Download Adjusted survival curves with inverse probability weights

Computer Methods and Programs in Biomedicine (2004) 75, 45—49
Adjusted survival curves with inverse
probability weights
Stephen R. Cole a,*, Miguel A. Hernán b
a
Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health,
615 N Wolfe St E-7014, Baltimore, MD 21205, USA
b
Department of Epidemiology, Harvard School of Public Health, Boston, MA, USA
Received 9 September 2003 ; received in revised form 17 October 2003; accepted 22 October 2003
KEYWORDS
Graphics;
Standardization;
Stratification;
Survival analysis
Summary Kaplan—Meier survival curves and the associated nonparametric log rank
test statistic are methods of choice for unadjusted survival analyses, while the semiparametric Cox proportional hazards regression model is used ubiquitously as a method
for covariate adjustment. The Cox model extends naturally to include covariates, but
there is no generally accepted method to graphically depict adjusted survival curves.
The authors describe a method and provide a simple worked example using inverse
probability weights (IPW) to create adjusted survival curves. When the weights are
non-parametrically estimated, this method is equivalent to direct standardization of
the survival curves to the combined study population.
© 2003 Elsevier Ireland Ltd. All rights reserved.
1. Introduction
In observational studies without random assignment of treatment, the unadjusted Kaplan—Meier
survival (1-probability of death) curves may be
misleading due to confounding. For example, if
patients with better prognostic values were more
likely to be assigned to the new treatment at baseline, then a higher survival curve could be found
in the treated group than in the untreated group,
even if treatment were not efficacious. A common approach to deal with this non-comparability
problem is to display a separate pair of survival
curves for each level of disease severity. Thus,
only treated and untreated subjects with the same
level of prognostic values at baseline would be
*Corresponding author. Tel.: +1-410-955-4342;
fax: +1-410-955-7587.
E-mail address: [email protected] (S.R. Cole).
compared. However, adjustment based on stratification of survival curves becomes unfeasible when
the number of baseline covariates is large or when
any of them are continuous. Note that the analytic issue motivating this problem is confounding;
therefore an adequate graphic that correctly summarizes over subgroups is needed. In contrast, if
the issue at hand were effect measure modification (i.e. statistical interaction), then summarizing
over subgroups may not be appropriate.
Several proposed alternative methods to produce adjusted survival curves suffer from numerous
shortcomings, including constraining the adjusted
survival curves to be proportional, computation
difficulty with numerous or time-varying covariates, and not allowing adjustment for continuous
covariates [1]. Below, we describe a method based
on inverse probability weights (IPW) that overcomes the above-mentioned problems and is easily
implemented with standard software (e.g. SAS).
0169-2607/$ — see front matter © 2003 Elsevier Ireland Ltd. All rights reserved.
doi:10.1016/j.cmpb.2003.10.004
46
2. Method
It is well known that standardization is a general alternative to stratification-based methods
of adjustment for covariates. In the simplest case
with one dichotomous baseline covariate, it can
be shown that direct standardization (to the combined study population) is equivalent to Robin’s
non-parametric g-computation algorithm [2], which
in turn is equivalent to non-parametrically estimated IPW [3]. Specifically, given n subjects with
m exposed, of whom a are diseased and c are
non-diseased, all of whom may be stratified into
j levels of a covariate, such that aj , mj , and nj
are the number of exposed and diseased, number of exposed, and overall number of subjects at
level j of the covariate, respectively.
TheIPW risk
proportion can be written as
j aj w j / j m j w j ,
=
(m/n)/(m
where
w
/n
).
Replace
the wj ,
j
j
j
a
[(m/n)/(m
/n
)]/
m
[(m/n)/(m
/n
)], facj
j
j
j
j
j
j
j
tor m/n out of the
numerator
and
denominator
and
a
[1/(m
cancel, leaving
/n
)]/
m
[1/(m
/n
j
j
j
j
j
j )],
j
j
rearrange as j aj (nj /mj )/ j mj [(nj /m
),
and
furj
ther rearrange the numerator as
j (aj /mj )nj /
m
(n
/m
).
Next,
the
m
’s
in
the
denominaj
j
j
j
j
tor cancel on each j, leaving j (aj /mj )nj / j nj ,
which is exactly a common form of the directly
standardized risk proportion [4].
Informally, analyses using IPWs weight each subject i by Wi , where Wi is equal to the inverse of
the probability of receiving his or her own treatment or exposure X conditional on the observed
covariate vector Z, rather than stratifying on, or
adjusting for, covariates Z by including them in
a regression model. Specifically, Wi = 1/f(X|Z),
where f[·] is by definition the conditional density
function evaluated at the observed covariate values for subject i. While the true weights Wi are
typically unknown, one can estimate these weights
ŵi non-parametrically from the corresponding sample proportions. However, just as non-parametric
standardization is unfeasible in the presence of
multiple or continuous covariates, so to is the
non-parametric calculation of IPW. Therefore, we
may use a semi- or fully-parametric model to estimate these weights. While it is not immediately
clear how to extend classic direct standardization to such a model, Robins and colleagues have
described how to do so for IPW [5].
Briefly, one fits a prior logistic regression of exposure X regressed on the covariate vector Z (the same
regression used to estimate the propensity score
[6]) and obtains estimates of the predicted probabilities Pr(X = x|Z) from the fitted model. These
predicted probabilities are used to calculate ŵi as
P̂r(X = x|Z)−1 . While asymptotically unbiased, in
S.R. Cole, M.A. Hernán
practice wi may be highly variable and notable stabilization may be achieved by replacing the numerator with the marginal probability of receiving the
exposure observed, SWi = f(X)/f(X|Z) [3]. A separate logistic regression model is used to estimate
the numerator of swi .
Using IPW to produce adjusted survival curves is
derivative of using such weights for the control of
confounding in general, as recently described by
Robins and colleagues [5]. A Cox regression model
weighted by the estimated stabilized weights sŵi
accounts for confounding by the covariate vector
Z because in the ‘‘pseudopopulation’’ created by
weighting the covariates Z are unrelated to exposure X. To obtain a variance estimate that is valid
under the null hypothesis and provides conservative confidence interval coverage, we use the robust
variance estimator of Lin and Wei [7]. A short SAS
program in the Appendix illustrates this method.
To adjust for potentially informative censoring,
one could multiply our IPW by time varying inverse probability-of-censoring
weights of the form
cwi (t) = tj=1 Pr[C(t) = 0]/Pr[C(t) = 0|Z], where
the product is over j and C(t) is an indicator of
censoring, and two logistic regression models analogous to the previous IPW are used to estimate the
numerator and denominator.
3. Example
Table 1 provides the data from a study comparing disease-free survival in 76 Ewing’s sarcoma patients, 47 of whom received a novel treatment (S4)
while 29 received one of three (S1—S3) standard
treatments [8]. Panel A of Fig. 1 displays the unadjusted survival curves for treatment S4 versus S1—
S3 combined. The estimated hazard (i.e. instantaneous risk) ratio comparing treatment S4 against
S1—S3 from a Cox model was 0.5 (95% confidence
interval: 0.3, 1.0; P = 0.03). The unadjusted analysis suggests that the S4 treatment was beneficial in
reducing the hazard of Ewing’s sarcoma recurrence
compared to the other treatments combined.
Serum lactic acid dehydrogenase (LDH) is an
enzyme thought to be related to tumor burden.
Indeed, in the present data abnormally high (i.e.
≥200 international units) pre-treatment LDH was a
strong predictor of recurrence (unadjusted hazard
ratio = 7.6; 95% confidence interval: 4.0, 14.5).
Further, high LDH levels were indicative of a lesser
likelihood of assignment to treatment with S4
rather than S1—S3 (odds ratio = 0.2; 95% confidence interval: 0.1, 0.5).
To implement the proposed method, two prior
logistic regression models were fit and sŵi were
Adjusted survival curves with inverse probability weights
47
Table 1 Recurrence of Ewing’s sarcoma by treatment and lactic acid level (N = 76)
Treatment
LDHa
Days to recurrenceb
S4
0
31, 335, 366, 426, 456, 578, 589, 762+, 792, 913+, 914+, 974, 1005, 1035,
1065+, 1096, 1107+, 1219+, 1250+, 1312+, 1403+, 1461+, 1553+, 1645+,
1706+, 1734+, 1826+, 1948+, 1949+, 1979+, 2222+, 2374+, 2435+, 2465+,
2526+
0, 91, 183, 334, 338, 365, 391, 518, 547, 608, 609, 851
1
S1—S3
0
1
153, 945, 1400, 1887, 2557+, 3134+, 3226+, 3348+, 3501+, 3743+
151, 152, 212, 214, 242, 243, 244, 245, 249, 273, 336, 337, 396, 427, 457, 761,
1249, 1310, 2708+
a
Serum lactic acid dehydrogenase, <200 vs. ≥200 international units.
b
Censored subjects indicated by +.
Fig. 1 Panel (A) presents the unadjusted survival curves for 76 Ewing’s sarcoma patients. Panel (B) presents the
inverse probability weighted survival curves.
constructed and added to the data file. Table 2 details the construction of the pseudopopulation for
the 76 patients. When the model for the denominator of the stabilized weights is non-parametric
(such
as in this example), the pseudo-sample size
sŵi equals the observed sample size (n = 76)
because the mean of the stabilized weights is unity
(3). In the pseudopopulation, the indicator of elevated LDH is no longer associated with treatment
(odds ratio = 1.0; robust 95% confidence interval:
0.4, 2.7). The estimated IPW hazard ratio com-
paring treatment S4 against S1—S3 was 1.1 (robust
95% confidence interval: 0.6, 2.1; robust P = 0.78),
indistinguishable from the results attained from
standard covariate adjustment (adjusted hazard
ratio = 1.1; 95% confidence interval: 0.6, 2.1).
However, since the final IPW model is not fit within
levels of the covariates Z, Panel B of Fig. 1 is a
straightforward plot of the survival curves for the
weighted data, which are not forced to be proportional, are computationally simple even with
numerous or time-varying covariates, and allow
Table 2 Stabilized inverse probability weights (sw) and pseudopopulation (N = 76)
LDH
S4
N
Pr(X = x)
Pr(X = x|Z)
sŵ
Pseudo N
1
1
0
0
1
0
1
0
12
19
35
10
0.62
0.38
0.62
0.38
0.39
0.61
0.78
0.22
1.60
0.62
0.80
1.72
19.2
11.8
27.8
17.2
Total
76
76.0
48
S.R. Cole, M.A. Hernán
for adjustment of continuous covariates. Indeed,
the full power of the weighting approach is best
appreciated with numerous time-varying and/or
continuous covariates as demonstrated by Hernán
et al. [9] and Cole et al. [10].
In conclusion, the present method based on IPW
has a direct and intuitively appealing interpretation. Under the assumption of no unmeasured
confounding, one survival curve represents the
experience of the entire sample had none been
exposed, while the other curve represents the experience of the entire sample had everyone been
exposed.
Acknowledgements
We thank Drs. Paul Allison, Sander Greenland, Alvaro Muñoz, James Robins and Patrick Tarwater
for constructive suggestions. Portions of this paper were presented at the 35th annual meeting of
the Society for Epidemiologic Research. Dr. Cole
was partly supported by the National Institute of
Allergy and Infectious Diseases by means of the
data coordinating centers for the Multicenter AIDS
Cohort (U01-AI-35043) and Women’s Interagency
HIV studies (U01-AI-42590). Dr. Hernán was partly
supported by NIH grant K08-AI-49392.
Appendix A. SAS Macro Code to implement method
/* Macro to implement weighted survival curves
** 1.
Must pass (a) data set name, (b) binary exposure variable, (c) covariate list, (d) survival time,
**
and (e) binary event indicator, in that order
** 2.
Two graphs are saved as "C:/temp/crude.png" and "C:/temp/weighted.png"
*/
%macro curves(data,exp,covs,time,event);
*PREPARE DATA;
data a; set &data; id=_n_; label &exp="Exposure" &time=”Time” &event=”Event”;
*CALCULATE WEIGHTS;
proc logistic descending data=&data noprint; model &exp=; output out=o1 prob=tn;
proc logistic descending data=&data noprint; model &exp=&covs; output out=o2 prob=td;
proc sort data=o1; by id; proc sort data=o2; by id;
data o12; merge o1 o2; by id;
data b(drop=_level_ tn td); set o12; by id;
if &exp=0 then do; tn=1-tn; td=1-td; end; tw=1/td; stw=tn/td;
*FIT CRUDE, ADJUSTED, AND WEIGHTED COX MODELS;
proc phreg data=b; model &time*&event(0)=&exp/rl; title1 "Crude";
proc phreg data=b; model &time*&event(0)=&exp &covs/rl; title1 "Adjusted";
proc phreg data=b covs; model &time*&event(0)=&exp/rl; freq stw/notruncate; title1 "Weighted";
*SET GRAPH OPTIONS;
goptions device=png targetdevice=png gsfname=grafout gsfmode=replace xmax=5 ymax=4
xpixels=1500 ypixels=1200;
axis1 label=(angle=90 font="Times" height=9 "Survival") width=3 order=(0.00 0.25 0.50 0.75
1.00) major=(height=2 width=3) minor=none value=(font="Times" height=5) offset=(0,0);
axis2 label=(font="Times" height=9 "Days") width=3 length=130 order=(0 500 1000 1500)
major=(height=2 width=3) minor=none value=(font="Times" height=5) offset=(0,5);
legend1 across=1 position=(top right inside) noframe label=none shape=line(10)
value=(justify=left font="Times" height=6 "Standard Tx" "Novel Tx");
symbol1 c=black l=1 w=9 v=none i=stepjs; symbol2 c=black l=3 w=9 v=none i=stepjs;
*PLOT GRAPHS;
proc phreg data=b noprint; model &time*&event(0)=; strata &exp; baseline out=g1 survival=s;
filename grafout "C:/temp/crude.png";
proc gplot data=g1; plot s*&time=&exp/vaxis=axis1 haxis=axis2 legend=legend1 noframe;
title1 move=(+25,+0) font="Times" height=6 "Unadjusted";
proc phreg data=b noprint; model &time*&event(0)=; strata &exp; freq stw/notruncate; baseline
out=g4 survival=s; title1;
filename grafout "C:/temp/weighted.png";
proc gplot data=g4; plot s*&time=&exp/vaxis=axis1 haxis=axis2 legend=legend1 noframe;
footnote1; title1 move=(+25,+0) font="Times" height=6 "Weighted"; run; quit;
%mend curves;
*CALL MACRO FOR EXAMPLE DATA;
%curves(a,s4,ldh,days,death); run;
Adjusted survival curves with inverse probability weights
References
[1] F.J. Nieto, J. Coresh, Adjusting survival curves for confounders: a review and a new method, Am. J. Epidemiol.
143 (1996) 1059—1068.
[2] J.M. Robins, A new approach to causal inference in mortality studies with a sustained exposure period: application to the healthy worker survivor effect, Math. Model. 7
(1986) 1393—1512.
[3] J.M. Robins, Marginal structural models, in: 1997 Proceedings of the American Statistical Association, Section on
Bayesian Statistical Science, 1998, pp. 1—10.
[4] K.J. Rothman, S. Greenland, Modern Epidemiology, Lippincott-Raven, New York, 1998.
[5] J.M. Robins, M.A. Hernán, B. Brumback, Marginal structural
models and causal inference in epidemiology, Epidemiology
11 (2000) 550—560.
49
[6] P. Rosenbaum, D. Rubin, Reducing bias in observational
studies using subclassification on the propensity score, J.
Am. Stat. Assoc. 79 (1984) 516—524.
[7] D.Y. Lin, L.J. Wei, The robust inference for the proportional hazards model, J. Am. Stat. Assoc. 84 (1989) 1074—
1078.
[8] R.W. Makuch, Adjusted survival curve estimation using covariates, J. Chronic Dis. 35 (1982) 437—443.
[9] M.A. Hernán, B. Brumback, J.M. Robins, Marginal structural
models to estimate the causal effect of zidovudine on the
survival of HIV-positive men, Epidemiology 11 (2000) 561—
570.
[10] S.R. Cole, M.A. Hernán, J.M. Robins, et al., Effect
of highly active antiretroviral therapy on time to acquired immunodeficiency syndrome or death using marginal
structural models, Am. J. Epidemiol. 158 (2003) 687—
694.