Download Lecture 5 - Categorical and Survival Analyses

Lecture 5 – Categorical Data and
Survival Analyses
OUTLINE
• Definition
• Common CDA
– Descriptive summaries
– Tests of Association
– Modeling
• Extensions
• Other examples in CDA
What is Categorical Data Analysis?
• Statistical analysis of data that are categorical
(cannot be summarized with mean +/- SD)
• Includes dichotomous, ordinal, nominal outcomes
• Examples: Disease prevalence, Discharge location,
Treatment adherence (yes/no)
Examples of Studies with CDA
• MI after CABG
• Diagnostic studies looking Sensitivity,
Specificity of a new test/procedure
• Discharge location after new surgical
intervention.
How to analyze words?
• Order vs. no order
• Breakdown mean +/- SD for two groups
• Do the same: Breakdown Outcome %’s for
two groups
How to analyze words?
• Comparing length of stay after CABG:
– New Trt = 19.2 +/- 2.7
– SOC
= 21.3 +/- 3.3
• Comparing prevalence of MI:
– New Trt = 16%
– SOC
= 24%
• Are these differences statistically significant?
clinically significant?
Choice of End Point
• Some designs have a binary response
variable
– MI after 3 years
– Overall Survival
– Time to CVD
– Time to recurrent MI
• Can Dichotomize as 1 year rate (Yes/No)
What is Categorical Data Analysis?
• paper example
Common CDA
• Descriptive summaries
• Tests for association
• Modeling
Descriptive summaries
Let’s Talk Data…
Descriptive Summaries in CDA
Nominal – Categorical Data Measured in
unordered categories
(summarize with %’s)
Ordinal – Categorical Data Measured in
ordered categories
(summarize with %’s)
Continuous – Quantitative Data Measured
on a continuum
summarize with
many measures
Types of Data
 Nominal – Categorical data measured in unordered
categories
 Race
 Blood Type
 Gender
 Ordinal – Categorical data measured in ordered
categories
Likert
(unlikely, neutral, likely)
 Cancer Stages
 Socio-economic Status (low, medium, high)
 Continuous – Quantitative data measured on a
continuum
 Serum Creatinine
 Height/Weight/BMI
Diastolic Blood Pressure
Tumor measurements
What the data might look like…
Compare Categorical Outcomes between groups
• How to assess if a predictor is associated with a
categorical outcome?
• Intuitive?: Get the %’s of the outcome prevalence
within each predictor group.
• Example: New Trt and MI.
– New Trt response rate = 16%
– SOC response rate = 24%
Contingency Tables
Group
MI
Yes
No
New
a
b
a+b
Old
c
d
c+d
a+c
b+d
n=a+b+c+d
Group
MI
Yes
No
New
12
8
20
Old
4
16
20
16
24
N=40
CDA Summary with Contingency Table
• Research question
Is there a relationship between Group and
Attacked Heart?
• Better to convert the table into percentages
(easier to see)
What the data might look like…
Step 1. Breakdown the frequencies
MI
No MI
New TRT
12
8
Old TRT
4
16
Step 2. Get the different %’s
(Cell %)
Row %
No MI
MI
New TRT
12
(30%)
60%
75%
8
( 20%)
40%
33%
Old TRT
4
(10%)
20%
25%
16
(40%)
80%
67%
20
Total
16
24
40
Col %
Total
20
Row vs. Column %’s: It’s your choice
• Row %’s:
– 40% of New trt patients had MI vs.
Old trt patients had MI
80% of
• Col %’s:
– 75% of No MI were in the New trt group vs.
MI were in New trt group
• P-value for test of association is the same!
33% of
Tests for Association
CDA tests for Association
• Is there a significant association between
Group and MI?
• What is a good way to test for an
association between the two?
Test for significant differences
• The most common tests are the Chi-square test and
Fisher’s Exact test.
• Research question: Is there an association between
treatment group and MI?
• To answer this:
Compare what you would expect if there was no
association to what you observed
Expect if no relationship?
No MI
New TRT
MI
Total
20
Old TRT
20
Total
40
Expect if no relationship?
No MI
MI
20
New TRT
Old TRT
Total
Total
20
16
24
40
Same % with MI by Group
(Cell %)
Row %
No MI
MI
New TRT
8
(20%)
40%
50%
12
( 30%)
60%
50%
Old TRT
8
(20%)
40%
50%
12
(30%)
60%
50%
20
Total
16
24
40
Col %
Total
20
Test for significant differences
• Have exact same response % would favor “no
association”
• There is another general way to calculate what you
“expect”
• Use Row totals, Column totals, Grand total to
calculate “Expected” frequencies
Observed vs. Expected Frequencies
• Observed frequencies = actual counts
• “Expected” frequencies:
= Row total x Column total / Grand total
(why?)
What you actually observed in Study
Actual
No MI
MI
Total
New TRT
12
8
20
Old TRT
4
16
20
Total
16
24
40
“Expected” frequencies
Actual
Expected
No MI
MI
Total
New TRT
12
8
8
12
20
Old TRT
4
8
16
12
20
Total
16
24
40
Chi-square test
• Quantify if the actual frequencies are far enough
away from the Expected (assuming no association)
• We can quantify using the Chi-square test statistic
• We can get the p-value to determine if there is a
significant association.
Chi-square test for association in RxC table
• H0: There is no association between row and columns
TotalROW * TotalCOL
expected ij 
TotalOVERALL
• The classic Pearson’s chi-squared test of independence
2
2

(Observedij  Expectedij ) 2

 12
dist
Expected
• Fori a12x2
(2-1) x (2-1)
ij = 1
j 1 table, df =
• Conservatively, we require expected ≥ 5 for all i, j
Chi-square Test
2
2
2
TS
 
(Observedij  Expectedij ) 2
Expectedij
i 1 j 1


12  82 8  12 2 4  82 16  12 2




8
12
8
12
 6.67
•Associated P-value for this Chi-square value is p=0.0098.
Thus, we conclude group and MI are significantly associated
(given α = 0.05).
“Expected” frequencies
Actual
Expected
No MI
MI
Total
New TRT
12
8
8
12
20
Old TRT
4
8
16
12
20
Total
16
24
40
Fisher’s Exact Test
• Fisher’s Exact test will test similar hypotheses as
the Chi-square test.
• Use Fisher’s Exact test when assumptions of Chisquare test are not satisfied.
• That is, when you have Expected < 5
(basically implying when cell sample size is small).
Confidence Intervals for
%’s
Confidence Interval for %’s
• You conduct your follow-up after CABG study and accrue
40 patients.
• After 3 years 20 out of all 40 patients have had a MI.
• Q1. What is your best guess at the true (population) MI
rate at 3 years?
A. Based on your sample, 20/40 = 50%
Sampling Variability
Inference
MI = 50%
MI at 3 yrs = ?
Sample
Population
Sampling Variability
Inference
MI = 44%
MI at 3 yrs = ?
Sample
Population
Confidence Interval for %’s
• A good way to make inference about what the
range of plausible values of the population % is
to calculate a Confidence Interval (CI).
• Q2. How much precision do you have in terms
of estimating the MI rate at 3 yrs. in the
population based on your sample?
95% Confidence Intervals
• 95% Confidence Interval for Mean:
sd
X  2
n
• 95% Confidence Interval for Proportion
(Standard “Wald” CI):
pˆ 1  pˆ 
pˆ  2 
n
Confidence Interval for %’s
• Q2. How much precision do you have in terms of estimating the
MI rate in the population based on your sample? (Remember, 20
of 40 total had MI)
A. A 95% Wilson CI for population MI rate is (35.2%, 64.8%).
Thus, if we have repeated our study over and over again, each
time drawing a sample of 40 patients, then the true population
MI rate at 3 yrs. would be between 35.2% and 64.8%
approximately 95% of the time.
Confidence Interval for %’s
• What’s interesting is that there are “lucky” and “unlucky”
combinations of p (response rate) and N (sample size)
• That is, for a given sample size:
* for some p you may higher ability to make inference
* for some p you may have less ability!
• Not to scald the Wald, but not all CI’s are created equal
• Paper
Modeling in CDA
Modeling in CDA
• Modeling is done with variations of
Logistic Regression:
•
•
•
•
•
•
Dichotomous
Ordinal (Proportional odds)
Nominal (Generalized logit)
Conditional (Matched-pairs)
Exact (small sample size/rare outcome)
Longitudinal (GEE, GLMM)
• Simple (1 predictor) vs.
Multivariable (>1 predictor/adjusted)
Why use adjusted analysis?
• Do you think patient demographics or clinical
characteristics at baseline would affect MI?
• What if half of the patients are all <30 yrs. old and half
are all >80 yrs. old?
• What are some possible confounders of response? Effect
modifiers?
• These are testable in adjusted analyses.
You may not need adjusted.
• Typically have well-defined specific patient populations
of interest.
• Thus, inclusion/exclusion criteria might have removed
variability from potential confounders
• A well designed, well executed trial usually does not
require intensive and complex analysis.
What is Logistic Regression?
• In a nutshell:
A statistical method used to model dichotomous
or binary outcomes (but not limited to) using
predictor variables.
Used when the research method is focused on
whether or not an event occurred, rather than
when it occurred (time course information is not
used).
What is Logistic Regression?
• What is the “Logistic” component?
Instead of modeling the outcome, Y, directly,
the method models the Pr(Y) using the
logistic function.
What is Logistic Regression?
• What is the “Regression” component?
Methods used to quantify association between an
outcome and predictor variables. Could be used to
build predictive models as a function of predictors.
What can we use Logistic Regression for?
• To estimate adjusted prevalence rates, adjusted
for potential confounders (sociodemographic
or clinical characteristics)
• To estimate the effect of a treatment on a
dichotomous outcome, adjusted for other
covariates
• Explore how well characteristics predict a
categorical outcome
Fig 1. Logistic regression curves for the three drug combinations. The dashed reference line represents the probability of DLT of
.33. The estimated MTD can be obtained as the value on the horizontal axis that coincides with a vertical line drawn through the
point where the dashed line intersects the logistic curve. Taken from “Parallel Phase I Studies of Daunorubicin Given With
Cytarabine and Etoposide With or Without the Multidrug Resistance Modulator PSC-833 in Previously Untreated
Patients 60 Years of Age or Older With Acute Myeloid Leukemia: Results of Cancer and Leukemia Group B Study 9420”
Journal of Clinical Oncology, Vol 17, Issue 9 (September), 1999: 283. http://www.jco.org/cgi/content/full/17/9/2831
Logistic Regression quantifies “effects”
using Odds Ratios
• Does not model the outcome directly, which leads to
effect estimates quantified by means (i.e.,
differences in means)
• Estimates of effect are instead quantified by “Odds
Ratios”
Logistic Regression &
Odds Ratio (OR)
• The odds ratio is equally valid for retrospective,
prospective, or cross-sectional sampling designs
• That is, regardless of the design it estimates the
same population parameter
(not true for Relative Risk)
Relationship between
Odds & Probability
Odds  event  =
Probability  event 
1-Probability  event 
Probability  event  
Odds  event 
1+Odds  event 
The Odds Ratio
Definition of Odds Ratio: Ratio of two odds estimates.
Example:
Suppose 16 out of 40 people in the trt group had a
MI and only 5 out of 25 in the placebo group had a MI.

16
Pr  MI | trt group  
 0.40
40

5
Pr  MI | placebo group  
 0.20
25
The Odds Ratio
Example Cont’d:
So, if Pr(MI | trt) = 0.40 and Pr(MI | placebo) = 0.20
Then:

0.40
Odds  MI | trt group  
 0.667
1  0.40

0.20
Odds  MI | placebo group  
 0.25
1  0.20

0.667
 OR  Trt vs. Placebo  
 2.67
0.25
Interpretation of the Odds Ratio
•Example cont’d:

Outcome = MI,
OR Trt vs. Plb   2.67
Then, the odds of a MI in the treatment group
were estimated to be 2.67 times the odds of
having a MI in the placebo group.
Alternatively, the odds of having a MI were
167% higher in the treatment group than in the
placebo group.
Odds Ratio vs. Relative Risk
• An Odds Ratio of 2.67 for trt. vs. placebo
does NOT mean that MI is 2.67 times as
LIKELY to occur.
• It DOES mean that the ODDS of MI are 2.67
times as high for trt. vs. placebo.
Odds Ratio vs. Relative Risk
• The Odds Ratio is NOT mathematically equivalent
to the Relative Risk (Risk Ratio)
• However, for “rare” events, the Odds ratio can
approximate the Relative risk (RR)
 1-P  MI | plb  
OR=RR 
 1-P  MI | trt  


The Logistic Regression Model
Logistic Regression:
 P Y 
ln 
  0  1 X 1   2 X 2 

 1-P  Y  


 K X K
Linear Regression:
Y   0  1 X 1   2 X 2 
 K X K  
The Logistic Regression Model
predictor variables
 P Y 
ln 
  0  1 X 1   2 X 2 

 1-P  Y  


 K X K
dichotomous outcome
 PY  
 is the log(odds) of the outcome.
ln 
 1  PY  
The Logistic Regression Model
 P Y 
ln 
  0  1 X 1   2 X 2 

 1-P  Y  


intercept
 K X K
model coefficients
 PY  
 is the log(odds) of the outcome.
ln 
 1  PY  
The Logistic Regression Model
 P Y 
ln 
  0  1 X 1   2 X 2 

 1-P  Y  


P Y 
exp   0  1 X 1   2 X 2 
1  exp   0  1 X 1   2 X 2 
 K X K
 K X K 
 K X K 
In this latter form, the logistic regression model directly
relates the probability of Y to the predictor variables.
Application of Logistic Regression:
• paper example
Extensions of
Logistic Regression
• Outcomes with more than 2 categories
(polytomous or polychotomous)
• Cumulative logit model – Proportional odds model for
ordinal outcomes (ordered categories)
• Generalized logit model for nominal outcomes or nonproportional odds models (unordered categories)
Extensions of
Logistic Regression
• Ordinal Logistic Regression model:
– Fits a logistic regression model with g-1
intercepts for a g category outcome and one
model coefficient for each predictor
– Models cumulative probability of being in a
“higher” category
Discharge Location as Ordinal
(Died, Assisted, Home)
• There is no law that says you can’t model all categories of
Discharge Location
• Ordinal logistic regression example:
Predictor
OR
P-value
Trt vs. Control
1.24
0.036
M vs. F
0.87
0.163
Ordinal Outcome
(Died, Assisted, Home)
Predictor
OR
P-value
Trt vs. Control
1.24
0.036
M vs. F
0.87
0.163
OR(Trt vs C) = 1.24 means there was 24% higher odds of
being in a higher DL category for Treatment vs. Control
(adjusting for gender).
OR(M vs F) = 0.87 means there was 13% lower odds of
being in a higher DL category for Males vs. Females
(adjusting for Trt group).
Extensions of
Logistic Regression
• Nominal Logistic Regression Model:
– Fits a logistic regression model with g-1 intercepts
and g-1 model coefficients for a g category
outcome
– Model captures the multinomial probability of
being in a particular category using generalized
logits
Nominal Logistic Regression
• Doesn’t make “Proportional odds” assumption
• Separate OR’s for C-1 categories of C category outcome
(get OR for every group except Referent)
• Example:
Predictor
Trt vs. Control
Home vs. Died
Assisted vs. Died
OR
P-value
1.22
1.56
Overall=0.048
0.236
0.034
Nominal Logistic Regression
Predictor
Trt vs. Control
Home vs. Died
Assisted vs. Died
OR
P-value
1.22
1.56
Overall=0.048
0.236
0.034
• Thus, there was 56% higher odds of being discharged to
Assisted Living compared to Dying for Trt. vs. Control.
Extensions of
Logistic Regression
• Longitudinal data / repeated measures data /
Clustered data with binary outcomes
• Multilevel models (nested data structures)
GEE (Generalized Estimating Equations)
GLMM (Generalized Linear Mixed Models)
Repeated Measures /
Longitudinal data
• Longitudinal data = data on subjects over time
• Repeated measures need to be taken into account
when testing for differences
• Need to investigate correlation of repeated measures
Extensions to
Logistic Regression
• Exact Logistic Regression
• Small Sample Size
• Adequate sample size but rare event
(sparse data)
Questions?
Part II. Analysis of Time-to-event
Data
(A.K.A., Survival Analysis)
What do we mean by Time?
•
•
Length of follow-up till the event of interest
occurs
Follow-up can start at (for example)
1. Randomization into a clinical trial
2. Time of employment
3. First contact on record in retrospective cohort
•
Age of the individual at the time of the event
What is Survival Analysis?
• Survival analysis is a collection of statistical
analysis techniques where the outcome is
time to an event.
• Survival or time-to-event outcomes are
defined by the pair of random variables (ti, δi)
that give the observation time and an
indicator of whether or not the event
occurred
What do we mean by Event?
• Usually we mean death – thus the name
“survival” analysis
• Other examples:
– Cancer relapse or recurrence
– Disease incidence
• Can also be a positive outcome:
– Discharge from psychiatric counseling
– Normalization of WBC count
(in these examples, death would be a censored
outcome)
Censoring
• In the pair of random variables (ti, δi) that
constitute survival outcomes:
– ti is an observed variable representing time (e.g.,
actual time until death or time until last follow-up)
– δi is a Bernoulli random variable (0,1) or indicator
of whether the observation is censored or not – 1
if we observed a failure, 0 if we have a censored
observation
Censoring Occurs
• When we have incomplete information about the
exact survival time due to a random factor
– Non-informative censoring – whether an observation is
censored or not is independent of the value of the
observation.
– Informative censoring – whether an observation is
censored or not is dependent on the value of the
observation
– E.g., we use dates seen in clinic provide censoring times
(without attempted phone contact to verify vital status)
• We will require non-informative censoring
mechanisms. If censoring is informative, then these
methods will generate biased results.
Types of censoring
• Right censoring – true survival time is greater
than what we observed
• Left censoring – true survival time is less than
what we observed (less common)
• Interval censoring – subjects are not observed
continuously and we only know the event
happened between time A and time B (e.g.,
annual testing of partner of an HIV+
individual)
Three common reasons for
right censoring
• Person does not experience the event before
the study ends
• Person is lost to follow-up during the study
period
• Person withdraws from the study because of
death (if death is not the outcome of interest)
or some other reason (e.g., adverse drug
reaction)
What do the data look like?
end of study
1
2
drop out
3
4
event occurred
5
0
5
10
15
20
Example: Survival of Patients With
Renal Resistive Index ≥ 0.8
• 86 hypertensive patients with open or
percutaneous repair of RVD
• RA resistive index (RI) defined as 1-EDV/PSV in
the D-segment
• RI dichotomized: <0.8 or ≥0.8
• Outcome of interest is time to death (any
cause)
Example: Survival of Patients With
Renal Resistive Index ≥ 0.8
Preop RI
Hosp
UNITNO
006-52-90
<08 or ≥ 0.8
1
013-39-46
1
77.1
0
014-57-80
0
112.3
0
022-39-81
0
90.3
0
023-80-78
0
88.3
0
026-66-88
0
104.3
0
028-75-46
0
57.8
0
030-10-56
0
26.8
0
033-83-68
0
38.1
1
034-30-42
1
90.9
0
RI=1 if ≥ 0.8
Time to Death
Event Indicator
41.0
1
δi = 1 if death
δi = 0 if censored
RI=0 if < 0.8
Survival Distribution
• Distribution of times to event – called “survival
times,” even when the “event” is not “death”
• Let T = survival time (T ≥ 0)
t = specified value for T
• Survival times follow a continuous distribution
with times ranging from zero to infinity
• Ordinary methods for estimating and comparing
continuous distributions cannot be used with
survival data due to the presence of censoring
Probability Density Function f (t )
1
f (t )  lim P[t  T  t  t ]
t 0 t
Difficult to estimate density directly because of
censoring – histogram not direct estimate of f(t)
Cumulative Distribution Function F(t )
t
F (t )  P[T  t ]   f ( s )ds
0
Defined in the same way we would any CDF
Survival Function S(t )

t
t
0
S (t )  Pr[T  t ]   f (u )du  1   f (u )du  1  F (t )
• Monotone non-increasing function
• S(0) = 1
• S(+∞) = 0
Hazard Function λ(t)
Prob event in (t, t  t ) given survived to t
 (t )  lim
t 0
t
Pr(t  T  t  t | T  t )
 lim
t 0
t
Instantaneous death rate at time t, given alive at time t
Hazard Function λ(t )
• So, you survived to time t, what is the probability
that you survive another increment of time t?
• Standardize this conditional probability to a per unit
of time.
• As unit of time gets very small (i.e., goes to 0) this
conditional probability becomes an instantaneous
rate.
• Some simple features of λ(t)
– λ(t) takes on values in the interval (0, ∞)
– λ(t) could be instantaneously increasing, decreasing, or
constant
Survival Distribution
• Any one of these four functions is enough to
specify the survival distribution. There exists
an equivalence relationship between the
them.
• Survival analysis techniques focus on survival
distribution S(t) and hazard rate λ(t)
– When λ(t) is high, S(t) decreases faster.
– When λ(t) is low, S(t) decreases slower.
How do censored cases
affect survival estimation?
•
Censored patients do not make the survival
curve drop in steps
•
Censored cases do reduce the number of
patients left who are contributing to the
survival curve
•
Thus every event after that censored case
will result in a “larger” step down than it
would have been without the censored case
How do censored cases
affect the survival estimation?
•
The reduction in sample size due to amount
of censored cases present will result in
reduced reliability of the estimates of
survival.
•
That is, larger amount of censored cases
make wider CI’s about survival estimates.
•
End of the survival curve is most affected
yet is of great interest
Survival Estimation: The KaplanMeier (K-M) Method
• Also known as “Product-limit” Method
• Most popular method of estimating
survival / time-to-event
• Good statistical properties: estimates
converge to true survival distribution
as sample size grows
• Nonparametric - does not require
knowledge of the underlying
distribution
K-M Estimation: How it Works
• Order death/censoring times from smallest to
largest
• Update survival estimate at each distinct
failure time
j
Sˆ (t( j ) )   Pˆ [T  t( i ) | T  t( i ) ]
i 1
 Sˆ (t( j 1) )  Pˆ (T  t( j ) | T  t( j ) )
K-M Estimation: RI Example
Product-Limit Survival Estimates
Time
Censored
Survival Number Number
(months)
Survival
SE
Failed
Left
(*)
0.000
1.0000
0
0
27
2.628
*
.
.
0
26
7.392
0.9615 0.0377
1
25
13.470
0.9231 0.0523
2
24
14.587
*
.
.
2
23
16.164
0.8829 0.0636
3
22
16.296
0.8428 0.0722
4
21
ˆ = (prop. alive after this death)  (surv. estimate at prior death)
S(t)
21
=
 0.8829
22
Plot of K-M Survival Estimate
Group with RI≥0.8
1.0
Proportion Alive
0.9
0.8
0.7
0.6
0.5
0.4
0
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
Months Post-surgery
Survival estimate updated
at each death time
Open diamonds mark
censoring times
Assumptions of Kaplan-Meier
•
Non-informative Censoring:
The probability of being censored does
not depend upon a patient’s prognosis
for the event.
•
Deaths of patients in a sample occur
independently of each other
•
Does not make assumptions about the
distribution of survival times
Before Kaplan-Meier
•
•
•
•
•
Life-table (“actuarial”) method of
estimating time to death
Break follow-up time into pre-defined
intervals
Number of subjects alive at beginning of
interval
Number of subjects dying during interval
Estimate survival in similar fashion to
Kaplan-Meier
Testing for difference between two
survival curves: Log-rank test
• Are two survivor curves the same?
• Use the times of events: t1, t2, ... (do not include censoring
times)
• Treat each event and its “set of persons still at risk” (i.e., risk
set) at each time tj as an independent table
• Make a 2×2 table at each tj (i.e., each distinct death time)
Event
No Event
Total
Group A
aj
njA- aj
njA
Group B
cj
njB-cj
njB
Total
dj
nj-dj
nj
Log-rank test for comparing survivor
curves
• At each event time t j, under assumption of equal
survival (i.e., SA(t) = SB(t) ), the expected number of
events in Group A out of the total events (dj=aj +cj) is
proportional to the numbers at risk in group A to the
total at risk at time tj:
E(aj)= dj x njA / nj
• Differences between aj and E(aj) represent evidence
against the null hypothesis of equal survival in the
two groups
Log-rank test for comparing survivor
curves
• Use the Cochran Mantel-Haenszel idea of pooling
over events j to get the log-rank chi-squared statistic
with one degree of freedom
2


 a j  E (a j ) 
j
2

 ~ 2
 
1
ˆ
V
a
r
a

j
j
Log-rank test for comparing survivor
curves
• Idea summary:
– Create a 2x2 table at each uncensored failure time
– The construct of each 2x2 table is based on the
corresponding risk set
– Combine information from all the tables
• The null hypothesis is SA(t) = SB(t) for all times t
(i.e., tests for differences across entire
distribution)
Resistive Index example
1.0
0.9
0.8
RI < 0.8
Proportion Alive
0.7
0.6
0.5
RI ≥ 0.8
0.4
0.3
Log-rank Χ2= 15.2
p-value<0.0001
0.2
0.1
0.0
0
(N=84)
(N=76)
(N=63)
(N=48)
(N=37)
(N=30)
(N=15)
(N=8)
(N=3)
12
24
36
48
60
72
84
96
108
Months Post-surgery
120
Other Tests to Compare Survival
Curves You May Encounter
• Wilcoxon (a.k.a., Peto) Test
– Weights analysis by the number of subjects at risk
at each distinct death time
– More sensitive than log-rank to early differences
in survival curves; log-rank is more sensitive to
late differences in curves
• Likelihood Ratio Test
– Assumes exponential distribution
– Optimal if survival is, in fact, exponentially
distributed
From Stratification to Modeling
• Goal: extend survival analysis to an approach
that allows for multiple covariates of mixed
forms (i.e., continuous, ordinal and nominal
categorical)
• We have two options for our expansion
– Model the survival function or time
– Model the hazard function (between 0 to ∞)
We will model the hazard function
What are Proportional Hazards?
 (t | x1 )
C
 C  S (t | x1 )  S (t | x 2 )
 (t | x 2 )
• The constant C does not depend on time
• The model is multiplicative
Cox Proportional Hazards Model
• D.R. Cox assumed proportionality was constant
across time and proposed the following model:
p
 (t; X )  o (t ) exp(  i xi )
i 1
where λ0(t) is the baseline hazard and involves t but
not X
p
• exp(   i xi ) is the exponential function; involves X’s
i 1
but not t (as long as the are time independent)
Cox Proportional Hazards Model
• The regression model for the hazard function as a
function of p explanatory (X) variables is specified as
follows:
log hazard:
log λ(t; X) = log h0(t) + 1X1 + 2X2 + … + pXp
hazard:
λ(t; X)  λo (t)e
β1X1
e ...e 
β2X2
βpXp
Cox Proportional Hazards Model
β1
• Interpretation of e
– The relative hazard (i.e., hazard ratio) associated
with a 1 unit change in X1 (i.e., X1+1 vs. X1),
holding other Xs constant, independent of time
– The relative (instantaneous) risk for X1+1 vs. X1,
holding other Xs constant, independent of time
• Other ’s have similar interpretations
Cox Proportional Hazards Model
•
e
1
“multiplies” the baseline hazard λ0(t) by the
same amount regardless of the time t, thus we have
a “proportional hazards” model – the effect of any
(fixed) X is the same at any time during follow-up
•  is the focus whereas λ 0(t) is a nuisance variable
• Cox (1972) showed how to estimate  without
having to assume a model for λ 0(t) (e.g., estimating
λ 0(t) with a step function)
• Let # steps get large —partial likelihood for 
depends on , not λ0(t)
Partial likelihood
• The likelihood function used in Cox PH models
is called a partial likelihood
• We use only the part of the likelihood function
that contains the ’s
• It depends only on the ranks of the data and
not the actual time values.
Partial likelihood
• Let the survival times (times to failure) be:
t1 < t2 < ... < tk
• And let the “risk sets” corresponding to these times be:
R1, R2, ..., Rk where Rj = list of persons at risk just before tj
• The “partial likelihood” for  is
 1 X1i   2 X 2 i ...  p X pi
 e
L(  )   
1 X 1 j   2 X 2 j ...  p X pj
i 1
 e
 jRi
k





(Assumes no ties in event times)
• To estimate , find the values of s that maximize L()
above.
Partial likelihood
• Why does the partial likelihood make sense?
e
1X 1i   2 X 2 i ...   p X pi
e
jRi
1X 1 j   2 X 2 j ...   p X pj
1X 1i   2 X 2 i ...   p X pi
0 (ti )e

X

(
t
)
e
 0 i
1 1 j   2 X 2 j ...   p X pj
jRi
hazard of failedperson

hazards of ones who could have failedat ti
• Choose  so that the one who failed at each time
was most likely - relative to others who might have
failed!
Some General Comments Thoughts
• Similar to logistic regression, a simple function of
the β has a particularly nice interpretation
•
ˆ
e can be interpreted as a relative risk (risk ratio)
for a one unit change in the predictor
ˆ  0.60  e0.60  0.55 (protective effect)
0.60
ˆ
  0.60  e  1.82 (increased risk)
Some General Comments Thoughts
• Estimates of βs are asymptotically normal (i.e., are
normally distributed)
• Two important implications of asymptotic normality
– We can use the likelihood ratio, score, and Wald tests to
make inference about our data
– Wald test: “thing/SE(thing)”
– We can use the usual method to construct a 95%
confidence interval
e
ˆ
ˆ
 1.96 SE (  )
Resistive Index: Univariable PH Regression
Summary of the Number of Event
and Censored Values
Percent
Total Event Censored Censored
86
22
64
74.42
Testing Global Null Hypothesis: BETA=0
ChiTest
Square DF Pr > ChiSq
Likelihood Ratio 12.6328 1
0.0004
Score
15.2341
1
<.0001
Wald
12.6195
1
0.0004
“Score” test is equivalent to log-rank
Analysis of Maximum Likelihood Estimates
95% Hazard
Ratio
Parameter Standard
Hazard Confidence
Parameter
DF Estimate
Error Chi-Square Pr > ChiSq Ratio
Limits
PRERIGEP8 1
1.56144 0.43954
12.6195
0.0004 4.766 2.014 11.279
β̂
e
ˆ
e
ˆ 1.96 SE ( ˆ )
Resistive Index: Multivariable PH Regression
Analysis of Maximum Likelihood Estimates
Parameter
RI >=0.8
History of Coronary Disease
History of PVD
Preop - Postop Num Meds
Failed BP Response
DF
1
1
1
1
1
Parameter
Estimate
1.89910
1.65099
0.78338
-0.46046
1.43046
Standard
Error Chi-Square Pr > ChiSq
0.48164
15.5471
<.0001
0.75558
4.7745
0.0289
0.44869
3.0483
0.0808
0.20628
4.9830
0.0256
0.57694
6.1474
0.0132
Hazard 95% Hazard
Ratio Ratio Confidence
Limits
6.680
2.60 17.17
5.212
1.19 22.92
2.189
0.91
5.27
0.631
0.42
0.94
4.181
1.35 12.95
• Each effect is estimated controlling for other effects in model
• Note: increase in hazard ratio for RI in multivariable model
• Proportional hazards assumption should be verified
 Examine survival plots by covariate values
 Plot –log(-log(S(t)) vs. time, should be parallel if hazards are
proportional
Resistive Index example
1.0
0.9
0.8
RI < 0.8
Proportion Alive
0.7
0.6
0.5
RI ≥ 0.8
0.4
0.3
Log-rank Χ2= 15.2
p-value<0.0001
0.2
0.1
0.0
0
(N=84)
(N=76)
(N=63)
(N=48)
(N=37)
(N=30)
(N=15)
(N=8)
(N=3)
12
24
36
48
60
72
84
96
108
Months Post-surgery
120
Example of proportional hazards violation
Unadjusted all-cause mortality survival curve, by annual hospital volume of
Medicare breast cancer cases: United States, 1994–1996
Remedial Measures for Non-proportionality
• Stratified analysis
– Uses stratification to control for the nonproportional factor
– Removes factor as covariate (i.e., get no effect
estimate)
• Add time dependent covariate
– Time dependent covariates change value over
time
– Can use indicator to fit new effect after change
point (e.g., at 40 months in previous plot)