if ¢
<
1 it is
=¢
~eferred
to
as underdispersion.
For examples of overdiapened data see Crowder (1979), Williams (1982), and Cox and
Snell (1989).
Smith and Heitjan (1993) present a score test for assessing whether
overdispersion is present in generalized linear models.
the score function for
P is given by
18
-h
S(P) =
K
-h
= i=1
L:
=
=
t
i=1
t
t
{I
8P
+
lJ9. t y.I - b'(9.)
I
}
a(tIl)
(1j)
a9 i
(ap)
i=1
=
~)
y.9 - b(9.)
I
a(tIl)
8
i=1
=
log f(Yi; 9i'
t
C(Yi' til)}
(chain rule lJL
~
lJL)
= (89)t
~ 89
Yi - E[Yi]
}
{ a(tIl)
t
i=1
( 3.8)
Thus the score function depends on the density function, f(Yi)' only through J.li
Canonical
= E(Yi) and
LiW
As defined by McCullagh and NeIder (1972) the link function is the function that relates
the linear predictor " to the expected value J.l of the datum y.
As shown before there are special links for the various distributions of interest. These links
occur when
Normal
Poiaaon
(I
=
f]
and are eaUed canonical links.
" =
J.l
19
" = In (1-(p)p)
Binomial
Gamma
Inverse Normal
The generalized linear models are fitted by the method of maximum likelihood, using an
iterative weighted least squares algorithm.
the parameter
Thi.e yields maximum likekihood estimates ~ of
I!, and their corresponding standard errors.
Hence estimates ij for the linear
predictor and Pi for the fitted values can be obtained. Details are given in Neider and
Wedderburn (1972).
Just as important as the parameter estimates, are assessments of goodness of fit for
the model fitted to the data. Given K observations one can fit models that contain up to K
parameters. One extreme model is the null model which gives one common value for all the
p'a.
At the other extreme is the maximal model with K parameters, one parameter per
observation.
The maximal model assigns all variation between the Yi's to the systematic
component of the model. It is said to be a completely specified model with the error term
equal to zero. Even though the maximal model does not do much in terms of summarizing
the data, it does provide a baseline against which models with fewer parameter:s can be
compared and contrasted. Thi.e comparison is simply the difference between the maximum
log likelihood achievable under the fujI model and that achieveable under the reduced model.
Let ley, rP, y) = maximum likelihood achievable under the full model,
l(p,
and let
tP, y) = maximum likelihood achievable under the reduced model,
8 and 9 be
the estimated canonical parameters under the reduced and the full model
respectively. Then the scaled deviance is:
5(y,
M = - 2 log { I(p, tP, y)}
ley,
,pi y)
20
= 2( l(y, tP, y) - 1 (p, tP; y)]
K
= 2 ~ (Yi(i - i) - b(i) + b(i)] / a(tP)
i=1
=2
t
wi(Yi(i - i) - b(i)
+ b(i)] /
t/J
i=1
(3.9)
where D(Yl
p» is referred to as the deviance of the reduced model relative to the full model;
tP is the scale parameter.
For the full model
l(y, tPi y) =
=f
In (211'(7'2)
For the reduced model
l(p , tP; y)
=-(2(7'2 r l
t
(since y = p).
(Yi - Pi)2 -
i=1
Therefore, the deviance D(Yi' p)
=
t
In (211'(7'2).
.t
(Yi - p)2 which corresponda to the
1=1
residual sum of squares under the reduced model.
Similarly, if the error probability distribution is the Poisson distribution then
which corresponda to the log-likelihood ratio statistic often uaed as a test statistic in the
analysis of contingency tables.
21
For the g&IIlma distribution
K
D(y, jl)
=2L
[-In (Yi / f£i)
+
(Yi - f£i) / f£i]
i=1
For Normal distributions the scale deviance haa a X2 11
II
distribution with
= number of parameters in the full model minus the number of linearly independent
parameters in the reduced model.
For non-normal distribution&, the scaled deviance is distributed aa X211
onlyasymptotieally,
thus little is known about the distribution of non-normal deviance for small sample sizes
(Adena and Wilson 1982).
Now suppoee we wish to compare two models, ml and m2, where m2 has 112 - III more
terms than ml.
Suppose
S (ml, y)
S (m2, y)
= the scaled deviance of modell,
= the scaled deviance of model 2;
then S(ml, m2) = S(ml, y) - S(m2, y) has a
x2112 _ III
distribution if the two models have
Normal errors.
For other distributions S(ml, m2) haa an approximate x2112- 1I 1
distribution.
Another measure of goodness of fit is the generalized Pearson X2 statistic, which is given by
where V(jl) is the estimated variance function for the distribution of interest.
3.2
Quasi-likelihood
In this eeetion we review quasi-likelihood met.hods and point. out lOme of the important
22
concepts and properties of quasi-likelihood techniques. This section is based upon the concept
and expositions given by Wedderburn (1974), McCullagh (1988), and Liang and Zeger (1991).
Unlike the likelihood approach of Section 3.1, in quasi-likelihood, analogues to equation
3.2 are missing, that is we are unsure about the random mechanism from which the data
come from.
Instead of assuming that y comes from an exponential family, we only specify
the relationship between the mean and variance.
This means that only a second moment
assumption is required in quasi-likelihood analysis - thus avoiding the complete specification
or the underlying distribution.
This is a particularly useful approach if one does not have
enough data to estimate a distribution form but sufficient enough to estimate the relationship
of the first two moments (Pregibon. 1984).
Hence the assumptions for quasi-likelihood
analysis are:
(1)
g (1'.)
1
= x.p
1
(2) Var (Yi)
= tP V(J'i) where V is a known function
Definition:
Suppose we have independent observations Y i (i = 1.....K)
with E(Y.)
=1" and
1
1
where V is some known function. Furthermore. suppose for each observation J'i
some known function of a set of parameters
Pl' P2..... 13 p •
.= E(Yi) is
Then for each observation
Wedderburn defines the quasi-likelihood function. or more correctly. the log quasi-likelihood.
for the ith observation.
p.
Q(Yii J'i) =
j
(3.10)
Yi
or equivalently the quasi-likelihood function Q. can be defined as a system of partial
differential equations:
23
(3.11)
The quaai-likelihood is the sum of components of (3.10) over the K observations. The
statistical properties of quasi-likelihood functions are similar to those of ordinary likelihood.
Details are given by Wedderbum (1974) and McCullagh (1983).
Some of the major results are summarized below.
(1)
The quasi-score function is simply the first derivative of the log quasi-likelihood with
respect to fJ and is given by
(3.12)
V(Pi) is referred aa the "working" variance of Yi.
(2)
.j(S(fJ)
variance
is asymptotically distributed aa multivariate normal with mean
= 0 and
=D
where
~ () po t i l
8 Pi
D = L..J (#)
y- (Pi) Var(Yi) y- (Pi) 8! /K
i=1
(3) Among linear estimators of fJ , quaai-likelihood estimates have minimum variance (Firth
1987).
(4)
..fK (iJ - fJ)
is also asymptotically distributed as multivariate normal with mean 0 and
variance-covariance matrix I;1 D
E- 1
where
All the above results are baaed on the first-order Taylor Series expansion of the equations
24
determining
p, S(P) = 0 in (P - p).
Next is a table of quasi-likelihood, quasi-score functions, and variance functions for some of
the familiar distributiona (Liang and Zeger, 1990).
Distribution
YUU
Q(u;y)
Normal
_(Y-JJ)2/2
Poisson
1
Y In JJ-JJ
Gamma
JJ
JJ2
-Y/JJ - InJJ
log (l-JJ)
Ylog~
-JJ +
binomial
JJ(l-JJ)
As mentioned before, the only assumption we have in quasi-likelihood analysis is that we
have to know the relationahip between the first two moments. Liang and Zeger extended this
concept to longitudinal data where they view it as a multivariate version of the quasi-
•
likelihood function. (i.e. yt is now an n x 1 vector) .
i
3.3
Extension of GLM to Repeated Measures
In this section we summarize the major results of Liang and Zeger (1986) and. Zeger and
Liang (1986).
Recall that for quasi-likelihood to be applied to repeated measures data we
require only that the variance of r be a known function of J!
= E(r).
Furthermore, the
correlation structure must be taken into consideration when estimating the regression
parameter P and its variance-covariance matrix.
Let Yi t be the ni repeated measurements on individual i, i.e.
...·..·'Y·mi ) 1·-12
Yit = (Y·1'Y·2'
1
1
, ,
Let
~t
= (~t'1
Xit ,.....,
2
.....,
K
Xit ) be the vector of covariate8 corresponding to Yit' the tth
p
•
25
outcome in the ith individual where 1 = 1, 2,......, K and t
= 1, 2,....., ni' Further let E(Yit)
= lJit and R i (0) be ni x ni 'working' correlation matrix for each Yi' where 0 is an unknown
vector of order s, by which R(o) can be fully specified.
It is possible that the working
correlation matrices can be different for different subjects. The varianee-covariance matrix of
Yi is given by :
V.
1
=A. 1/ 2 R..(o)A. 1/2 I fJ
1
1
(3.13)
1
where Ai = diag {var(Yit.)} is an ni x ni diagonal matrix which is a function of the mean, say
g«lJit) as the tth diagonal element.
Again R i (0) is an ni x ni 'working' correlation matrix that describes a postulated
correlation structure of Yi' after having adjusted for the effects of the covariates. The quasi-
•
likelihood analogue of the score function is given by
k
S(P)
= L:
i=1
ot.1 Vi.1 (y.1 - IJ.)
= 0
1
(3.14)
where
Note that this equation would be identical to equation 3.8 (regular score function) if y came
from an exponential family and the obeervationa for each subject were independent.
Ui (P, 0) = 0i Vi - 1 (Yi -lJi) is equivalent to the estimating function suggested by
Wedderburn (1974) except that the Vi's here are functions of
0
and p. The only difference
is the introduction of an additional parameter a to account for the aerial correlation of the
data.
0
is consistently estimated by & which is a function of
Furthermore
,p is consistently estimated by
estimate of p, say
PG
consistent estimators of
p and
the scale parameter
,po
~ which itself is a function of p. Therefore an
is obtained by iterating between aolutioDB to equation (3.14) and
0
and
,p as described by Liang and Zeger (1986). That is, PG is the
26
solution of
t
Ui
i=1
{P,
Q
(P,
~(P»)
= 0
(3.15)
where Ui(P, a) = Di t Vi -1 (Yi - Pi) and K is the number ofsubjects or clusters.
Equation 3.15 is called a generalized estimating equation (GEE). Liang and Zeger show that
as K goes to infinity,
PG
is a consistent estimator of P
and that
.JK
(P G - P) is
asymptotically multivariate normal with varianc:e-covariance matrix VG given by:
(3.16)
K
t
-1
where II = ""
LJ D I. V.I D·I
and
i=1
A consistent estimator of VG' say
estimators,
PG'
Q
Io=t
i=1
VG'
D. t V.- 1 Cov(y.) V.- 1 D..
I
I
I
I
I
is obtained by evaluating Di and Vi at the
and ~, and replacing Cov(Yi) by (Yi - i'i) (Yi - i'i)'. Thus estimation of
the parameters {3 G' a, and 4J is performed as follows:
1.
Assume Ri (a) is the identity matrix and 4J = 1. Then estimate P using Fisher's scoring
method.
.
2. For any gIven
/3-
calcae
ul t resl'd UA&lt
.1.
{
' } / (_ii)I/2
'Yit=
Yit - Pit
vi
(3.17)
where i'it depends on the current value of Pand iliii is the jJth element of the inverse of 11'
3.
Estimate the nuisance parameter tP by
~- 1
- Lk Lni
i=l t=l
t
2
it / (N -p)
(3.18)
•
27
k
L:
N=
where
ni
and p is the number of covariates.
i=1
4.
Estimate the 'working' correlation matrix
K
Ruv =
L: 1iu1iv / (N - p)
(3.19)
i=1
The two Iteps, i.e. (3.15) and (3.17) - (3.19), are iterated until convergence.
As stated by Liang and Zeger, incorrect specification of the correlation structure does not
violate the consistency of the estimates
PG
and
VG.
However as shown by Liang (1986)
specifying a working correlation matrix dose to the actual one increases the efficiency of the
parameter estimates.
(1)
Some of the most common choices of the working correlation are:
Identity matrix: This specification limply ignores the serial correlation structure of the
data and treats the repeated observationa as if they are independent. In this situation one
may use PROC LOGISTIC or PROC CATMOD to obtain estimates of regression coefficients.
The resulting estimates will be consistent but not efficient if the correlation is large.
1
0
0
0
0
o
1
0
0
0
o
0
0
0
0 1
28
(3)
Exchangeable correlation:
This correlation specification assumes that the repeated
observations share a common random component i.e. Corr(yit' Yiv)
=
Q
for all
t~v.
The
random effects model assumes this correlation structure:
1
Q
Q
Q
Q
Q
Q
Q
Q
1
Q
Q
Q
1
Q
Q
1
Under the assumption of exchangeable correlation,
t i:
a= ¢
(3)
tit tit' I{
t>t'
i=1
t
Q
is estimated by:
.5 ni (ni - 1) - p}
(3.20)
i=1
Autoregressive Correlation: under autoregressive correlation structure the correlations
decrease as the time spaces between the observations increases, i.e. Corr (Yit' Yia) =, Q It-sl:
1
Q
Q
1
Q
Q
1
Q
a
1
t-2
29
Under autoregressive correlation structure
Q
can be estimated by the slope from the
regression of log E ("rit "ris) on It-sl.
(4)
Besides the 3 correlation structures given above, the
Unspecified correlation:
correlation structure
CAD
al80 be totally unspecified if ni = n for all i.
Then the 1/2 n(n-1)
elements of R are estimated by
It
=~ / K ~
£-
i=1
A.- 1/ 2 S. S.T A: 1/ 2 .
I
I
I
1
(3.21)
AB pointed out before, the GEE approach provides a powerful way of analyzing repeated
measures data that arise from a wide range of distributions.
I t also allows for
accommodation to different choices of correlation structure as well as varying number of
measurement times for subjects.
Resulting estimates are consistent, asymptotically unbiased,
and asymptotically normally distributed. However, all these nice properties are based upon
large sample size assumptions. Paik (1988) investigates the small sample properties of the
GEE technique for gamma error and AR-l correlation structure based on average
cover~ge
probability, i.e., the fraction of simulations whose confidence interval based on the t-statistic
included the true parameter. Paik's simulation results indicate that the confidence intervala
for the GEE estimator do not provide desirable coverage probability of the true' parameter
due to considerably biased point estimates. Paik's Table I is shown on the next page.
30
Table I
Average coverage probabilities for GEE interval estimation baaed on t-statistics and average
interval length (parentheses) for
on~parameter
models baaed on 500 simulation samples
nominal level = .95
correlation structure: AR-l
a: .9 (upper entry) .3 (lower entry)
number of parameters: 1
Sample SiR
Working Correlation
Independent
Exchangeable
AR-1
.924 • (2.37)
.950 (1.49)
.973 • (1.19)
.950 (1.06)
.980 • (.825)
.917. (1.65)
.940
.908 • (1.69)
.934 (1.28)
. 943 (1.11)
.961 (.881)
.897 • (1.55)
.943
(1.21)
.941 (1.02)
.973. (.800)
.930. (1.17)
.937 (.999)
.953 (.784)
.920 • (1.23)
.947 (1.02)
.913. (.800)
.890 • (1.59)
.910 • (1.56)
~
.938 (1.72)
(1.26)
mentioned by Paik, the average interval length for sample size 20 and a
= .9 under
AR-l is close to that for sample size 50 under independent correlation, indicating that the
same efficiency can be achieved with 40% of samples by specifying the correct correlation
structure.
If the sample size is small, say less than 30, some authors suggest that further steps be
•
31
taken to obtain estimates with reduced bias.
Thus Paik proposes three kinds of jackknife
techniques, namely, the standard jackknife, the weighted jackknife, and the linear jackknife,
which seem to reduce the bias and provide correct coverage probabilities.
Asymptotic relative efficiency when the working correlation is misspedfied was examined
by Liang (1986) when repeated measurements are binary and the true correlations are .3 and
.7.
When t.rue correlation was .3 and the 'working' correlation was misspecified as
independent, AR-l, or exchangeable, the estimates were still efficient with lowest asymptotic
relative efficiency of 0.95. However, when t.he t.rue correlation was 0.7 the asymptotic relat.ive
efficiencies
ranged from 0.23 (true correlation=exchangeable and working correlation=!-
dependence) to 0.99 (true correlation=exchangeable and working correlation=independence).
Chi.Squared Tests fQr ~ Regression Parameters
Most of this section is based upon the concepts given by Rotnitzky and Jewell (1990),
Rao and Scott (1981), and Rao and Scott (1987). From previous sections we have that:
Var (f3G) = 11- 1 10 11- 1
(3.22)
where 11 and 10 are defmed in Section 3.3. Equation 3.22 above is sometimes called as the
sandwich estimator. Let us now consider partitioning the regression coefficients such that:
a_G-_[~]
I-'
CI
where {3G is a pxl vector and 'Y is a qx1
vector of the first q components of {3 G [1 ~ q ~ p ] and 6 is the remaining (JHl) components
of {3G' We are interested in testing the hypothesis that:
Ho : 'Y = 'Yo
versua
Ha : 'Y ::f; 'Yo·
Then from the asymptotic normality of
PG , the generalized Wald test statistic is given by
32
Tw
(3.23)
P
where t is a qxl vector of the first q component of G and
V"T
is the qxq sub-matrix of
V{J
which is given in Equation 3.22. If we usume that the "working" correlation is the true
correlation structure then we may use a modified "working" Wald statistic,
(3.24 )
where
W"T is the qxq submatrix of W {JI where
W Q = K (~D. V.- 1 D·r 1
~
L...tll
I
Rotnitzky and Jewell (1990) then proceed to provide an important result in the form of a
They prove that the modified "working" Wald statistic is simply a weighted
Theorem.
combination of q independent x21 random variables I Le.,
T * w --
where
2: c·J X·J2
cl~c2~
( 3.25 )
I
... ~Cq~O are the eigenvalues of a matrix Q which itself is a function of Dj'
Vi' and Ai defined previously. In particular
(3.26)
where
1 -, -1·
Qo=1('" ~D.V. D.
L...t I I
I
Q 1 = 1('"1""
.LJ
(3.27)
n.v.- 1 Cov (y.) V.- 1 f>.
I
I
I
I
I
(3.28)
33
ifq<p
(3.29)
ifq=p
where D (1) and D (2) are the ni xq and nix(M) matrices of the first q and last M columns
i
i
of Di respectively. The implication of the above result is that the coefficients of the linear
combination (Equation 3.25) are identically one when the "working" conelation is gueeeed
correctly. That is to say if the "working" correlation is guessed conectly then c1 ~1 and
~ 1 where c 1 = q-l
L
Cj and
C2
C2
= q-l LCj-2. However, note that cl ~1 and c2~1 does
not necessarily imply that the "working" correlation matrix is indeed the true but unknown
conelation matrix.
Since clustering can have substantial effects on the distribution of T w • similar to that of
the standard Pearson chi-squared test statistic,
some adjustment of T w· is necessary.
Following Rao and Scott (1981) Rotnitzky and Jewell propose adjusting T w • as follows.
(3.30 )
In this dissertation the GEE backbone program was obtained from Dr Karim, Johns Hopkins
University. We then modified this program to obtain the generalized Wald test statistic, T w ,
and the modified "working" Wald statistic, T W ·' IML codes to obtain the matrices QO' Q1'
and Q given above have been added to the program.
This will help users of the GEE
program to verify the adequacy of their guessed "working" correlation matrices.
3.4
Logistic Regression
Logistic regression is used widely to study the relationship between a dependent
dichotomous variable and eeveral independent variables which may be all continuous, all
34
categorical or a combination of the two.
The logistic regression model is given by the
mathematical function:
w-(x) =Prob(Y=llx)= 1+expt-Q-~x) where
where
Q
and
~
-00
< x < 00
(3.31)
are unknown regression parameters.
The function w-(x) is monotonic:ally increasing from 0 to 1 .. x increases from
-00
to
+00.
Since w-(x) h.. a sigmoid shape like that of many epidemiologic events, the logistic function is
often used by epidemiologists and statistici&nl to model risk or probability of disease
development during lOme specified time interval .. a function of several independent
variables (Kleinbaum, Kupper and Morganstern, 1982). The function is approximately linear
for 0.2
< ",(x) < 0.8. There are two methods of estimating the logistic risk function - the
discriminant function (DF) approach suggested by Truett, Cornfield, and Kannel (1967) and
the iterative weighted least squares method which is equivalent to the maximum likelihood
•
method (ML) suggested by Walker and Duncan (1967) and Day and Kerridge (1967).
The DF approach assumes normality of the independent variables while the ML approach
requires no assumptions on the form of the distribution of the independent variables. Next
we discuss the two estimation approaches.
1.
The Discriminant Function Approach
In the case of the discriminant method, the posterior probability of an event occurring
based on the logistic model is developed directly from the assumption of normality of the
independent variables and Bayes' rule.
That is,
Prob[y.=lIX.]
=
I
1
(3.32)
35
where the conditional distribution of X.1 given y.1 = 1 is assumed to be normal (PI' 0'2) and
the conditional distribution of X. given y. = 0 is assumed to be normal (PO' 0'2); that is, the
1
1
conditional distributions have different means, but the same variance.
By equating (3.31)
and (3.32) it can be shown that
Q
= -In[(I-p)/p] - [1/(20'2)][(P1-PO)(P1 + PO)]
fJ
=(PI - pO)/0'2
and p is the unconditional probability that Y
i
= 1.
Sample estimates of
Q
and fJ are
obtained by substituting the maximum likelihood estimates of PI' PO' P and 0' 2
Specifically
where
where
So 2 and S2 1 are sample variances of Xi given Yi = 0 and Yi = I, respectively and KO and K 1
are the sample sizes.
2.
The Maximum Likelihood Approach
The ML approach (Walker and Duncan, 1967 and Halperin, Blackwelder, and Verier,
1971) requires no assumption about the distribution of the Xi's
36
The likelihood function for an event occurring is
The logarithm of the likelihood is
Taking the derivative of the logarithm of the likelihood function with respect to
Q
and f3 we
obtain the maximum likelihood equations
810 L = ~
y. _ J... x. p. = 0
~
2...,
vQ
i=1
81n
L_
'"'a'ir op
t
•
1=1
I
i=1
I
I
k
y.x.
- "x.P..
I I k..J 1 1
These equations are solved iteratively using the Newton-Raphson technique, which is based on
the first order Taylor series expansion of
T(6) = 81n L
8(Q,P)
•
6 r +1
such that
= 6• r + [T'(6• r )r 1 T(I)•
where 6
=(Q,P)'
The asymptotic variance-covarianee matrix is the Frechet-Cramer-Rao lower bound which is
- {E[T'(6)]r 1 (Brooks et al 1988).
37
Halperin, Blackwelder and Verter (1971) did a comparative study of the DF and ML
approaches to logistic regression using both theoretical and empirical development techniques.
They showed, theoretically, that for the case of one independent variable, if P
= 0, the DF
approach gives correct estimates of a and p. If, however, P :;:. 0, the estimates of a and P
may differ markedly from the true population values of a and P even when the logistic
regression model holda. Empirically, these authors concluded that, the DF approach works
reasonably well, even for nonnormal data such as thoee from the Framingham Study.
However, they found that the ML method usually gave a slightly better fit to the model, as
evaluated from the observed and expected number of cases per decile of risk. The overall
conclusion of these authors was that the ML approach is preferable for most situations
commonly encountered in epidemiology.
Brooks, Clark, Hadgu and Jones (1988) performed a Monte Carlo simulation study to
compare the robustness to the assumption of normality of the discriminant function and
maximum likelihood methods of estimating the logistic regression coefficients for the
simplified case of one independent variable.
These authors generated data from four
distributions - the normal, exponential, Bernouilli, and Poisson. The overall conclusion of
this study is that when the data are normally distributed, both the DF method and the ML
method provide good estimates of a and
/3. For nonnormal distributions and moderate to
large sample size, the ML estimates are preferred; however, for very nonnormal distributions
and small sample sizes and highly significant beta, alternate methods to both procedures
should be sought. A:I1 example of such a case is attached in the next page which is adapted
from Brooks et al.
Table 3.1
Comparison of the Discriminant and Maximum Likelihood Estimates of the Logistic Regression Coefficients
Poisson po
Uncondi
tional
Probability
Sample
Size
0.05
500
0.2.5
Model Parameters
50
100
500
0.50
50
100
.500
=0.5
Maximum Likelihood
Estimates
Disl..Timinant Estimates
,..
Rei-bias
6
,..a
S.E.~
-2.94
-2.740
-2.512
0.0
-0.5
2.0
-2.983
-2.012
-2.709
1I.01lS
0.007
11.007
0.001
-0.406
-0.903
0.010
0.000
0.004
-1.099
-0.902
-0.666
0.0
-0.5
-2.0
-1.148
-0.981
-0.800
11.014
0.014
0.013
OJlO4
-0.434
-1.151
0.016
0.015
0.012
-1.099
-0.902
-0.666
0.0
-0.5
-2.0
-1.146
-0.940
-0.788
0.010
0.003
0.008
0.013
-0.453
-1.133
0.011
0.010
0.008
----
0.000
0.197
0.432
0.0
-05
-2.0
-1.100
-0.931
-0.784
0.004
0.004
0.004
-0.004
-0.439
-1.109
0.005
0.004
0.004
-----
0.000
0.197
0.432
0.0
-0.5
-2.0
-0.006
0.210
0.466
0.012
0.011
0.013
0.017
-0.506
-1.663
0.014
0.015
0.021
---0.011
-0.168
0.000
0.197
0.432
0.0
-0.5
-2.0
0.00.5
0.205
0.450
0008
0.008
0.009
0.007
-0.493
-1..581
0.009
-----
O.oII
0.012
0.015
-0.209
0.000
0.197
0.432
0.0
-0..5
-2.0
-0.005
0.196
0.433
0.004
0.003
0.004
0.010
-0.492
-1..533
0.004
0.004
0.005
-0.015
-0.233
a
Roundml wu DeTfonned after
ccrnOUIatIOOS.
6
S.E.(~
or8'
-----0.189
-0.549
-----0.131
-0.424
-0.093
-0.433
-0.122
-0.446
----
Rei-bias
...a
S.E.~
it
-2.963
-2.761
-2.528
0.008
0.008
0.007
-0.086
-0.621
-2.204
0.021
0.023
0.033
---0.242
0.102
-1.123
-0.93.5
-0.673
0.014
0.014
0.013
-0.116
-0.676
-2.313
0.036
0.040
0.05.5
0.352
0.156
-1.132
-0.904
-0.663
0.009
0.009
0.008
-0.036
-0.637
-2.301
0.024
0.027
0.040
0.273
0.150
-1.097
-0.906
-0.664
0.004
0.004
0.004
-0.020
-0.527
-2.046
0.010
0.010
0.015
0.254
0.023
-0.007
0.211
0.447
0.012
0.011
0.011
0.038
-0..5.57
-2.430
0.030
0.039
0.056
---0.110
0.213
0.005
0.20.5
0.444
0.008
0.008
0.007
0.014
-0..516
-2.1.53
0.019
0.026
0.031
0.037
0.076
-0.00.5
0.196
0.432
0.004
0.003
0.003
0.019
-0.510
-2.02.5
0.008
0.010
0.013
--0.020
0.013
S.E.<f)
ort
----
----
----
----
Extracted from Broots. Clar .. HJJdgu nd Jones
w
00
CHAPTER 4
THE EPIDEMIOLOGY OF PELVIC INFLAMMATORY DISEASE
AND ECTOPIC PREGNANCY
The fIrst part of this chapter summarizes the descriptive epidemiology of ectopic: pregnancy
and di.scuues proposed risk factors for ectopic pregnancy.
The second part diacusses the
epidemiology of pelvic inflammatory disease including the etiology and diagnosis of PID . It
also addresses the biological plausibility of PID as a risk factor for the development
of
ectopic pregnancy.
4.1
Ectopic: Pregnancy
Ectopic: pregnanc:y is a major health problem world wide. Ectopic: pregnancy occurs when
a fertilized egg is implanted outside the uterine corpus. The most common implantation sites
are the fallopian tubes, with less frequent occurrence in the ovaries, cervix and other
abdominal areas (Pritchard et ai, 1985).
Ectopic pregnancy increases have been documented in many parts of the world (Beral,
1975; Glebatis et al, 1983; Meirik 1981; Hockin et al, 1984; Westrom et al 1981). In the
United States alone the annual incidence of ectopic pregnancy increased from 4.5 per 1000
pregnancies in 1970 to 14.3 per 1000 pregnancies in 1983 (CDC; MMWR, 1986),
increase.
&
three fold
In Lund, Sweden the rate of ectopic: pregnancy per 1000 diagnosed pregnancies
increased from 5.8 during the early 1960's to 11.1 in the late 1970's for women of
reproductive age.
Ectopic: pregnancy has been referred to as an unqualified disaster in human reproduction
40
(DeCherney
1983).
It presents a life-threatening event, accounting for about 10% of all
maternal mortality (Schnider, Berger,
Cattell, 1977).
Although the annual maternal
mortality rate from ectopic pregnancy in the US has declined over the years, it remains as the
leading cauae of maternal deaths during the first trimester or pregnancy (Atrash et al, 1986;
Frau et al 1986; Budnick and Parker, 1982).
The recunence rate of ectopic pregnancy is 10 to 20 percent (BroDJOn, 1977) and 20 to 40
percent of women with ectopic pregnancy will not be able to conceive again (Mueller et al,
1987, Sherman et al, 1987).
Furthermore the consequences or ectopic pregnancy are alao
associated with great emotional stress, can have a major effect on a woman's sexual as well as
reproductive health by threatening her fertility.
In the developing nations of Africa, Asia,
and South America, where health care services are limited or unavailable for the majority of
the population, the morbidity and mortality of ectopic pregnancy are even more serious.
Thus the prevention of ectopic pregnancy by identifying its poeeible risk factors is a
significant and useful public health effort.
Etiology
A. Pelvic Inflammatory Disease
A number of ease-control studies have shown an association between pelvic infection and
ectopic pregnancy, with adjusted odds ratios ranging between 2.0 to 7.5 (Bren et al, 1974;
Levin et al, 1982; WHO, 1985).
A cohort study of pelvic inflammatory disease undertaken in Lund, Sweden provided t.he
strongest evidence of an association between ectopic pregnancy and pelvic inflammatory
disease. The hypothesis that pelvic inflammatory disease is associated with ectopic pregnancy
is based upon the fact that PlO-causing organisms such as Neisseria gonorrhea, Chlamydia
trachomatis, uroplasmas and mycoplasmas ascend to the fallopian tubes causing tubal
infection and eventually tubal occlusion thus preventing the implantation of a fertilized ovum
in the uterus.
~
41
B.
Intrauterine Device
A number of case-control studies have shown an association between IUD's and ectopic
pregnancy, with adjusted odds ratios ranging between 0.5 to 32.2 (Table 4.1). The hypothesis
that IUD's are associated with ectopic pregnancy is based upon studies that have shown that
a device in utero affects tubal motility and ovum transportation (Mastroianni, Rosseau, 1965;
Bengtsson, Mowad, 1966)
C. Oral Contraceptives
The uaociation between ectopic pregnancy and oral contraceptives has been studied for
both prior to and at time of conception. A. .hown in the Table 4.2 except the Women'.
Health Study and the WHO study with non-pregnant matched controls there does not seem
to exist an association between ectopic pregnancy and oral contraceptives as evidenced by the
95% confidence intervals for the odds rati06.
4.2
Pelvic Inflammatory Disease
Pelvic inflammatory disease is a serious and common health problem for women of
reproductive age.
Every year an estimated one million women in the United States alone
experience an episode of acute pelvic inflammatory disease; resulting in 1.7 million private
office visits (Blount et al, 1983) and 260,000 hospitalizations (Washington A, et ai, 1984).
Pelvic inflammatory disease has long been known as one of the major causes of female
infertility (Bedbug, Epeyz, 1958; Westrom, 1975) and increases a woman's risk' for chronic
abdominal pain, ectopic pregnancy and recurrent pelvic inflammatory disease (Westron, 1975;
Westrom, 1980).
According to Westrom, 11% of women with one episode of pelvic
inflammatory disease develop tubal occlusion, and 23% women with two episodes of
pm
develop tubal occlusion. The tubal occlusion rate increases to 54% for women with three or
more episodes of PID.
Table 4.1
Selected case-control studies to evaluate the relation between thc use of an intrautcrine device (IUD) and the occurrence of
ectopic pregnancy
Study location and
time author
No. of
cases
No. and type of
controls
Odds ration
(95% Confidence interval)
IUD use at
conception
Past IUD use
Controlled variables
Finland, 1972-1974
Rantakyala
71
ROO legal
abonion
6.4 (1.1-13.0)
Not available
None
Australia,
1971-1977
Pagano
287
3,200 obstctric
32.2 (21.8-47.7)
Not available
None
Finland, 1976
Savolainen ct aI
136
196 legal
abonion
155 delivery
16.3 (9.6-27.8)
1.0 (0.3-3.0)
None; excluded current
users when estimated risk
for past use
Boston, 1976-1978
Levin et al
85
498 delivery
Not available
1.4 (0.7-2.5)
13 variables; all
multigravida; excluded
current IUD users
Women's Health
Study, 1976-1978
Ory
615
3,453
nonpregnant
hospital
0.8 (0.6-1.0)
1.4 (1.1.-1.7)
Age, race, parity, previous
ectopic pregnancy pelvic
inflammatory disease
Seattle, 1975-1980
O1ow et aI
155
456 populationbased delivery
Not available
1.7 (1.1-2.6)
Race. gravidity. smoking
status, reference year,
condom us douching;
excluded current IUD users
World Health
Organizations,
1978-1980
1,108
1; 108 pregnant
1.108
nonpregnant
matched triplets
6.4 (3.3.-12.2)
0.5 (0.3-{).7)
0.7 (0.4-1.2
0.7 (0.1-3.9)
Age, parity, marital status,
hospital of treatment
WHO
Adopted from ChOw et aI
~
~
N
Table 4.2
Selected case-eontrol studies oral contraception (OC) use and the occurrence of ectopic pregnancy
Odds ration
(95% Confidence interval)
OC use at
conception
Past OC use
Study location and
time (author)
No. or
cases
No. and type
or controls
Finland, 1972-1974
Rantakyala, et al
61
780 legal
abortion
3.1 (0.9-10.4)
Not available
None; excluded current users
of intrauterine devices
Finland, 1976
Savolainen, et al
81
182 legal
abortion
155 delivery
2.1 (0.6,7.1)
0.5 (0.2-1.5)
None; excluded current users
of intrauterine devices
Boston, 1976-1978
Levin, et al
85
498 Delivery
Not available
0.6 (0.3-1.2)
13 variables; all multigravida
475
2,923
nonpregnant
hospital
0.1 (0.1-0.2)
Not available
Age, race, parity, previous
ectopic pregnancy, pelivic
inflammatory disease;
excluded past IUD users
1,108
1,108
pregnant
1,108
nonpregnant
mathced
triplets
2.2 (1.0-5.0)
0.1 (0.1-02)
Not available
Not available
Age, parity, marital status,
hospital of treatment
Women's Health
Organizations,
1978-1980 (48)
WHO
World Health
Organizations,
1978-1989 (48)
WHO
Controlled variables
Adopted from Chow et al
J::V-J
Economically, PIO and its consequences also represent a substantial economic burden to
society. The direct and indirect costs including PIO-related ectopic pregnancy and infertility
costs are staggering - an estimated 3.5 billion dollars per year (Peterson, et al, 1991).
Both Neisseria gonorrhea and Chlamydia trachomatis, which are prevalent sexually
transmitted diseases, have long been recognized as agents causing pelvic inflammatory
disease.
Other microbial isolates such as mycoplasmas and uroplasmas have also been
implicated as agents causing pelvic inflammatory diaeues.
Thus the ascent of these
microorganisms from the lower genital tract to the upper genital tract is, by definition, a
cause of pelvic inflammatory diaeale.
Oespite PIO's enormous medical and public health implications the clinical diagnosis
of PIO by clinical measures alone is difficult and has a high degree of inaccuracy.
In a
Swedish study in 1969 acute pelvic inflammatory disease was correctly diagnosed clinically in
only 65% of 814 consecutive cases admitted to the University Hospital, Lund, Sweden
(Jacobson I, Westrom L, 1969). A poor correlation between the clinical diagnosis of pelvic
inflammatory disease and the "correct" diagnosis (laparoscopically confirmed diagnosis) was
also reported in two other studies. (Farr, Findlay, 1928; Chaparo, Snehanshu, et al 1978) .
Hadgu et al (1986) provide a mathematical model that provides a more accurate
diagnostic algorithm for classifying women as having PIO or not.
According to this model
the best predictors of PIO include:
(1) purulent vaginal discharge
(2) erythrocyte sedimentation rate >, 15 mm/hr
(3) positive gonorrhea result
(4) adnexal swelling on bimanual examination and
(5) temperature> 38 0 C
(6) age less than 25 years
(7) single marital status
The logistic probability estimates for vanous combinations of the diagnostic variables
45
(Table 4.3) indicate that if a woman is positive for all seven of the predictor variables, then
her estimated probability of having PID is 97%. If a woman is positive for all the variables
except for purulent vaginal diKharge, then her probability of PID is estimated to be 86%. If
all the predictor variables are negative, then the probability of acute PID drope to 7% (Badgu
, Westrom, Brooks, 1986).
Furthermore Badgu et al developed hybrid models which combine these simple clinical
parameters and laparoscopy ( the Gold Standard).
For the hybrid model the sensitivity,
specificity, and overall classification rates were 100%, 67.2%, and 89.2% •
~
pointed out before, the hypothesis that PID is associated with ectopic pregnancy is
based on the theory that PID may prevent or retard migration of the fertilized ovum to the
uterus, thus predisposing a woman to ectopic gestation.
46
Table 4.3
Probability estimates of acute pelvic inflammatory disease for various
combinations of symptoms based on a logistic regression analysis with no interaction
Variables
Value
Value of prediction variable
Intercept
-2.5191
1
1
1
1
1
1
1
1
Discharge
1.5333
0
0
0
1
0
1
0
1
Erythrocyte
1.3458
0
0
0
0
0
0
1
1
Gonorrhea culture
results
0.9091
0
1
1
0
1
1
1
1
Adnexal
0.6646
0
0
0
0
1
1
1
1
Fever
0.6602
0
0
0
1
1
1
1
1
Age
0.3392
0
0
1
1
1
1
1
1
Marital status
0.3302
0
0
1
1
1
1
1
1
.07
.17
.29
.60
.61
.87
.86
.97
sedimentation
rate
Probability (acute pelvic
inflammatory disease)
0= No
I = Yes
Extracted from Hadgu et al
CHAPTERS
ANALYSIS OF THE ECTOPIC PREGNANCY DATA
Section 5.1 Description of The Lund Study
In this Section we
provide
longitudinal study consisted of
&
&
simple description of the Lund study.
The Lund
cohort of 2501 women, under 36 years of age, who
underwent a laparoscopic examination at the Department of Obstetrics and Gynecology,
University Hospital, Lund, Sweden, Crom January 1, 1960 to December 31, 1984.
The
catchment area of the Department includes the City of Lund with surrounding rural districts
in Southern Sweden.
It is the only Department for in-patient treatment of gynecological
disorders in women residing in the area. It also serves female non-residents studying at the
University of Lund. In 1960 the female population 10-35 years of age was close to 40,000
women, increasing to just over 50,000 in 1985.
All women in this study were inpatients with
signs and symptoms of acute pelvic inflammatory disease who underwent index (initial)
laparoscopy within 24 hours of admission. The laparoscopic technique used was
pr~nted
in
earlier reports (Westrom L, 1975) and remained constant for the period of the study. The
intra-pelvic findings of the laparoscopy were graded as follows:
Mild £In:.
Fallopian tubes reddened and swollen, but freely movable and with normal
morphology of the fimbriated ends.
Presence of an infectious purulent or seropurulent
exudate in the pelvic cavity or leaking Crom the fimbriated ends of the tubes.
Moderately severe PID:
Tubes not freely movable. Fimbriated ends abnormal or not clearly
visible. Infectious exudate an! fibrin deposits on serosal surfaces.
48
~
fm:. Pelvic peritonitis and/or abecess formation. Laparaseopic visibility significantly
impaired by inflammatory masses.
Each woman had to satisfy the minimum criteria for clinical diagnosis of PID before she
underwent laparoecopy to confirm the clinical diagnosis. The minimum criteria for clinical
diagnosis of PID were: From
196~1969:
Lower quadrant bilateral abdominal pain or pelvic
pain of shorter duration than three weeks plus two or more of the following:
1) abnormal vaginal discharge
2) rectal temperature> 380 C
3) vomiting
4) menstrual irregularities
5) urethritis symptoms
6) proctitis symptoms
.
7) marked tenderness of the pelvic organs on bimanual examination
8) adnexal maaa
9) erythrocyte sedimentation rate >15 mm per hour
From 1970 • 1984: All three of :
1) low bilateral abdominal or pelvic pain of shorter duration than three weeks
2) purulent vaginal discharge diagnoeed by direct microscopy of a wet mount of the vaginal
contents or painful intermenstrual bleeding
3) increased motion tenderness of the uterus and adnexa at a bimanual pelvic examination.
There were 3,250 women who fulfilled the clinical diagnosis of PID. However 749 women
were excluded from the follow-up study for the following reasons.
a)
Women subjected to index laparoec:opy after their 36th birthday. These women were
excluded because earlier studies (Westrom; 1975) had revealed that
pregnancy seeking
behavior was rare after the age of 35.
b)
Women in whom the index laparoeeopy revealed signifieant fertility reducing
(large fibroida, endometriosis, genital malformations, etc)
factors
49
c) Women who suffered from malignant diseases, chronic diseases of the heart, lung/chest,
kidneys, or who had chronic metabolic disorders.
d) Women who had consulted for infertility before the index laparoecopy.
e) Women who were known drug addicts or suffered from chronic psychotic diseases
such as
schizophrenia.
f) Women who refused laparoscopy or in whom a laparoscopy was contraindicated
for other
medical reasons.
g) Women given a diagnosis of puerperal PID, ie., disease episode within six weeks of
delivery.
Treatment
All patients with PID were treated in hospital with bed rest and antibiotics for at least
ten days. During the 25 year period of this study, a number of different antibiotic treatments
were used.
These include streptomycin/penicillin during the 1960-65 time period;
chloramphenicol/penicillin during 1966-71; and ampicillin-doxycycline during the 1971 to
1984 time period.
Follow-Up
Out of the 2501 women, 1004 were further excluded from the study for the following
reasons:
a) 514 women who did not expose themselves for the chance of pregnancy, i.e.,
women who
avoided pregnancy voluntarily.
b) 221 women who tried to conceive but were unsuccessful due to proven tubal or
factor infertility.
c) 168 women were lost to follow-up.
d) 19 women had history of ectopic pregnancy at initial pregnancy (at baseline).
e) 82 had incomplete examination.
non-tubal
50
Of the 1497 women who became pregnant after initiallaparoscopy, 744 were pregnant
once, 382 twice, 242 three times, 88 four times, and 41 more than four times. Thus a total
of 2806 pregnancies are included in this analysis. Of the 2806 pregnancies 1497 were firstorder, 753 were eecond-order, 371 were third-order, and 185 were Courth-order or higher order
pregnancies.
The major research questions in this analysis include:
1. Does the presence oC pelvic inflammatory disease predispoee a woman to have ectopic
pregnancy?
2. Bow has ectopic pregnancy changed since 1960?
3. What other factors predict high risk for ectopic pregnancy?
5.2
Descriptive Analysis
Table 5.1 shows the number of pregnancies contributed by each woman during the 19601984 time period. Meet women contributed only one or two pregnancies in this analysis; the
mean number of pregnancies contributed by each woman is 1.9 (Table 5.1).
One could
analyze a single pregnancy from each woman, thus using only 1497 (53%) of the 2806
pregnancies, and resulting in a loes of information and loes of statistical power.
The
magnitude of the loes in statistical power would be even greater if there were only a small
correlation among the pregnancies from the same woman. Furthermore if we limit the study
to the outcome of the first pregnancy only we can not distinguish changes in ectopic
pregnancy rate over pregnancy order.
Biologically, the damage to the fallopian tubes resulting from pelvic inflammatory disease
may be irreversible, and if it is serious enough to cause an initial ectopic pregnancy it may
also be serious enough to cause subsequent ectopic pregnancies.
Thus it is important to
study the impact of pelvic inOammatory disease on ectopic pregnancy for each pregnancy
order separately as well as all pregnancies together.
51
Simple descriptive statistics are presented in Table 5.2. The risk of ectopic pregnancy
increases with increasing age at pregnancy, prior adnexa.l operations and time-period of index
lapar08COpy.
More importantly the risk of ectopic pregnancy appears to increase markedly
with prior PID scores and number of PID episodes.
The PIO-score index was constructed by accounting for the number of previous PIO
episodes and their severity. A PIO-score of 0 indicates no previous history of PIO; a score of
1 indicates a history of mild PID, and a score of 2 indicates a history of moderately severe to
severe PID. Previous ectopic pregnancy wu alao a..ociated with current ectopic pregnancy
status. Of the microorganisms isolated at index laparoec:opy only the presence of Mycoplasma
at index laparoscopy was significantly associated with ectopic pregnancy. None of the birth
controls used at index laparoecopy (IUD, pill, barrier methods) was associated with ectopic
pregnancy. However, the information on birth control methods is severely inadequate since it
wu obtained only at initial laparoecopy and not during the follow-up time.
Note that
strictly speaking standard-c:hi squared tests are not appropriate for these data since
pregnancies within a woman (cluster) are correlated.
52
Table 5.1.
Number of Pregnancies Contributed by Each Woman
During the 1960-1984 Time Period.
Number of Pregnancies
Conoibuted
Number of Women
Total number of
pregnancies
1
744 (50%)
744 (26%)
2
382 (25%)
764 (27%)
3
242 (16%)
726 (26%)
4
88 (6%)
352 (13%)
5
29 (2%)
145 (5%)
6
9 (1%)
54 (2%)
7
3 (0%)
21 (1 %)
Total
1497
2806
Average number of pregnancies conoibuted by each women
~= 1.9
1497
53
Table 5.2 Univariate Anal ysis of Ectopic Pregnancy that Occurred During 1960-1984 Time
Period at theDepanment of Obstetrics and Gynecology. University Hospital. Lund. Sweden.
* Ec«mics
* Pregnancies
Variable
••
Percent
P-Value
.000
Mother's age at Pregnancy
S24
41/1107
3.7
25-30
101/1132
8.92
31 +
51/567
8.99
S 19
7011259
5.56
20-24
8811039
8.47
25+
35/508
4.7
1960-1967
ron92
7.58
1968-1976
10211267
8.05
1977-1984
31n46
4.16
84/1215
6.91
97/1377
7.04
8/118
6.78
0
104/1494
6.96
1
5'2n54
6.90
2
271372
7.26
3+
10/186
5.38
0
158/2640
5.98
1
261132
19.70
2+
9/34
26.47
Age at index laparoscopy
0.023
Index Laparoscopy Time Period
.003
Marital Status
Married or Cohabiting
Single
-
Other
.98
Prior Pregnancies
.86
Prior adnexal operations
.000
54
Variable
P-Value
# EctQpics
# Pregnancies
Percent
Prior PIO episodes
0
18n89
2.28
1
148/1789
8.27
2
21/189
11.11
6139
15.38
3+
.000
Previous Still Births
0
188/2769
1+
5/37
6.79
0.108
13.5
Previous Ectopic Pregnancy
0
168/2639
6.24
1
25/113
22.12
2+
.000
28.57
Prior PIO Score
0
18n89
2.28
1
91/1457
6.25
2
84/560
15.0
.000
Microorganisms Isolated at Index Laparoscopy
Gonorrhea (GC)
Yes
51n05
7.23
No
142/2101
6.76
Yes
13/190
6.84
No
180/2616
6.88
Yes
44/444
9.91
No
149/2362
6.31
0.66
ECOLI
.98
Mycoplasma (MYCO)
0.006
55
Variable
P-Value
# Ectopics
# Pregnancies
Percent
Strep
Yes
42/499
8.42
No
151/2707
6.55
.134
Birth Controls Used at Index laparoscopy
IUD (ever)
Yes
47n18
6.55
No
14612088
6.99
Yes
51/800
6.38
No
14212006
7.08
Yes
15/246
6.10
No
17812500
6.95
.684
Pill
.508
Barrier
** = Chi-squared test
.613
56
Figure 5.1 shows the longitudinal ectopic pregnancy rates by PID-score (PIO severity)
and pregnancy order.
Ectopic pregnancy was related to PID-severity cross-sectionally.
Women with no PIO (score = 0) showed lower ectopic pregnancy rates compared to women
with PIO (score 1 • acore2) at all pregnancy orden.
increased with the severity of tubal damage.
The ectopic pregnancy proportion
Moreover, the damage did not decline
longitudinally; rather, it seemed to stay constant acrou pregnancy order, thus indicating
that the risk of ectopic prepancy from PID does not seem to decrease with increasing
pregnancy order.
Though the ectopic pregnancy proportion for severe PIO decreases at 3rd order pregnancy,
a chi-squared teat indicates no
statistically significantly diff'erentce (p=0.337) from the
proportion of first order or second order pregnancy. However, for women with NO-PIO a
chi-squared test indicated a significant increase (p=0.04) in the ectopic pregnancy proportion
from fint-order pregnancy to third order pregnancy.
Thus though the ectopic pregnancy
proportion for women with no PID is less than those of women with mild or severe PIO c.rosssectionally, proportions for the no-PIO groups tend to increase longitudinally, unlike those of
PIO women.
Figure 5.1 Ectopic Pregnancy Rates by
Severity and Pregnancy Order
E
c
t
20
o
P
I
c
15
I:.
P
r
e
g
n
10 .-
a
n
c
y
R
a
t
e
s
5 .-
o
I
I
I
I
P1
P2
P3+
I
Pregnancy Order
No PIO
---*- Mild PIO
~ Severe PIO
P3+-Pregnancy Order 3 or more
P1-Pregnancy Order 1
P2-Pregnancy Order 2 Bars are standard errors
-S-
Vl
'-l
58
5.3 Model fitting
5.3.1 Analysis Using Standard Logistic Regression Model (Ignoring Dependence)
In this aection we analyze the ectopic pregnancy data using ordinary logistic regression
techniques assuming the pregnancies within a cluster (women) are statistically independent.
First our model-building strategy involves seeking the most parsimonious model that explains
the data but also includes the scientifically relevant variables (such as PIO and
g~nital
mycoplasmas) whether they are statistically significant or not. There!ore our model-building
strategy and particularly our variable selection proc:esa we take the following into account:
1. A univariate logistic regression analysis of each variable will be undertaken.
2.
Any variable whose univariate test has a p-value of
< 0.25 will
be considered as a
candidate for the multivariate logistic regression. The cut-off' point of 0.25 is based upon the
work of Bendel and Afifi (1977) for linear regression and Mickey and Greenland (1989) for
logistic regression. The problem with this approach is that an explanatory variable mAy be
weakly associated with a dependent variable in a univariate analysis; however, it may become
an important variable in the presence of other covariates.
3.
We will then fit a multivariate logistic regression model and the importance of each
variable included in the model will be verified. This will be done by the examination of the
Wald statistic for each variable.
Variables that do not contribute to the model are then
eliminated and a new model is fitted.
The new model will be compared to the old model
through the likelihood ratio test.
4.
Once we have obtained the model that we feel conta.ins the essential variables, we will
then consider the need for including interaction terms among the variables.
Finally we also use stepwise logistic: regression methods to see if the variables selected
using the above variable selection procedure cillIer from those selected by a stepwise logistic
regression technique.
In order
to
make the logistic regression analysis easier, an abbreviated code sheet is given
59
in Table 5.3. which describes the independent variables and provides the code names for the
variable that will be used.
The results of fitting the univariate logistic regression models to these data are given in
Table 5.4. In Table 5.4 we show the following:
1. the estimated regression coefficient(s) for the univariate logistic regression
model
containing only this variable.
2. the estimated standard error of the regression coefficient.
3. the estimated odds ratio, which is obtained by exponentiating the estimated regression
coefficient.
4. the 95% CI for the odds ratio.
5. the value of the log-likelihood for the model.
6. the likelihood ratio test statistic, G, for the hypothesis that the slope
.
coefficient is zero. This is obtained by:
G- -2 1 {likelihood without the variable(constant only)}
og
likelihood with the variable
-
Under the null hypothesis this quantity will follow the chi-squared distribution with 1
degree of freedom for the independent variables with two categories and 2 degrees of freedom
for the independent variables with 3 categories.
From Table 5.4 we see that with the exception of NPREV (# previous pregnancies) and
ISTATUS (marital status at index laparoscopy), there is evidence that each of the variables
has some association with the outcome variable, namely ectopic pregnancy. This observation
is based upon an inspection of the confidence interval estimates, which for most variables
either do not contain 1 or just barely so.
Therefore, based on the univariate results, we begin the multivariate model with all the
variables except NPREV and 1STATUS. On the basis of the results shown in Table 5.5 it
appears that all of the variables except AGE and PREVSTlL show considerable importance
60
in the multivariate logistic regression model. This suggests that we fit a new model which
does not contain AGE and PREVSTIL. The results of this reduced model are displayed in
Table 5.6.
The likelihood ratio test for the difference between the models in Table 5.5 and
5.6 yields a value of G = 1256.9 - 1256.7 = 0.2.
Comparing this to the chi-squared
distribution with 2 degrees of freedom yields a non-significant
~value,
demonstrating that the
variables AGE and PREVSTIL add little to the model once the other variables are retained.
Also note that the inclusion or exclusion of AGE and PREVSTIL to the model does not alter
the values of the other regression coefficients by much.
An alternative approach to variable selection is to use the stepwise method in which
variables are selected either for inclusion or exclusion from the model in a sequential fashion
based on a pre-specified inclusion/exclusion criterion.
We used the PROC LOGISTIC
procedure with the STEPWISE option and obtained variables identical to the ones shown in
Table 5.6.
Table 5.3 Code Sheet for the Variables in the Lund Ectopic Pregnancy Data
Variable
Abbreviation
Identification Code
ID
Ectopic Pregnancy
ECf
1 = ectopic pregnancy
o = not ectopic pregnancy
Age at Index Laparoscopy
0= Age
~
19
1 = Age
~
20
Age at Pregnancy
0= Age
~
24
1 = Age
~
25
Marital Status at Index Laparoscopy
AGE
AGEGP
ISTATUS
o = Married or Cohabiting
I = Single
2 = Other
Year initial laparoscopy was perfonned
LAPID
0= 1977-1984
1 = 1960-1976
Genital Mycoplasma at Index laparoscopy
MYCO
0= No
1 = Yes
Prior Pregnancies
NPREV
o = no prior rregnancy
1 = 1 prior pregnancy
2 = 2 or more prior pregnancies
Prior adnexal surgeries
o = no previous adnexal surgery
1 = 1 or more adnexal surgery
NSG
61
62
Variable
Abbreviation
PID Score (Laparoscopy)
SCORE
SCORE1=1 (Mild PID)
SCOREI
SCORE1=O (Otherwise)
SCORE2 =1 (Severe PID)
SCORE2 =0 (Otherwise)
SCORE2
Table 5.4
Variable
Univariate Logistic Regression Models for Ectopic Pregnancy
Parameter Standard Odds Ratio
Estimate
Error
Estimate
95%
c.1.
1) SCOREI
1.0485
.2619
2.85
(1.71,4.77)
SCORE2
2.0227
.2662
7.56
(4.49,12.74)
2) AGE
.1624
.1090
1.1 H
3) AGEGP
.9379
.1H04
4) ISTATUSI
.0518
ISTATUS2
-2 log L
G
p-value
1326.045
79.670
.0001
(0.95,1.46)
1402.941
2.723
.0989
2.55
(1.79,3.64)
1374.52
31.137
.0001
.1526
1.05
(0.78,1.42)
1405.55.
.117
.943
0.0107
.3829
1.01
(0.48,2.14)
.6773
.2009
1.97
(1.33,2.92)
1392.70
12.964
.0003
6) NPREVI
-.0100
.1760
.99
(0.79,1.40)
1405.594
.07
.9655
NPREV2
-0.0522
.1982
.95
(0.64,1.40)
7) NSG
1.4344
.2072
4.20
(2.80,,6.30)
1367.15
38.474
.0001
8) MYCO
.4909
.1800
1.63
( 1.15,2.32)
1398.777
6.887
.0087
9) PREVSTILL
.7632
.4868
2.14
(0.83,5.57)
1403.608
2.057
.1515
5) LAPID
0\
W
64
Next we proceed to assess interactions among the variables in the model. A total of 10
.
possible two-way interactions were formed from the model in Table 5.6.
However the
interaction between PIn score (SCORE) and age at pregnancy (AGEGP) could not be
assessed because BOme of the cells contained zero observations.
At this point we are interested in assessing the contribution of each interaction term to
the previously developed model Table 5.6.
model are presented in Table 5.7.
The result of adding each interaction to this
Upon inspection of Table 5.7 we see that there is one
interaction term that is statistically significant. This is the SCORE x MYCO interaction.
The addition of the SCORExMYCO interaction term appears to provide a significant
improvement over the main effects only model (p=0.0497). Therefore the inclusion of this
interaction
into the model offen the possibility to better describe the effect of PIO and
genital mycoplasma on ectopic pregnancy.
The estimated coefficients and their standard
errors are given in Table 5.8.
We now turn to the interpretation of the fitted model in Table 5.8. For covariates not
involving interaction terms the estimated odds ratios are obtained by simply exponentiating
the estimated regression coefficients. For example, the odds of developing ectopic pregnancy
increases by 2.4 fold (=exp(O.8801» for women whose age at pregnancy was 25 or more.
Estimated odds ratios and 95% C.I. for the variables not involving interaction terms are
given in table 5.9.
A modification of the above approach is necessary to estimate the odds ratios for
variables involving interactions. Specifically we would like to:
(1)
Estimate the odds ratio of PIO (SCORE) separately among those women with genital
mycoplasma and those without mycoplasma.
(2)
Estimate the odds ratio of mycoplasma separately among those women with PIO and
those women with NO PIO (SCORE1 = 0 and SCORE2 = 0).
(3) A fifth odds ratio of interest may be the odds ratio for both PIO and mycoplasma
(SCORE=l and MYCO = 1) relative to women with no genital mycoplasma and No-PID
65
(MYCO =0 and SCORE=O ) controlling for other predictors.
Each of these odds ratios is obtained by exponentiating an appropriately calculated logit
difference. We will first present the logit and the differences in general terms, and then use
the values of the estimated regression coefficients in Table 5.9 to get numeric values. First let
p·X·
represent the contribution by the terms containing AGEGP, LAPIO and NSG, terms
that do not involve interactions. Then the values of the estimated logit differences are given
in Table 5.10 For example, the logit difference d 1 (Table 5.10) represents the effect of
SCOREI (mild pm) among women without genital mycoplasma (MYCO=O).
We now turn to the interpretations of the estimated odds ratios in Table 5.11. From
Table 5.11 we see that PIO (whether it is mild or severe) by itself is an important risk factor
when the woman does not have genital mycoplasma.
The estimated adjusted odds ratios for
mild and severe PIO among women with no genital mycoplasma are 4.10 and 10.19
respectively.
However, among women with genital mycoplasma the odds ratios for mild or
severe PIO are not as impressive. The estimated odds ratios for mild and severe PIO among
women with genital mycoplasmas at index laparoscopy are 1.08 and 2.60 respectively. These
differences in the odds ratios for PIO among those with mycoplasma infection compared to
those without mycoplasma infection results from the negative interaction between PIO and
mycoplasma. This indicates that a positive finding for PIO (or mycoplasma) is important if
the finding for mycoplasma (or PIO) is not positive.
Similarly we see that genital mycoplasma is also a significant risk factor among women
who do not have PIO. The estimated odds-ratio for mycoplasma among women with no PIO
is 5.10. The estimated odds-ratios for mycoplasma among women with mild PIO and severe-
PIO are 1.34 and 1.30 respectively. When taken together, the odds ratio for both mild PIO
and mycoplasma (SCOREI=1 and MYCO=I) relative to
women with no PIO and no
mycoplasma (SCORE1=0 and MYCO=O) is 5.51. The odds ratio for both severe PIO and
mycoplasma (SCORE2=1 and MYCO = 1) relative to women with no pm and no
mycoplasma (SCORE 2 = 0 and MYCO=I) is 13.27.
Table 5.5
Estimated Coefficients and Standard Errors for the Multivariate
Model Containing Variables Identified in the Univariate Analysis
Variable
Parameter
Estimate
Standard
Error
Wald
Oli-Square
Pr>
Oli-Square
INTERCPT
-4.9152
0.3303
221.5086
0.0001
SCOREI
1.0159
0.2639
14.8184
0.0001
SCORE2
1.9127
0.2707
49.9265
0.0001
AGEGP
0.8813
0.2053
18.4194
0.0001
AGE
-.0217
0.1151
0.0356
0.8504
LAPID
0.5462
0.2078
6.4056
0.0086
NSG
0.9608
0.2210
18.8962
0.0001
MYCO
0.4465
0.1898
5.5316
0.0186
PREVSTlLL
0.1703
0.5258
0.1048
0.7461
-2log L
= 1260.394
66
Table 5.6
Estimated Coefficients and Standard Errors for the Multivariate
Model Containing Variables Significant in Table 5.5.
Parameter
Estimate
Standard
Error
Wald
Oli-Square
Pr>
Variable
INTERCPT
4.9266
0.3267
227.3643
0.0001
SCORE1
1.0169
0.2638
14.8656
0.0001
SCORE2
1.9174
0.2699
50.4633
0.0001
AGEGP
0.8677
0.1855
21.8819
0.0001
LAPID
0.5506
0.2073
7.0540
0.0079
NSG
0.9664
0.2200
19.2888
0.0001
MYCO
0.4505
0.18902
5.6816
0.0171
-2log L = 1260.540
Chi-Square
67
68
Table 5.7 Log-likelihood, Likelihood Ratio Test Statistic (G), Degree of Freedom.
and P-Value for Possible Interactions of Interest to be Added to the Main Effect Model.
Interaction
-2 log-likelihood
G
d.f
p-value
Main Effects Only
1260.5
SCORE x LAPIn
1260.1
0.4
2
.8183
SCORE x NSG
1259.5
1.0
2
.6065
SCORE x MYCO
1254.5
6.0
2
.0497
AGEGP x LAPID
1260.5
0.0
1
1.0000
AGEGP x NSG
1260.4
0.1
1
0.7518
AGEGP x MYCO
1260.5
0.0
1
1.0000
LAPID x NSG
1260.4
0.1
1
0.7518
LAPID x MYCO
1257.2
3.3
1
0.0693
NSG
1259.2
1.3
1
.2542
x MYCO
•
.
Table 5.8
Estimated Co-efficient and Slandard Errors for the Multivariate
Model Containing Main Effects and Significant Interactions.
Variable
Parameter
Estimate
Standard
Error
Wald
Chi-Square
Pr>
Chi-Square
INTERCPT
-5.3073
0.4111
178.4401
0.0001
SCORE1
1.4112
0.3647
20.4032
0.0001
SCORE2
2.3215
0.3709
47.0801
0.0001
AGEGP
0.8801
0.1862
24.0466
0.0001
LAPID
0.5652
0.2086
7.7541
0.0054
NSG
0.9375
0.2207
16.7725
0.0001
MYCO
1.6290
0.4992
9.5599
0.0020
SCORE1 x MYCO
-1.3336
0.5708
4.9463
0.0261
SCORE2 x MYCO
-1.3653
0.5828
4.9977
0.0254
Log-likelihood = 1254.5
Table 5.9
Estimated Odds Ratios and 95% C.I for variable not
involving interation terms:
Variable
Estimated Odds Ratio
95% C.I.
AGEGP
2.41
(1.67.3.47)
LAPID
1.76
(1.17.3.47)
NSG
2.55
(1.66.3.94)
69
70
Table
~lO
Expression for Ole Logita and Logit Differences in Termll of
the Estimated Parameters for the P08lIible Combinations of MYCO
and SCORE.
Score (pm)
MYCO
o
1
o
Logit Difference
d 1 =iJ 1 SCORE
1
•
+ iJ*X*
+
iJ 3(SCORE x MYCO)+ iJ*X*
iJ 3(SCORExMYCO)
Logit
DiB'.
iJ 3(SCORExMYCO)
71
Table 5.11
Values of the Estimated Legit Differences and Odds Ratios
Effect
Among
Legit Difference
Odds Ratio
SCOREI = 1
MYCO=O
1.4112
4.10
SCOREI = 1
MYCO = 1
(1.4112-1.3336)
1.08
MYCO = 1
SCOREI = 0
1.6290
5.10
MYCO = 1
SCOREI = 1
1.6290-1.3336
1.34
(1.4112 + 1.6290 - 1.3336)
5.51
SCOREI + MYCO
SCORE2 = 1
MYCO =0
2.3215
10.19
SCORE2 = 1
MYCO= 1
(2.3215-1.3653)
2.60
MYCO = 1
SCORE2 = 0
1.6290
5.10
MYCO = 1
SCORE2 = 1
(1.6290-1.3653)
1.30
(2.3215 + 1.6290 - 1.3653)
13.27
SCORE2 + MYCO
72
Analysis !.Wllg Q.EE Techniques
5.3.2
In this section we use the GEE method explained in Chapter 3 to fit the ectopic
pregnancy data, with the logit link function h(p) = log-IP and the variance function V(p) =
-p
p(l-p).
The GEE method was used to model the marginal dependence of E(Yit) on the
covariate:s. Table 5.12 shows estimated regression coefficients and Z-statistics for a binomial
error and a logit link model.
Three different correlation structures , namely independent,
exchangeable, and I-dependent correlation structures, are fitted to investigate the effect of
specifying different correlation structures. One-dependent correlation structure assumes that
correlations one occasion apart are the same (but non-zero) and correlations two or more
occasions apart are zero. From the magnitude of the regression coefficients of the model with
one-dependent correlation structure, it can be seen that pelvic inflammatory disease is the
strongest predictor of ectopic pregnancy, followed by history of genital mycoplasma at index
laparoscopy and age at pregnancy.
The presence of severe pelvic inflammatory disease
increases the odds of having ectopic pregnancy by 10.93 ( = exp(2.3918)) fold for women
without genital mycoplasma. This odds increases to 14.38 (=exp(2.3918+1.654o-1.3803) for
women with genital mycoplasma. Similarly, the odds of developing ectopic pregnancy after
mycoplasma are 5.23 times higher (=exp(1.654) for women without PID. The estimate of the
'working' correlation,
Q,
is 0.089. Similarly, the estimate of the scale parameter is 0.9532 for
the I-dependent correlation structure. One can note that for dichotomous outcomes the true
value of 4J is one. Thus a way in which the results of Table 5.12 could be adjusted to a scale
parameter of one is through division of regression estimates and their standard errors by the
square root of the estimated scale parameter, e.g., for I-dependence division by the square
root of 0.9532 would be applied. Since such results would be similar to those already shown
no such adjustment has been made. Furthermore, we find the computational procedure to be
reassuring by providing a scale parameter estimate which is so close to the known value of
one.
73
For the time-independent variables note that the coefficients and the z-statistics for the
three
correlations are similar. However, for the time-dependent variable, NSG, the
coefficients and their associated robust z-statistics are different under the three correlation
structures.
As pointed out by Zeger and Liang (1986)
" the effect of assuming a non-
independent correlation structure is to use weighted linear combinations of both Y's and X's
for each subject in GEE. It is therefore sensible that covariates which vary with time may
experience larger changes in their coefficients than do time-independent covariates for finite
sample."
74
Table 5.12
GEE Analysis of Ectopic Pregnancy with Legit Link
and Binomial Variance
Correlation
Structure
Estimate
Intercept
independent
exchangeable
I-dependence
-5.3073
-5.3108
-5.3410
SCORE1
independent
exchangeable
1- dependence
SCORE2
S.E. - Robust
z- Robust
0.382
0.401
0.397
0.421
0.429
0.425
-12.60
-12.39
-12.56
1.4112
1.4284
1.4326
0.336
0.355
0.349
0.363
0.370
0.365
3.89
3.86
3.92
independent
exchangeable
l-dependence
2.3215
2.4263
2.3918
0.342
0.360
0.355
0.370
0.376
0.373
6.27
6.45
6.42
LAPID
independent
exchangeable
I-depcndence
0.5651
0.6085
0.6024
0.204
0.214
0.211
0.217
0.222
0.219
2.61
2.74
2.74
AGEGP
independent
exchangeable
I-dependence
0.8801
0.8818
0.8910
0.182
0.179
0.183
0.189
0.184
0.189
4.65
4.79
4.71
NSG
independent
exchangeable
I-dependence
0.9375
0.3495
0.6097
0.215
0.247
0.234
0.222
0.291
0.252
4.23
1.20
2.42
MYCO
independent
exchangeable
I-dependence
1.6270
1.6295
1.6540
0.483
0.523
0.506
0.527
0.544
0.540
3.09
2.99
3.06
SCORElx
MYCO
independent
exchangeable
I-dependence
-1.3336
-1.3375
-1.3575
0.554
0.596
0.578
0.588
0.610
0.604
-2.27
-2.19
-2.25
SCORE2 x
MYCO
independent
exchangeable
I-dependence
-1.3655
-1.3899
-1.3805
0.566
0.605
0.588
0.615
0.634
0.629
-2.22
-2.19
-2.19
Covariate
For independent correlation structure
scale parameter = 0.9618
For exchangeable correlation structure
scale paramenter = 0.9441
working correlation = 0.1043
For l-depcndence correlation structure
scale pararnenter = 0.9532
working correlation = 0.089
S.E Naive
75
5.3.2.1
Selection Between
~
IIu:tt Correlation Structures
Comparison of the naive z-statistics and robust z-statistics can be used to evaluate
whether a specified correlation structure is incorrect.
Note that naive standard errors are
calculated assuming that the 'working' correlation structure is the true one. Recall that from
equation 3.22 we have that
When the correlation structure is assumed to be correctly specified then
iO
= iI-I. One can therefore expect relatively large differences between the robust and naive
z-statistics if the working correlation is far from the true correlation. However, if they are
similar it does not necessarily imply that the assumed correlation structure is similar to the
true correlation structure.
Table 5.13 shows the absolute relative difference between the robust and naive z-statistics
for the two correlation structures. For the independent correlation structure, naive z-statistics
show larger values than robust z-statistics. This suggests that the variance is underestimated
since correlated data are treated as independent and counted as separate information. For
almost all covariates the independent correlation structure showed relatively higher relative
difference in absolute terms. Both the exchangeable and one-dependent correlation structures
appear to have similar relative changes. Perhaps another way of choosing among 'correlation
structures is to use the technique of Rotnitzky and Jewell given in Section 3.3. Table 5.14
gives the eigenvalues of the matrix Q defined in Section 3.3. From Table 5.14 we see that
(~1' ~2) values for the three different correlation structures are similar with the I-dependent
correlation structure having values slightly closer to one, but not by much. Rec.all that i l is
the average of the eigenvalues of matrix Q and ~2 is the average of the squared eigenvalues
of matrix Q given in equation 3.26.
Table 5.13
Absolute Relative Difference Between Robust and Naive Statistics for the Independent and
Exchangeable Correlation Structure.
Exchangeable
Independent
Covariate
Z~Robust
Z-Naive
Relative
Change
Z-Robust
Z-Naive
INTERCEPT
-12.60
-13.88
10.1%
-12.39
-13.25
SCOREI
3.89
4.21
8.2%
3.86
SCORE2
6.27
6.80
8.4%
LAPID
2.61
2.77
AGEGP
4.65
NSG
MYCO
I-Dependence
Relative
Change
Relative
Change
Z-Robust
Z-Naive
6.9%
-12.56
-13.46
7.2%
4.02
4.1%
3.92
4.10
4.6%
6.45
6.75
4.6%
6.42
6.74
5.0%
6.1%
2.74
2.85
4.0%
2.75
2.86
4.0%
4.84
4.1%
4.79
4.92
2.7%
4.71
4.88
3.6%
4.23
4.35
2.8%
1.20
1.41
17.5%
2.42
2.61
7.9%
3.09
3.37
9.1%
2.99
3.12
4.3%
3.06
3.27
6.9%
INTI
-2.27
-2.41
6.2%
-2.19
-2.24
2.28%
-2.25
-2.35
4.4%
INTI
-2.22
-2.41
8.6%
-2.19
-2.30
5.0%
-2.19
-2.35
7.3%
INTI = SCOREI X MYCO
INTI = SCORE2 X MYCO
.....
0\
Table 5.14
The Eigenvalues of Malrix Q for the Independent and Exchanageable Correlation Structures.
Eigenvalues
Independent
Exchangeablc
I-Dependent
c.
1.4268
1.5030
1.3689
C2
1.2835
1.3357
1.3319
cJ
1.2412
1.2015
1.2208
e.
1.1809
1.1133
1.1587
C5
1.0689
1.0459
1.0410
e6
].0332
0.9979
1.0039
C7
0.9832
0.9910
0.9780
c8
0.8997
0.8995
0.9107
c9
0.8380
0.8181
0.8247
c.
1.1062
1.1117
1.0932
c2
1.2563
1.2740
1.2264
......
......
78
5.3.3
Analysis Using Rosner's Beta-Binomial Model
In this section we provide a summary of Rosner's (1984,1989) regression technique for
correlated binary data.
Rosner's model was derived as an extension of the beta-binomial
model of Williams (1975).
For simplicity, we first restrict our attention to the case where
each primary sampling unit has exactly two subunits. Let Y1 and Y2 be the binary outcomes
associated with the two units in the ith pair. Associated with each member of the pair is a pdimensional vector of covariatea xi'
Let us now consider a primary sampling unit with x
=Xo = (0 0 •.• 0).
Roener then
the beta binomial distribution to model the probability of the four disease states
U8e8
(++,
+-,
-+, -) for such a PSU or cluster. Specifically, he assumes that this cluster has
probability Pi of having any particular sub-unit affected, where Pi follows a beta distribution
with parameters a and b over the class of all individuals with covariates xO' It then follows
that the probability of disease states for such a PSU is given by
1
P(++ I xO) =
of
p.2 p.a-1 (1- p.)b-1
1
1
1
r(a+b) dp.
r(a)r(b)
1
(a+l)a
= (a+b+1)(a+b)
Similarly
P(+ -
I xa) =
P(-+)
= (a+b+l)ab (a+b)
(b+1)b
P(- I xO) = (a+1>+1) (a+b)
Then the probability distribution of disease states for a PSU with arbitrary covariates x. is
1
given through two conditional logistic regression models:
79
In
where
crl
= In ( b ~ 1) , and
cr2
= In (b+1)b
(a+1)a
Rosner then shows that, using (5.1), the joint probability distribution of Y 1 and y 2
It is clear that when cr2
18
= 2cr1 ' Rosner's model reduces to the standard, uncorrelated logistic
model for two independent observations. Rosner shows the odds ratio for association between
y 1 and Y2' controlling for x, is given by:
=
(a+1)(b+1)
ab
Parameter estimation in (5.2) is then undertaken by the method of maximum likelihood.
Since there does not exist a closed form of (5.2), an iterative procedure based on the NewtonRaphson method may be employed. The interpretation of a regression parameter of Rooner's
model, {3x, is as follows:
f3x is the increase in the log-odds of response for
subunit holding all other covariates and the response of the fellow subunit fixed.
a particular
80
5.3.3.1
Extension to
~
General
~
Let us now consider the general case, where the i th primary unit has ni subunits, i
= 1, ..,
K. Let Yij = 1 if the jth subunit of the primary unit is affected, and 0 otherwise, j = 1,....,
ni' Le t ! i
!i
(1)
(0)'
,.... !i
4
(ni)
(1)'
,...., Xi (ni)' '
h
(0) .
. f'
. d
d
.
d
were Xi
IS a matnx 0 tlme-lD epen ent covarlates an
.: f'
d - d
.
are matrlcles 0 tlUle- epen ent covanates.
Rosner (1984, 1989) generalize the beta-binomial model to accommodate both timedependent and time-independent covariates using a particular polychotomous logistic
regression model. Under this model the likelihood for the i th primary unit is given
by
(0)
ni
&si bo.-s.
P(y-'1
1
exp
1
[.E
J=1
YlJ" (.B!i
+ 1'!i
0)
)]
I !1') = --------....::...-.;;..w-:----------
E
Zi
(0)
ni
a,rz..
IJ
b(n._Ez..)
1
exp
IJ
[E
j=1
Zij (.B:i
where a and b are the parameters of a beta distribution,
+ ~ i
.B and
(5.3)
0)
)]
•
l' are regression coefficients
corresponding to time-independent and time-dependent variables, Yij=1 if the jth subunit in
=
·
. IS
. auecte
- er
d an d Oot
h erwlse.
.
the 1.i hpnmary
uDlt
bm ~~
(b + j), m
~
1, aO
F urt h ermore,
am
m-l
= j=O
11'
( a+J,
.)
= b O = 1, and the summation in the denominator. is over all
possible permutations z· = (z.l' z'2' .., z· ) of 0'5 and l's. And s· denotes the number of
1
1
1
lDi
1
subunits of the the i th primary unit which are affected. If no covariates are present, then
according to Rosner (1989) the denominator in (5.3) is (a+b)n.1 and
Prob(y.)
reduces to
_1
lsi' bo.-s.
/ (a+b)n.. which is a beta model with parameters a and b. Moreover for dusters
111
of size 1, the model reduces to ordinary logistic regression and the coefficients of fJ and l'
have a marginal interpretation.
The full likelihood is then given by L
= II P(Yi I Xi)
i
and the Newton-Raphson technique
81
is used to obtain maximum likelihood estimates and perform significance tests concerning a,
b, p, and "'(. The measure of dependence in this model is the pairwise odds ratio between
subunits, (a
+ 1) (b + 1)/ ab, where a and b are the parameters of a beta-binomial
distribution. Rosner dermes the pairwise odds ratio to be the odds of disease for one cluster
member given that another cluster member is also affected, divided by the comparable odds
given that the other cluster member is not affected.
5.3.3.2
Comparison with Ordinary Logistic Regression
AD. important distinction between Roener's model (5.3) and the ordinary logistic
regression model is that Rosner's model explicitly includes the outcome of previous visits as
additional predictor covariates, whereas ordinary logistic regression treats the visits within a
cluster as separate and independent observations. In a subsequent paper Rosner and Milton
(1988) have presented an approach for obtaining appropriate ordinary logistic regression
estimates and standard errors from Rosner's regression estimates:
below:
l:
.
1
=
n·1 ( a+b+n.)
1
(a+b+l)l:
n·1
•
1
l:
n.[a.+b+1+(n.-1)p]
.
1 1
1
=
Var(pp)OL
Var(pp)R
1
(a+b+l)
=
=
1
a+b+l
E ni
Ei
n. (a+1>+n. )[x. (0)]2
l l
Ip
1
a+b+l
~{[a+1>+1 +(ni-1)Pe]4= [xie(j)]2}
1
J
These results are given
82
for balanced data (ni = n), we have
= Var(re)OL = a+b+l+{n-l)Pe
Var(re)R
a+1>+1
where {{3 P)OL and {{3 P)R are the time-independent regression coefficients from the ordinary
logistic regression model and Rosner's model. Similarly respectively {re)OL and ("Ye)R are
the time-dependent regression coefficients Crom the ordinary logistic regression and Rosner'.
model respectively. The above formulae are obtained by assuming
all the covariates are
independent and that the regression of the eth time-dependent covariate,
#- k)
Xe 0) on Xe (k)
(j
is approximately linear (Rosner and Milton, 1988). Glynn and Rosner (1993) in a
simulation study investigated the performance of Rosner and Milton's conversion formula for
both time-dependent and time-independent covariates. These
authors conclude that the
conversion formula performed well in terms of coverage probability, consistency, and bias.
5.3.3.3
Application of Rosner's Model to the Ectopic Pregnancy Data
We now present the application of Rosner's beta-binomial model to the ectopic. pregnancy
data described in Section 5.1. Estimates of these regression coefficients that condition on the
status of previous pregnancies were obtained by using the polychotomous logistic regression
program peHLE of Rosner which was written in FORTRAN IV.
Table 5.15 summarizes results of the beta-binomial modelling approach with the use of
the Lund ectopic pregnancy data set. From the magnitude of the regression coefficients of the
model, it can be seen that pelvic inflammatory disease is the strongest predictor of ectopic
pregnancy I followed by history of genital mycoplasma and age at pregnancy.
The conditional pairwise odds ratio of ectopic pregnancy between two pregnancies is
,
83
(a
+
1) (b
+
1)/ (ab) = 2.2. The conditional pairwise odds ratio of 2.2 means that on the
average the odds of a subsequent ectopic pregnancy increases by 2.2 fold if the prior
pregnancy was ectopic after having adjusted for the effects of the covariates given in Table
5.15.
The formal interpretation of the regression parameters for Rosner's beta-binomial model
is as follows:
/l
= odds of developing ectopic pregnancy at pregnancy order t for an exposed (e.g. PID)
versus an unexposed (e.g. no-PID) woman at pregnancy order t conditional on:
(a) values of all other covariates being the same at pregnancy order t.
(b) outcome status at pregnancy 1,2,..., t-I the same (or
#
ectopic pregnancies among the
previous t-I visits must be the same) for exposed and unexposed women.
Thus for binary longitudinal data of the Lund ectopic pregnancy type we have:
logit (Pr (ectopic pregnancy at the tth pregnancy order
I ~ , outcome status at
visit 1, ..., t.-
1)]
+
f3 x
(0)
- -
+
r x (t)
--
where a and b are the two parameters from the beta-binomial distribution and S_t is the
number of ectopic pregnancies out of the previous t.-I pregnancies. Thus the interpretation of
regression coefficients is not invariant with respect to cluster size unlike the coefficients of the
marginal model, where the interpretation of fJ is the same regardless of cluster size.
84
Table 5.15
Variable
Results of the Ectopic Pregnancy Data Using Rosner's
Beta-Binomial Modeling Approach
P-value
Parameter
Standard
Estimates
Error
2 sided
PARISE
a
0.89963
0.40438
2.22468
0.2610
b
18.25537
8.41525
2.16932
0.03006
LAPID
0.47757
0.19926
2.39676
0.01654
MYCO
1.45227
0.40534
3.58279
0.00034
SCOREI
1.28516
0.30068
4.27417
0.00002
SCORE2
2.08186
0.30703
6.78053
0.00000
AGEGP
0.82971
0.18338
4.52463
0.00001
NSG
0.79874
0.21230
3.76239
0.00017
SCIMYCO
-1.18991
0.46765
-2.54445
0.01094
SC2MYCO
-1.21646
0.46238
-2.63087
0.00852
85
5.3.3.4 Comparison with Ordinary Logistic Regression
Ami ~ Models
Table 5.16 summarizes the results from ordinary logistic regression, generalized estimation
equations, ud Rosner's beta-binomial as applied to the ectopic pregnucy data set.
Both
levels of statistical signifiC&llce ud estimated odds ratios appeared to vary according to the
estimation procedure.
Fitting the studard logistic regression model to all available
pregnucies without consideration of the correlation between the outcomes of the pregnancies
within a woman produced unbiased estimates of effects but underestimated SE's ud
overstated statistical signifiC&llce of covariates slightly.
Standard errors, obtained from the
GEE approach of Liug ud Zeger, were up to 14% higher thu the unadjusted SE's obtained
from the studard logistic regression model. The estimated regression coefficients and their
SE's that were obtained from Rosner's model were uniformly smaller thu the corresponding
estimates that were obtained from the estimating equations approach with the exception of
the NSG variable.
However, the levels of statistical significance were in generally good
agreement between these two models.
The fact that estimated logistic regression coefficents and their associated SE's from
Rosners model were closer to the null thu the estimates from the GEE approach is
attributable to the underlying difference in the models with regard to the inclusion of
outcome for previous pregnucy as u additional predictor variable. The GEE model which is
a marginal model, excludes the status of previous pregnancies as predictors, whereas Rosner's
model (which is a conditional model) includes the status of previous pregnucies as
covanates. Thus, if the goal of the study is to estimate marginal effects of a covariate (e.g.
effect of PID on ectopic pregnucy irrespective of the status of previous pregnancy outcomes),
then the results of the estimating equation approach are more appropriate.
On the other
hud if the focus is, say, individual patient counseling of a specific womu, then it is useful to
Table 5.16
86
--
Estimated Logistic Regression Coefficient from Alternative
Model Predicting Ectopic Pregnancy in 1497 Women
Variable
Standard
Logistic
Regression
GEE
Rosner's
Model
SCORE1
(SE)
1.4112
(0.3647)
1.4326
(0.365)
1.2852
(0.30058)
SCORE2
(SE)
2.3918
(0.373)
2.4263
(0.3764)
2.08186
(0.30703)
AGEGP
(SE)
0.8910
(0.189)
0.8818
(0.1841)
0.82971
(0.18338)
LAPIDI
(SE)
0.6024
(0.219)
0.6085
(0.2223)
0.47757
(0.19926)
NSGI
(SE)
0.6097
(0.252)
0.3495
(0.2910)
0.79874
(0.21230)
MYCO
(SE)
1.6540
(0.540)
1.6295
(0.5443)
1.45227
(0.40534)
SCIMY
(SE)
-1.3575
(0.604)
-1.3375
(0.6097)
-1.18991
(0.46238)
SC2MY
(SE)
-1.3803
(0.629)
-1.3900
(0.6341)
-121646
(0.46238)
0.089
0.0496
2.2
Intraclass Correlation
Pairv.ise odds ratio
87
condition on the outcome of previous pregnancies, since conditional models such
88
Rosner'.
model provide estimates of risk in the population of women that have the same history
specific woman under consideration for counselling.
88
the
Furthermore, Rosner's approach
estimates the joint distribution of outcomes and thus enables one to assess goodness of fit of
specific models since the model specifies a likelihood (Equation 5-3) assuming the bet&binomial model is correct.
5.3.3.5 Comparison Qf Th Intercepts Qf The Models
Let Pi denote the random variable for the probability that any of the ni pregnancies of
woman i will result in ectopic pregnancy at baseline.
This probability has subscript i,
indicating that it is specific to woman i. Insufficient information precludes estimating Pi for
each woman, but it is possible to estimate the distribution of the Pi across women. Some
women have higher probabilities of ectopic pregnancy and their Pi is higher (closer to 1),
whereas other women have lower probabilities of aborting and their Pi is smaller (closer to
•
0). Assuming Pi has a beta distribution we have,
Var(Pi)
=
P(1-P) (a+t+1)
=
P(1-P) r
Thus the expected value for a woman's probability of ectopic pregnancy is P, and the
variance among these probabilities can be high or low depending on the value of r. Here the
parameter r plays the role
88
the intraclass correlation coefficient.
A comparison of the intercept terms for the standard logistic regression and Rosner'.
model show that they are the same (Table 5.17).
For the standard logistic model there is
only one parameter, BO' which is estimated to be -3.0078. This corresponds to a baseline
probability of ectopic pregnancy of 0.0471.
For Rosner's model, the beta distribution
88
parameters are estimated as a = 0.89963 and b = 18.25537. The baseline probability is thus
estimated as 0.0522.
The intraclass correlation coefficient is r
= a+~+1
0.0496.
The
hypothesis that the pregnancies from the same woman are statistically independent of each
other (B O : r = 0) is tested by comparing the difference in the -2 log (L) between the two
models to a chi-squared distribution with one degree of freedom (Rosner, personal
communication). In this data set, this hypothesis is rejected (log likelihood difference =9.8, p
< 0.001),
thus we conclude that pregnancies from the same woman are positively correlated ,
though this correlation is small. The low correlation among pregnancies from the same
woman coupled with the low average number of pregnancies contributed by each woman (1.9)
are perhaps the reasons why the correlation does not have a large effect on parameter
estimation and testing.
In this work we have not investigated the behavior of the log likelihood ratio test statistic
when the true parameter is near or on the boundary of the hypothesis region. Feder (1968)
has studied the behavior of -2 log A near the boundary of the hypothesis region; additional
work is probably needed.
•
Table 5.17 Comparison of the Intercepts of the Standard Logistic Regression and Rosner's Model
Standard Logistic
Model
---
.----
._._-----------~---._---
Parameter estimate
--
Rosner's Model
_. - - ----
80=- 3.cX>78
a= 0.89963
b= 18.25537
Filled probability of ectopic Pregnancy
when all predictors = 0
_I_
= 0.0471
I +exp(3.0078)
P= il = 0.0522
a+b
r=_I_ = 0.0496
a+b+l
210g L
1254.5
1244.7
co
\0
90
5.3.4
Alternative Approach
An alternative strategy for binary dependent outcomes is Bonney's (1987) regressive
Bonney'. approach expresses the likelihood of a set of binary
logistic regression approach.
dependent outcomes as a product of conditional probabilities each of which is assumed to be
logistic.
Thus, the dependent problem is transformed to independent univariate logistic
regression as follows:
Pr
(Y I ~) =Pr (yl' Y2'· .. , Yn I ~ )
= Pr
=
(Yll~) Pr (Y21 Yl'~)··· Pr (Ynl Y 1, Y2,···, Yn-l'~)
Q
+ "Yl
Z·1
1
+ "Y2
Z·2
1
+ ... +
"Y'1- 1 Z..
1,1- 1 + {3 X.1
•
where,
Z.. = 2Y. - 1
1J
J
o
if j
<i
ifj~i
Then, the likelihood for the n dependent observations is:
n
Pr
(Y I ~) =
rr
exp(O.Y.)
1
1
Where the product is over observed Y's.
Thus, in the case of the Lund ectopic pregnancy data both Rosner's and Bonney's model
structures have some similarities in the sense that both include the status of previous
pregnancies as predictors, though the likelihood functions are different. As mentioned before,
91
the likelihood for Rosner's model is derived as an extension of the beta-binomial model,
whereas, the likelihood for Bonney's model is simply the product of conditional probabilities
each of which is assumed to be logistic. Secondly, Rosner's model assumes that Yl' Y2"'"
y.~ 1 have equal and additive effect on Y.,
whereas Bonney's model can accommodate both
1
equal and unequal effect of YI' Y2,m, Yi- 1 on Vi' Thirdly, Rosner's model describes only
one pattern of dependence, namely the exchangeable correlation structure of the Y's given the
X's.
On the other hand, Bonney's model can accommodate more specialized patterns of
dependence including Markov dependence of order 1. Finally, Rosner's model provides direct
estimates of pairwise odds--ratio whereas with Bonney's model estimates of pairwise oddsratios are complicated to obtain. In the case of clusters of size 2 or 3 formulas are given in
Bonney (1987 formulas 2.8 and 2.9).
CHAPTER 6
ANALYSIS OF MOUTHRINSES IN INHIBITING THE DEVELOPMENT
OF SUPRAGINGIVAL DENTAL PLAQUE
6.1 Descriptive Analysis
In this clinical trial subjects were generally healthy adult male and female volunteers, age
18-55, with pre-existing plaque and gingivitis but without periodontal disease. Prior to entry
subjects were screened for a minimum of 20 sound natural teeth, a minimum plaque mean of
2.0 and a minimum gingivitis mean of 2.0.
Subjects with gross oral pathology or on
antibiotic, antibacterial or anti-inflammatory therapy were excluded from the study.
Subjects were then randomized to two new mouthrinses (A & B) or a standard
mouthrinse with double blinding.
6.1.1
~fouthrinse
Dispensation
Subjects began their assigned regimen of rinsing vigorously with 15 ml of the assigned
rinse for 30 seeonds, twice daily, morning and afternoon, for six months. All rinsings were
under supervision of the representative of the Principal Investigator on week days. Subjects
were also given 16 ounce bottles and graduated plastic cups and were instructed to follow
their assignment at home over weekends and to maintain a diary. During the study, subjects
followed their usual oral hygiene and dietary habits but were instructed to refrain from using
commercial mouthrin.ses and to advise the investigators of the use of antibiotic or antiinflammatory drugs.
6.1.2 Examinations
Subjecta were instructed to refrain from all oral hygiene and use of experimental products
93
on examination days until after the plaque collections and/or examinations, at baseline and
at 3 and 6 months following the initiation of rinses. Separate case report forms were used at
each examination interval.
Gingivitis was scored at baseline, 3 months and 6 months by the Modified Gingival
Index.
Plaque was also scored at baseline, 3 months and 6 months by the Turesky
modification of the Quigley-Hein Index (1970).
A summary of the demographic characteristic of each treatment group is given in Table
6.1.
Randomization appears to have produced treatment groups which are reasonably
comparable with respect to the distribution of demographic characteristics such as smoking
status, age, baseline plaque index and baseline gingivitis index. However, the sex distribution
was different in that experimental monthrinse A had significantly more females than the
standard mouthrinses or B.
§.J.....a.
Descriptive Analysis
In the next sections we will study the effectiveness of the 3 mouthrinses in inhibiting the
development of
dental~.
Table 6.2 provides summary statistics of the Modified
Quigley-Hein Plaque index (PINDEX) at 3 months and 6 months.
From Table 6.2 we see
that for all subjects, the mean plaque index decreased over time and the variance decreased
as mean decreased.
However the coefficient of variation was constant.
The sample
correlation among the observations within individual was relatively high (correlation
0.6275).
=
From the attached plot of time since study enrollment versus mean plaque index
(Figure 6.1), the experimental mouthrinses A and B appear to be more effective in inhibiting
the development of dental plaque. There was a 17.8% reduction in mean plaque from the 3
month to 6 month period for experimental mouthrinse B and only a 4.1% reduction for
experimental mouthrinse A.
Table 6.1
Summary Statistics of Demographic O1aracteristics by Treatment Group
Treatment Group
Covariate
Standard
(n= 39)
A
(n=34)
B
(n=36)
p. value
Sex
Female
Male
15 (38.5)
24 (61.5)
24 (70.6)
10 (29.4)
19 (S2.8)
17 (47.2)
0.023
Smoking Status
Nonsmoker
Smoker
33 (84.6)
6 (15.4)
28 (82.4)
6 (17.6)
31 (86.1)
S (13.9)
0.910
Age
Median
Mean
Range
27
27.5
(23. 38)
26
29.1
(22.48)
2S
27.4
(23.47)
0.3843
Baseline Plaque Index
Median
Mean
Range
2.5
2.56
(2.07.3.3)
2.51
2.57
(2.04, 3.57)
2.43
2.48
(2.02.3.13)
0.S364
Baseline Gingivitis Index
Median
Mean
Range
2.15
2.22
(2.02. 2.81)
2.16
2.19
(1.98. 2.88)
2.13
2.16
(2.01. 2.54)
0.3797
\0
~
..
...
..
•
Table 6.2
Descriptive Statistics for Plaque Index Date
3-months
6-months
Mean
1.464
1.346
Correlation
1
0.6275
0.6275
1
Variance
0.4856
0.2925
0.2925
0.4473
Coefficient of Variation
0.476
0.497
\0
VI
96
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97
Therefore mouthrinse B appears to be both effective (reduces plaque quickly) and is longlasting (continues to reduce plaque over the entire time of the clinical trial) whereas
mouthrinse A seems to be effective but its long-lasting effect, ie., its ability to continue to
reduce plaque after 3 months is not impressive.
Figure 6.2 and 6.3 show the histogram of the plaque measurements at 3 and 6 months
respectively. The histograms were skewed to the right. The Shapiro-WUk test of normality
at the two time points showed w
= 0.969 (p = .101) and w = .964 (p =0.043), which suggest
that these data do not represent samples from normal distributions.
To account for the
skewness in the distribution of plaque measurements at 3 and 6 months, a gamma
distribution was evaluated. Moreover when a gamma distribution was superimposed on the
histogram it appeared to provide an adequate fit to the data. This observation is further
confirmed by the ~? goodness of fit test (Snedecor-Cochran, 1980) which yielded a p-value of
0.4723 for the 3-month measurement and 0.1689 for the 6-month measurement.
The null
hypothesis tested is that plaque observations are randomly drawn from a gamma distribution.
Maximum likelihood parameter estimates were obtained using PROC CAPABILITY in the
SAS-QC System (1989). Gamma quantile-quantile plots are shown in Figures 6.4 and 6.5. In
general the points on the plot fallon or near the line with slope equal to the scale parameter,
although, three or four observations appear to be outliers.
98
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Figure 6.3: Fitted Gamma Distribution for Dental Plaque
MONTH-'
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102
6.2 Model Fittmg
6.2.1 Analysis Using GEE Techniques
Since characteristics of the plaque data appear to be distributed as gamma, a generalized
linear model is fit with gamma error and log link. Log link has the physical interpretation
that the covariates have multiplicative effects.
Thus we used the GEE method to model the marginal dependence of E(Yit) on the
independent variables. Table 6.3 shows estimated regression coefficients and z-statistics for a
gamma error and log link model. A. in the case of the pregnancy data exchangeable and
independent correlation structures are fitted to investigate the effect of specifying different
correlation structures.
From the magnitude of the regression coefficients in Table 6.3 and their associated robust
confidence intervals it is clear that the variables AGE, SEX, and SMOKE are not important.
The Wald chi-squared value for testing the hypothesis that the coefficients for AGE, SEX,
and SMOKE are jointly equal to zero was 2.2405 with 3 degrees of freedom (p
= 0.5240).
Thus we fitted a parsimonious model without these variables. The results of this model are
displayed in Table 6.4.
Table 6.4 also shows the results of a grossly misspedfied
exchangeable correlation of 0.9, rather than its estimated value of 0.4299. Our purpose here
is to show the effect of grossly misspecifying the correlation on the regression estimates and
their robust z-statistics.
103
Table 6.3
GEE Analysis of the Dental Clinical Trial with Log Link and Gamma Variance
Correlation
Structure
Estimate
S.E. (Naive)
Intercept
Independent
Exchangeable
-1.3078
-1.3114
.2718
.3220
.2921
.2915
-4.47
-4.50
Rinse A
Independent
Exchangeable
-0.3547
-0.3582
0.0729
.0863
0.0896
.0892
-3.96
-4.01
Rinse B
Independent
Exchangeable
-0.3672
-0.3677
.0699
.0829
.0700
.0696
-5.25
-5.28
Baseline
Independent
Exchangeable
0.6265
0.6282
0.869
.1029
0.0890
.0886
7.04
7.09
Age
Independent
Exchangeable
0.0074
0.0075
.0057
.0067
.0083
.0083
.89
.90
Sex
Independent
Exchangeable
0.0546
0.0541
.0648
.0769
.0741
.0738
.74
.73
Independent
Exchangeable
0.0129
0.0153
.0817
.0968
.0980
.0973
.13
Independent
Exchangeable
-0.0812
-0.0823
.0571
.0436
.0439
0.0414
-1.85
-1.86
Covariate
Smoke
Time
For Independent Correlation Structure:
Scale parameter 0.1740
Mean square error 0.2916
c 1 1.36. c~ 2.3322
=
=
=
=
For Exchangable Correlation Structure:
Scale parameter .1745
Mean square error 0.2916
Working correlation 0.4148
c 1 1.0355. c~ 1.2748
=
=
=
=
=
S.E. (Robust)
Z-Robust
.16
104
For exchangeable correlation structure, the coefficient estimated for experimental mouthrin.se
B was -0.3641 (95% C.I. (-0.5023,-0.2259».
This means the plaque measurement for
mouthrinse B subjects are expected to be as the plaque measurements of the standard
mouthrinse multiplied by 0.6948 (=exp(-.3641».
Similarly the coefficient estimate for
experimental mouthrinse A was -.3244 (95% C.I.(-0.5043,-0.1445) , i.e. the plaque
measurements for mouthrinse A subjects were .7230 times the plaque measurement for
standard mouthrinse subjects. Furthermore, as time elapsed, plaque level is that of previous
time multiplied by 0.9178
(=exp(-0.0858).
The regression estimates and their robust z-
statistics did not change by much between the three correlation structures. This is consistent
with the theory of Liang and Zeger that z-statistics are robust to the mis-specification of the
correlation structure provided we have the correct link function.
6.2.2 Selection Among the Correlation Structures
Comparison of robust-z and naive-z statistics helps to suggest which
correlation
structures might be incorrect. Recall that naive z-statistics are calculated from the variance
estimate of the regression coefficients by assuming that the 'working' correlation sturcture is
indeed the true one.
Thus we can expect relatively large differences between these two
statistics if the working correlation is far from the true correlation structure.
Table 6.5 shows the robust z-statistic and naive z-statistic and their relative difference for
the three correlation structures. The grossly misspecified correlation structure showed more
marked difference between robust-z and naive-z. The least relative change was observed for
exchangeable correlation structure (a = 0.4299).
For independent correlation structure, naive z-statistics show larger values (in absolute
terms) than robust-z for the time-independent variables.
This indicates that variance is
underestimated for time-independent variables since correlated data are treated as
independent and counted as separate information. Note that ignoring the correlation leads to
incorrectly interpreting the data as weak evidence for the effect of time (z-naive = -1.47).
105
Using the technique of Rotnitzky and Jewell (Section 3.3) the (~1' ~2) values for the
exchangeable, independent, and a grossly misspecified exchangeable correlation structures are
(0.9824, 1.1755), (1.2379, 2.0479) and (1.7423, 7.2325) respectively. These results tend to
indicate the independent and grossly misspecified exchangeable correlations are inappropriate,
and
80
with it
our choice of covariate adjusted correlation is the exchangeable correlation structure
= 0.4299.
6.2.3 The Effect of Different Specification of Links
For gamma error three links are commonly used. These are:
1. Log link:
'1
= log (1')
2. Reciprocal link: '1
=b
3. Identity link:
=
'7
I'
Log link has the interpretation that the systematic part of the model is multiplicative on the
original scale, and hence additive on the log-scale. Identity link has the interpretation that
the systematic part of the model has an additive effect.
As pointed out in Section 3.1,
reciprocal link is the canonical link for gamma distribution in which sufficiency of resulting
statistics and convergence are guaranteed (McCullagh &. Neider). The reciprocal link has the
interpretation as the rate of a process or rate of change which may not be appropdate in this
case.
Table 6.4
Covariate
Intercept
GEE Analysis or the Dental Clinical Trial with Gamma Variance and Log Link
Correlation
Structure
Independent
Exchangeable
«=0.9
Rinse A
Independent
Exchangeable
«=0.9
Rinse B
Independent
Exchangeable
«=0.9
Baseline
Independent
Exchangeable
«=0.9
Time
Independent
Exchangeable
«=0.9
Estimates
S.E. (Naive)
S.E. (Robust)
Z-Naive
Z-Robust
-0.9890
-0.9935
-0.9982
.2315
.2753
.3159
.2581
.2573
.2565
-4.27
-3.61
-3.16
-3.83
-3.86
-3.89
-0.3203
-0.3244
-0.3287
.0709
.0844
.09683
.0922
0.0918
0.0916
-4.52
-3.84
-3.39
-3.48
-3.53
-3.59
-0.3631
-0.3641
-0.3651
.0699
.0834
.0959
.0701
0.0705
.0706
-5.19
-4.37
-3.81
-5.12
-5.17
-5.21
0.6132
0.6156
0.6181
.0878
0.1049
0.1204
.0939
0.0937
.0934
6.99
5.87
5.13
6.53
6.57
6.62
-0.0850
-0.0858
-0.0869
.0577
0.0438
0.0184
0.0438
0.0440
0.0443
-1.47
-1.96
-4.71
-1.94
-1.95
-1.96
For Independent Correlation Structure
Scale parameter = .1778
c1 = 1.2379, Cz = 2.0479
For Exchangeable Correlation Matrix
Scale parameter = .1783
c. = 0.9824
Cz = 1.1755
« = .4299
For grossly misspecified Correlation
Scale parameter = .1788
c. = 1.7423
Cz
=7.2325
« =.9
......
~
Comparison of Robust and Naive Z Statistics (Gamma Error Log Link)
Table 6.5
Exchangeable
Independent
Covariate
Z-Robust
Z-Naive
Intercept
-3.83
-4.27
Rinse A
-3.48
Rinse B
Baseline
Time
Mean
Relative
O1argc
Z-Robust
Z-Naivc
I J%
-3.86
-3.61
-4.52
30%
-3.53
-5.12
-5.19
1%
6.53
6.99
-1.94
-1.47
Grossly MisspeciOed
Relative
O1argc
Relative
O1argc
Z-Robust
Z-Naive
6%
-3.89
-3.16
19%
-3.84
9%
-3.59
-3.39
6%
-5.17
-4.37
15%
-5.21
-3.81
27%
7%
6.57
5.87
11%
6.62
5.13
22%
24%
-1.95
-1.96
1%
-1.96
-4.71
140
14.6
8.4%
42.8
....
o
.....,
108
Table 6.6 shows results of a gamma error and identity link model. For the exchangeable
.
correlation structure the predicted mean plaque measurements for the standard mouthrinse,
experimental mouthrinses A and B are 1.787 (= 0.3674
0.4490
+ .8408 X 2.562),
1.343 ( =-0.3674 -
+ 0.8408 x 2.568), and 1.235 ( =-0.3674 - 0.4817 + 0.8408 X 2.479) respectively.
The
corresponding predicted mean plaque measurements for the gamma error, log link and
exchangeable correlation structure are 1.793 (e(-0.9935
.9935 - 0.3244
+
+
.6156 x 2.567», 1.301 (=exp ( -
.6156 x 2.568», and 1.183 (exp(-9935 - 0.3641 - .6156 x 2.479». The
observed mean plaque measurements were 1.763, 1.288, and 1.144; thus the log-link model
seems to predict slightly better than the identity link. Figures 6-6 and 6-7 show observed and
predicted mean plaque values for each mouth-rinse group by time points. Comparison of the
predicted mean values and observed mean values shows that the two link functions provide a
good fit, with the log-link providing a slightly better fit at 6 months.
Furthermore, the fluctuations in the robust-z values across the different correlation
structures were more marked for the identity link than the log link but not by much.
Inconsistency with respect to robust-z values across the different correlation structures
contradicts the property of robustness against the misspecification of the correlation structure.
This could suggest that the log-link may be the more appropriate link since consistency of the
robust-z with respect to the misspecification of the correlation structure can be preserved only
when the model is correct.
"
•
Table 6.6
-
Estimated Regression Coefficients and Robust-Z Statistics for Gamma Error and Identity Link
Coefficient Estimates
Z-Statistics
Independent
Exchargcablc
Misspccilied
Independent
Exchargeable
Misspecified
Intercept
-0.3691
-0.3674
-0.3319
-0.87
-0.87
-0.70
Rinse A
-0.4385
-0.4490
-0.5061
-3.39
-3.51
-3.95
Rinse B
-0.4735
-0.4817
-0.5464
-4.60
-4.70
-4.64
Baseline
0.8385
0.8408
0.8449
5.18
5.19
4.63
Time
-0.1215
-0.1234
-0.1264
-1.99
-2.01
-2.04
a
0
0.4383
0.9
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•
CHAPTER 7
SENSITMTY AND SPECIFICITY FOR CORRELATED OUTCOMES
In this chapter a general estimating equations approach is uaed to obtain estimates of
sensitivity, specificity, predictive value positive (PVP), and predictive value negative (PVN)
when the data cODSist of conelated binary outcomes.
First order approximations to the
variances of estimated sensitivity, specificity, PVP, and PVN are given. Data from a dental
study are used to motivate and illustrate the methods.
7.1
Introduction
The use of diagnostic testa for the detection and evaluation of various diseases has grown
markedly in recent years. A survey of published material in this field was presented by Miller
et a1. (1977).
The performance of a new diagnostic test for predicting the presence or absence of a
medical condition is customarily evaluated by estimating its sensitivity and specificity with
respect to a traditionally used and accepted test which is regarded as a "gold standard" in
making the diagnosis.
In this context, sensitivity is the probability that the new test
indicates that the condition is present when the gold standard does, also. And specificity is
the probability that the new test indicates that the condition is absent when the gold
standard indicates it is absent, also. The positive and negative predictive values of a screening
test are the probabilities of the presence or absence of the disease given positive and negative
results of screening test, respectively. Fleiss (1981) discusses the problem of estimating
sensitivity and specificity when each observation may be regarded
&8
being independent of all
other observatioDS; Lachenbrucb (1988) discusses the situation where a new test is given to
113
each subject several times; and Hujoel, Moulton and Loesche (1990) discuss the use of the
specialized correlated binary models of Bahadur (1961) and Kupper and Haseman (1978) for
obtaining standard errors of sensitivity and specificity estimates when observations are
correlated.
.
In this Chapter we use the generalized estimating equations (GEE) approach developed
by Liang and Zeger described in Chapter 3 for estimating sensitivity, specificity, PVP and
PVN and their robust standard errors from clustered observations. Research presented in this
paper was motivated by a dental study. The purpoee of this study was to assess the efficacy
of a new test for predicting progression to gingivitis. For each subject in the study, data was
collected at each of 5 different tooth sites in the mouth. At each selected tooth site a small
flat rectangular strip (measuring approximately 0.0625 inch x 0.25 inch) was inserted into the
crevice between the gum and the tooth. These strips were specially designed for the collection
and evaluation of elastase concentration in gingiv81 crevicular fluid of individual teeth. Since
elevated concentrations of elastase are believed to be precursors of gingivitis (i.e. loa of
attachment between the tooth and the gingival tissue), indications of high elastase
•
concentration from the strips provide a new test for predicting progression to gingivitis of a
person.
Progression to gingivitis was measured by the change in attachment loss between an
initial baseline measurement and a second measurement 6 months later by X-ray. A change
in loss of attachment of greater than 0.6 mm was taken to indicate
progression and
represents the" gold standard".
In estimating the measures of test validity and their standard errors in the dental study,
it is reasonable to assume that the observations for progression or not to gingivitis within a
mouth are correlated; due to the
man~er
in which an individual cares for his/her teeth or
genetic factors, one would expect that if some teeth have progressed to gingivitis other teeth
within the same mouth are likely to progress also.
Therefore, statistical methods that
recognize and account for the correlation of observations within an individual are appropriate.
114
Zeger (1988) gives an overview of different statistical methods for analyzing correlated binary
data. Donald and Donner (1990) present results of a simulation study comparing estimators
of the common odds ratio when the data are correlated. In the next sections we provide first
order approximations to the variances of sensitivity, specificity, PVP and PVN.
7.2
Sensitivity, Specificity, PVP and PVN in Clustered Data.
This Section is concerned with estimating sensitivity, specificity, PVP and PVN by
treating both the "gold standard" and diagnostic test results as bivariate random variables.
In terms of the dental example the gold standard corresponcU to the determination of
whether the selected site in a subject's mouth progressed to gingivitis and the new test
corresponds to whether the test strip indicated an elevated elastase concentration in the
gingival crevicular fluid of corresponding teeth. Therefore, sensitivity and specificity may bP.
modeled directly. Let
T ij
=
1 if the "new test" is positive in the ith subject and jth tooth
o otherwise
G ij
=
1 if the "gold standard" is positive in the ith subject and jth tooth
o otherwise
A GEE model with Bernoulli variance function and a logit link is given by:
Prob [T..=l I G..=G]
lJ
=
lJ
exp(P O + PI G)
1 + exp (PO + PI G)
Thus sensitivity of the new test is given by:
Prob (T.. = I I G.. = 1]
lJ
and specificity
lJ
~11
115
= 1 + ~ (PO) = 11'00 •
Prob (Too = 0 I Goo = 0)
IJ
I
..
IJ
Here we further assume that,
1) Pr (Til = 1 I Gil = G) = Prob (Ti2 = 1 I G i2 = G)= .•. =Prob (TiS = 1 IG i5 =G)
for G=O,l
for all values of gl' g2' •.. , g5 and for all
i=l, 2, •.•, n
j
j=1,2, •..,5.
Now let
~
11 =
_ (exp(P O+P 1)
exp(P O+P 1»
- [1 + exp (PO + P )]2 ' [1 + exp (PO + P )]2
1
1
(811'11 811'11»
8P O '8P 1
811'00
~OO = ( 8P
O
8"'00 =
8P )
( - exp (PO) / [exp (1
1
+ pO)]2, 0)
d
an
.
Thus the first order Taylor series approximations to the variances of estimates of sensitivity
...
and specificity that account for clustering are
-
where
VR
.
.
•
~11 V R ~11
T
is the GEE sandwich variance estimator of
PG
given in Section 3.3.
Similarly PVP and PVN can be estimated indirectly using Bayes' rule. First let ¢ denote the
prevalence of gingivitis by the "gold standard" test ; also we assume E(G ij )=¢ for all i, j.
Then,
Prob (Too = 1 I Goo = 1) Prob (Goo = 1)
PVP= Prob [G ij = 1 I Tij = 1]
=
IJ
IJ
IJ
Prob (Too = 1)
lJ
116
11
Similarly,
PVN= Prob [Goo = 0 I Too = 0] =
lJ
lJ
=
Prob (Too = 0 I Goo = 0) Prob (Goo = 0)
lJ
P b{T _ 0)
1J
ro
ij-
11'00 (1- t/l)
11'00 ( 1- 9'» + 1I'01t/l
=
MOO'
where,
11'11 = Prob(T..=IIGoo=I)= 1
exp (PO + PI)
+ exp (P 0 + P1)
?rOO = Prob(T..=OIG..=O)=
I
+ exp1 (PO
I
+ exp (PO
lJ
IJ
lJ
IJ
?rIO = Prob(Too=IIG..=O)=
IJ
IJ
exp(P O)
)
)
.,
117
=
0o(1-tP)lI'OO(1 - 11'00)
+ (l-tP)
=
1I'00[ {1-tP) 11'00 (1 -11'00)
2
°0
I
01 = 11'11 tP
+ 11'10 (l-tP)
00 = 11'00 (l-tP)
+ tP
11'01(1-"'01)]
where
and
+ 11'01 tP
First order approximations to the variances of estimates of PVP and PVN that account for
clustering are:
Var (M n )
Var (MOO)
•
• •
- T
Al VR Al
-
•
•
and
- T
A O VR A O
where again VR is the GEE variance estimator of
PG given in Section 3.3.
Section 7.2 Sensitivity and Specificity Study of the Test Strips
To assess the sensitivity and specificity of the test strips, 30 subjects were recruited for a
prospective study. For each subject five tooth sites were selected and a test strip was applied
to each selection site. After 8 minutes the test strips were removed from the sites and "read"
by the dentist.
Six months later patients returned to the clinic where it was determined
whether the selected sites had progressed to periodontitis. When held under ultraviolet light,
strips fluoresce along the length of the strip. The extent to which each strip fluoresces along
its length depends upon the elastase concentration in the crevicular fluid.
Standardised
118
locations along the length of each strip enabled the dentist to determine the integer score for
the strip that could range between 0 indicating weak elastase concentration and 4 indicating
strong concentration.
Part of the aim of the study was to pick a strip score for use as a standard in practice by
dentists to predict patients' chances of progressing to periodontitis within 6 months with
acceptable sensitivity and specificity.
Table 7.1 lists the estimated sensitivity and specificity
of the strip device accounting for clustering of observations within subjects' mouths of the
prospective study.
Diagnostic performance of the strip was estimated at each of four cut
points.
Table 7.1 GEE study of sensitivity and specificity of the test strip at 8 minutes.
Specificity
Sensitivjty
9!! I!2i!ll L ~ Ql estimate !L ~ Q1i It ~ ~ estimate !L ~ ~
1+
0.89
0.96
1.00
0.12
0.19
0.26
2+
0.61
0.77
0.93
0.58
0.67
0.76
3+
0.16
0.40
0.64
0.72
0.80
0.88
4+
0.04
0.22
0.39
0.91
0.95
0.99
Table 7.1 shows that as strip cutpoint increases, sensitivity decreases from 0.96 to 0.21
and that specificity increases from 0.19 to 0.95. The strip cut point that provides moderate
diagnostic performance corresponds to a cutpoint of 2+.
At this level the estimates of
sensitivity and specificity are 0.77 and 0.68, respectively. Cluster adjusted 95% confidence
119
intervals are moderately broad.
Lower confidence bounds of these intervals suggest a less
than impressive diagnostic performance.
A dentist at the Centers for Disease Control
suggested that current knowledge in oral radiology indicates that measurement error
in
ascertaining extent of bone 1088 is large enough to prohibit accurate measurement in bone
..
height within 1-2 mm even after standardization of radiographic methods.
Further, the
method permits one to observe the proximal sites of teeth, only. In this regard, the CDC
dentist suggests that the results we have obtained portray a somewhat "optimistic"
performance of the sensitivity and specificity of the new test. Indeed, upon close scrutiny of
the radiologic data we found some bone height measurements taken at 6 months to be greater
than the baseline measurements indicating measurement error.
The estimated "working correlation" matrix is
1.00 0.10 0.10 0.10 0.10
0.10 1.00 0.10 0.10 0.10
0.10 0.10 1.00 0.10 0.10
0.10 0.10 0.10 1.00 0.10
0.10 0.10 0.10 0.10 1.00
indicating that the correlation between clustered observations within a mouth may be quite
small. In this case we may regard observations within subjects as being nearly independent
and our effective sample size as being more like 30 x 5
= 150 independent observations than 5
correlated observations taken from 30 subjects independently.
A. a last and interesting point, the CDC dentist suggests that the low observed
correlation may be more a function of untoward variation resulting from poor definition of
periodontal disease in its appropriate quantification and warns that our result pertaining to
the test may not be reproducible using more appropriate data.
120
7.4 SAMPLE SURVEY APPROACH
An alternative strategy for obtaining estimates of sensitivity and specificity is to use a
sample survey approach. In this approach sensitivity and specificity are defined in terms of
•
ratio estimates as follows:
Let,
Zl=l if both the 'new test' and the 'gold standard' indicate disease presence
o otherwbe.
Z2 = 1 if the"gold standard" indicates disease presence
o otherwbe.
Z3= 1 if both the "new test" and the "gold standard" indicate no disease
o
otherwise.
Z4= 1 if the" gold standard" indicates no disease
o
otherwise.
.
Thus sensitivity of the new test is given by
..
Prob (T=l
I G=I)
Prob(T=I, G=l)
=
Prob(G=I)
and specificity
I G-O) _ Prob(T=O, G=O)
P ro b (T-O
Prob(G=O)
=
E Z3
EZ4
Thus, this approach enables the lack of independence of multiple sites from the same subject
to be taken into account. Ratio estimates of sensitivity and specificity and their associated
standard errors were obtained with the PC-CARP software for survey data analysis (Fuller
WA, Kennedey W, Schnen D et ai, 1986). At the optimal cutpoint of 2+, the estimates of
sensitivity and specificity were 0.80
[95% C.I. (0.63,0.97)] and 0.67 [95% C.I. (0.58,0.76)]
respectively. These estimates and their 8S808iated 95% confidence intervals are similar to the
results obtained in the previous Section.
CHAPTER 8
SUMMARY AND CONCLUSIONS
8.1 Summary
In this dissertation, longitudinal data analysis in which outcome measurements are
repeatedly taken is investigated.
By incorporating the correlation structure among
measurements within individuals, NeIder's generalized linear models approach is extended for
the repeated measurements data using quasi-likelihood methods by Liang and Zeger.
The
Liang and Zeger method is appropriate when one has repeated non-normal data and the
primary interest of research is marginal dependence of response on covariates.
Hence this
modeling approach is useful in analyzing repeated measurements data in the following sense:
1.
'"
A wide class of error structures can be specified. It can be applied to various types of
response variables that have normal, Poisson, binomial, positive exponential, gamma, etc,
distributions.
2.
Any "working" correlation structure can be specified, and yet regression coefficient
estimates are still consistent even when the correlation is misspecified.
3.
It accommodates discrete and/or continuous covariates which may be time-dependent or
time-independent.
4.
It accommodates imbalances in the number of measurements across subjects.
5.
Regression coefficient estimates from generalized estimating equations are consistent and
asymptotically unbiased and normally distributed.
In Chapter 2, some of the procedures for analyzing repeated measures data were reviewed•
•
These included the classical methods of analysis for continuous response variables and those
based on parametric models assuming a multivariate normal error structure. The methods
122
for analyzing repeated measures data when the responae variable is categorical include the
weighted least squares approach (Koch et al, 1977), the Wei and Stram (1988) aproaeh with
time-dependent and/or time independent covariates, and the Stram, Wei, and Ware (1988)
approach for analyzing repeated measures data of an ordinal categorical responae variable.
In Chapter 3, we reviewed generalized linear models, quasi-likelihood, and Liang and
Zeger's extension of generalized linear models and quasi-likelihood to repeated measures data.
In Chapter 4, we provided a descriptive epidemiology o( ectopic pregnancy and discussed the
proposed risk factors for ectopic pregnancy including pelvic int1ammatory disease.
The
hypothesis that pelvic inflammatory diaeaae is a risk factor (or ectopic pregnancy is baaed
upon the conjecture that PID-causing organisms such as Neisseria gonnorrhea, Chlamydia
trachomatis,
uroplasmas and mycoplasmas ascend (rom the lower genital tract to the
fallopian tubes causing tubal infection and eventually tubal occlusion thus preventing the
implantation of a fertilized egg in the uterus.
In Chapter 5, the generalized estimated equations approach of Chapter 3 is implemented
to the ectopic pregnancy data from Lund, Sweden. From the Lund ectopic pregnancy data,
we learned that pelvic inflammatory disease is the strongest predictor of subsequent
•
development of ectopic pregnancy and that there is a dose-response relationship between PIO
severity and ectopic pregnancy. We also learned that the presence of Mycoplasma from lower
or upper genital tract sites at index laparoscopy is also a strong predictor. of ectopic
pregnancy.
Other correlates of ectopic pregnancy include age at pregnancy and history of
gynecological surgery.
We further analyzed the ectopic pregnancy data using Rosner's
conditional modeling approach which is a generalization of the beta binomial models for
correlated data. The GEE modeling approach was contrasted to Rosner's conditional model,
and recommendations for choosing between the two was dependent on the objective of the
study.
•
In Chapter 6, we used the GEE approach to study the efl"ectivenes8 of three
mouthrinses in inhibiting the development of dental plaque. From this study we learned that
both experimental mouthrinses A and B are more effective than the standard mouthrinse.
e.
123
In Chapter 7 we used GEE techniques to obtain estimates of sensitivity and specificity
when the data consist of correlated binary outcomes.
First order Taylor series
approximations to the variances of estimated sensitivity and specificity were given as a
function of the robust variance estimator given in Chapter 3.
8.2
Future Directions
In this section shortcomings of the GEE analysis approach and suggestions for future
research are outlined •
1.
The small-sample properties of the GEE methods have not been studied extensively.
Though regression estimates from GEE are consistent, unbiased and normally distributed
asymptotically, it is important to assess the performance of GEE's for small number of
clusters.
A simulation study of the GEE approach for analyzing correlated gamma data
indicated that parameters were quite biased and had poor coverage probability in small
•
samples (Paik, 1988).
•
2.
The effect of the quantity of, and mechanism for, missing data have not been
investigated in depth. Furthermore, in contrast with parametric normal theory methods and
the weighted least squares approach, which require only that the missing data mechanism be
missing at random,
the GEE methods require the stronger missing completely at random
assumption (Laird, 1988).
That is, whether an observation is missing can not 'depend on
previous outcomes nor covariate values.
3.
Though the Rotnitzky and Jewell approach is an important and first step, it is not a
necessary and sufficient condition for choosing the correlation matrix, R.
Furthermore
estimates of c 1 and c 2 by themselves without some kind of variance estimates may not
provide very useful information. Thus, sampling properties of the estimates ofc1 and c2 are
•
needed to quantify how far these estimates are from one•
4.
Further studies that look at bias and coverage probabilities of our Taylor series variance
estimates for sensitivity and specificity are warranted.
124
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