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ISSN: 2155-6180
Journal of Biometrics & Biostatistics
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Estimation and Sensitivity Analysis of the Survivor Average Causal Effect under the Monotonicity Assumption

Yasutaka Chiba*

Division of Biostatistics, Clinical Research Center, Kinki University School of Medicine, Japan

*Corresponding Author:
Yasutaka Chiba
Division of Biostatistics
Clinical Research Center
Kinki University School of Medicine
377-2, Ohno-higashi, Osakasayama
Osaka 589-8511, Japan
Tel: +81-72-366-0221
Fax: +81-72-368-1193
E-mail: [email protected]

Received date: November 01, 2012; Accepted date: November 03, 2012; Published date: November 09, 2012

Citation:Chiba Y (2012) Estimation and Sensitivity Analysis of the Survivor Average Causal Effect under the Monotonicity Assumption. J Biom Biostat 3:e116. doi:10.4172/2155-6180.1000e116

Copyright: © 2012 Chiba Y. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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In a previous editorial in this journal, we reviewed the identification assumptions of the Survivor Average Causal Effect (SACE) [1]. In this editorial, we discuss the estimation and sensitivity analysis of the SACE under the monotonicity assumption.

In randomized trials in which the outcome requires considerable follow-up, participants may die before the trial is complete. In such cases, the outcome is undefined. This situation is sometimes referred to as “truncated by death” or “censored by death” [2,3]. In these settings, a crude comparison between the survivors of each treatment arm may give misleading results because the randomization is no longer preserved by conditioning on a post-treatment event (survival), and thus the crude comparison is not a comparison of the same population comparing different treatments but is a comparison of different populations [4].

A treatment comparison that makes sense in this setting is the SACE, which is a comparison of the outcome between treated and untreated individuals in the subpopulation that would have survived in either arm [1,5]. Because this subpopulation is essentially an underlying characteristic of the individual, the SACE can circumvent the problem with the crude comparison. However, unfortunately, this subpopulation of interest is not identified. Evaluating the SACE from the observed data is a challenging subject.

We use identical notation to past literature [1]. Suppose that A denotes the binary treatment variable (A=1 for the treatment arm, and A=0 for the control arm), Y denotes an outcome of interest that is measured after some follow-up period, and S denotes an indicator of whether the individual survives (S=1 if alive, and S=0 if dead). For individuals who died (S=0), Y is undefined. For each individual, we can also consider potential outcomes [6] corresponding to what would have happened if an individual had been in an arm other than the one they were in. Let Sa denote the survival status if the individual were in A=a, and let Ya denote the outcome if the individual were in A=a. The variable Ya is defined only if Sa=1. Otherwise, the individual would have died, and Ya would be undefined. Here, we assume no interference [7], i.e., we assume that the outcome and survival status of an individual do not depend on the treatment status of other individuals. We also assume randomization of the treatment such that Ya A and SaA [8], where BC denotes that B is independent from C. These independencies also hold conditional on X denoting a set of baseline covariates that do not affect A but do affect both S and Y.

Using this notation, a crude comparison of the outcome, which compares the means of Y in each treatment arm among those who in fact survived, is formalized on a difference scale as E(Y | A=1, S=1)–E(Y | A=0, S=1), and the SACE is formalized as:

SACE≡E(Y1 | S1=S0=1)–E(Y0 | S1=S0=1).

To identify the SACE, we introduce the following two assumptions [1]:

Assumption 1: S1 ≥ S0 for all individuals.

Assumption 2: Y1S0 | {S1, X}.

Assumption 1 is sometimes referred to as the monotonicity assumption and implies that there is no individual with S1=0 and S0=1; i.e., Pr(S1=0, S0=1)=0. Under this assumption,

E(Y0 | S1=S0=1)

= E(Y0 | S0=1)

= E(Y0 | S0=1, A=0)

=E(Y | A=0, S=1)

and

E(Y | S =S =1)

= E(Y | A = 0, S = 1)

As this formula includes a potential outcome, the SACE cannot be identified under Assumption 1 only. However, by using Assumption 2, the following identification formula of E(Y1 | S1=S0=1) is derived:

E(Y |S =S = 1)

Thus, we can identify the SACE under Assumptions 1 and 2. Note that, as seen from (2) and (3), we can use the following assumption to identify the SACE [9] instead of Assumption 2:

Assumption 2*: Y1A | {S, X}.

This alternative assumption holds if all factors affecting both S and Y are observed and they are conditioned. Therefore, under Assumption 1, the SACE can be identified if all baseline covariates are observed.

Under Assumptions 1 and 2 (or 2*), the SACE can very simply be estimated because the identification formulas (1) and (3) commonly include S=1 in the conditions of expectations and probability. When we limit the analysis set to individuals with S=1 (i.e., survivors), we can apply the estimation methods developed in the context of observational studies to estimate casual effects of an exposure on the outcome by adjusting for measured confounders, where the target population is the unexposed group. These major methods (the model-based standardization approach [10], the inverse probability weighting [IPW] approach [11], and the doubly robust estimation [12]) have been summarized elsewhere [13].

Here, we present a marginal structural model [11] to yield an IPW estimate of the PSE. After limiting the analysis set to individuals with S=1, this analysis can be conducted using a weighted regression model of A on Y with the weights wi=1 for individuals with Ai=0 and wi=Pr(Ai =1 | Xi)/Pr(Ai=0 | Xi) for individuals with Ai=1, where i=1, …, N denotes an individual. The value of Pr(Ai=a | Xi) is often predicted using a regression model; e.g., a logistic regression model. We present an SAS code to implement this analysis in the appendix. For example, we consider the simple hypothetical data shown in Table 1, where only one binary baseline covariate is observed. The data yielded a SACE estimate of 0.30 (95% confidence interval [CI]: 0.25, 0.35), whereas the crude estimate was (480/800)–(300/600)=0.10 (95% CI: 0.05, 0.15).

    A=1 A=0
    X=1 X=0 Total X=1 X=0 Total
S=1 Y=1 300 180 480 250 50 300
  Y=0 20 300 320 200 100 300
S=0   70 130 200 290 110 400
Total   390 610 1000 740 260 1000

Table 1: A hypothetical data.

For the sensitivity analysis when Assumption 2 is violated, we can also apply the methods for unmeasured confounding developed in the context of observational studies [13]. We set a sensitivity parameter as [4,14]

δX=E(Y1 | A=1, S=1, X)–E(Y1 | A=0, S=1, X).

This sensitivity parameter is the average difference in the outcome that would have been observed under a treatment comparing two different subpopulations, in the stratum with X: the first is the subpopulation that would have survived with treatment (A=1, S=1); the second is the subpopulation that would have survived without treatment (A=0, S=1).

By substituting δX into (2), we obtain the following sensitivity analysis formula:

where is a SACE estimator, which is the difference between (1) and (3), under Assumptions 1 and 2. Specifically, under an assumption that the values of δX do not vary between the strata of X, this formula simplifies to

where δ=E(Y1 | A=1, S=1)–E(Y1 | A=0, S=1) (=δX). This sensitivity analysis formula implies that the sensitivity analysis can be easily conducted. The sensitivity parameter δ is set by the investigator according to what is considered plausible. The parameter can be varied over a range of plausible values to examine how conclusions vary according to different parameter values. The confidence interval of the true SACE for a fixed value of δ can be obtained simply by subtracting δ from the upper and lower confidence limits of . Therefore, we can readily display the results of the sensitivity analysis graphically, where the horizontal axis represents the sensitivity parameter, and the vertical axis represents the true SACE.

In some situations, it may be troublesome for investigators to determine the range of δ to examine. In such situations, the large sample bounds [2,15] can be used to determine this range. The bounds for E(Y1 | A=0, S=1)=E(Y1 | S1=S0=1) are derived from the number of individuals with S1=S0=1 in the subgroup with (A, S)=(1, 1). Because individuals in the subgroup with (A, S)=(1, 1) are limited to those with S1=S0=1 and those with S1=1 and S0=0, under Assumption 1, this number is:

where N11 is the number of individuals with (A, S)=(1, 1). For example, in the hypothetical data of Table 1, this number is 800×(600/1000)/(800/1000)=600. Using this number, the large sample bounds for E(Y1 | A=0, S=1) are calculated as:

As E(Y1 | A=1, S=1)=E(Y | A=1, S=1)=480/800, the range of δ becomes –0.20 ≤ δ ≤ 0.13. The result of the sensitivity analysis using (5) is shown in Figure 1, where the lower and upper limits of the SACE are 0.16 (95% CI: 0.12, 0.21) and 0.50 (95% CI: 0.45, 0.55), respectively. When the larger value of the outcome denotes that the individual is healthier, if investigators are sure that individuals who survived even without treatment are likely to be healthier overall than those who would have survived with treatment, the second subpopulation in (4) is larger than the first subpopulation. In this situation, δ ≤ 0, and thus, the lower limit of the SACE is improved to 0.30 (95% CI: 0.25, 0.35).

biometrics-biostatistics-survivor-average

Figure 1: Sensitivity analysis of the survivor average causal effect; the solid line indicates the survivor average causal effect and broken lines indicate 95% confidence intervals.

In this editorial, we have presented simple methods for the estimation and sensitivity analysis of the SACE under the monotonicity assumption (Assumption 1). The methods are essentially identical to the methods developed in the context of observational studies to estimate the casual effects of an exposure on the outcome by adjusting for measured confounders. The methods presented here can be extended in a straightforward manner to other effect measures. Although the monotonicity assumption is a strong assumption, whenever investigators consider that this assumption is plausible, the simple methods presented here can be applied.

Appendix

The following code is organized as follows. First, we used PROC LOGISTIC to fit the logistic model in the estimation of Pr(A=1 | X). Second, we used an SAS data step to calculate the weights for each individual from the estimated Pr(A=1 | X) of the previous logistic model. Third, we used PROC GENMOD to fit the weighted linear regression model that estimates the SACE. We note that the data file “IPW” contains only data of individuals with S=1 in the original data file.

/* Estimation of Predicted Values */

proc logistic data=IPW descending;

model A=X;

output out=PRED p=P;

run;

/* Calculation of the Weights */

data CIE;

set PRED;

if A=0 then WEIGHT=1; else WEIGHT=(1-P)/P;

run;

/* Weighted Analysis */

proc genmod data=CIE;

class i;

model Y=A/dist=normal link=identity;

weight WEIGHT;

repeated sub=i/type=ind;

run;

When the outcome is a binary variable, the SACE on the risk ratio scale can be estimated by replacing “dist=normal” and “link=identity” in the GENMOD procedure with “dist=poisson” and “link=log”. Furthermore, when the “estimate ‘beta’ A 1/exp” is added in this procedure, the risk ratio and the 95% CI are displayed on the output.

Acknowledgements

This work was supported partially by Grant-in-Aid for Scientific Research (No. 23700344) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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