Quantifying and Detecting Individual Level `Always Survivor' Causal Effects Under `Truncation by Death' and Censoring Through Time

TL;DR

This paper introduces a method to identify individual causal effects in the 'always survivor' group under death truncation, using only assumptions guaranteed by randomized trial design, demonstrated with prostate cancer data.

Contribution

It provides the first methodology to detect causal effects under death truncation with minimal assumptions, applicable to all outcome types.

Findings

Method successfully identifies individual causal effects in prostate cancer trial.

Applicable to all outcome types with minimal assumptions.

First approach to detect effects under death truncation using only trial design assumptions.

Abstract

The analysis of causal effects when the outcome of interest is possibly truncated by death has a long history in statistics and causal inference. The survivor average causal effect is commonly identified with more assumptions than those guaranteed by the design of a randomized clinical trial or using sensitivity analysis. This paper demonstrates that individual level causal effects in the `always survivor' principal stratum can be identified with no stronger identification assumptions than randomization. We illustrate the practical utility of our methods using data from a clinical trial on patients with prostate cancer. Our methodology is the first and, as of yet, only proposed procedure that enables detecting causal effects in the presence of truncation by death using only the assumptions that are guaranteed by design of the clinical trial. This methodology is applicable to all types of outcomes.

Tables

Table 1. Table 1: Survive and cancer progression at time t 𝑡 t ( Y ​ ( t ) = 1 , S ​ ( t ) = 1 ) formulae-sequence 𝑌 𝑡 1 𝑆 𝑡 1 (Y(t)=1,S(t)=1)

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
$X=1$	6	30	72	86	116	175	144
$X=0$	40	94	146	150	164	147	121

Table 2. Table 2: Did not survive until t , 𝑡 t, ( Y ​ ( t ) = 2 , S ​ ( t ) = 0 ) formulae-sequence 𝑌 𝑡 2 𝑆 𝑡 0 (Y(t)=2,S(t)=0)

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
$X=1$	3	10	14	22	41	88	139
$X=0$	3	8	15	25	47	114	182

Table 3. Table 3: Censored at time t , 𝑡 t, ( Y ​ ( t ) = 3 , S ​ ( t ) = 2 ) formulae-sequence 𝑌 𝑡 3 𝑆 𝑡 2 (Y(t)=3,S(t)=2)

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
$X=1$	9	9	9	9	9	10	11
$X=0$	15	15	15	15	16	17	19

Table 4. Table 4: Results without monotonicity assumptions

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
Estimate	$0.07$	0.12	$0.15$	$0.10$	0.00	-0.37	-0.51
95% CI	$(0.02,0.11)$	$(0.07,0.20)$	$(0.08,0.23)$	$(0.03,0.18)$	$(-0.08,0.08)$	$(-0.44,-0.30)$	$(-0.58,-0.44)$
99% CI	$(0.01,0.12)$	$(0.05,0.22)$	$(0.06,0.25)$	$(0.00,0.20)$	$(-0.11,0.10)$	$(-0.46,-0.28)$	$(-0.59,-0.42)$
p-value	$0.0018$			$0.004$	0.51	1	1

Table 5. Table 5: Results Monotonicity S 1 [ 0 ] ​ ( ω , t ) + S 0 [ 1 ] ​ ( ω , t ) ≤ 1 superscript subscript 𝑆 1 delimited-[] 0 𝜔 𝑡 superscript subscript 𝑆 0 delimited-[] 1 𝜔 𝑡 1 S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1 for all ω ∈ Ω 𝜔 Ω \omega\in\Omega

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
Estimate	$0.07$	$0.16$	$0.19$	$0.17$	$0.12$	$-0.11$	-0.10
95% CI	$(0.03,0.12)$	$(0.10,0.23)$	$(0.12,0.27)$	$(0.10,0.24)$	$(0.04,0.20)$	$(-0.19,-0.03)$	$(-0.18,-0.02)$
99% CI	$(0.02,0.13)$	$(0.08,0.24)$	$(0.10,0.29)$	$(0.07,0.27)$	$(0.02,0.22)$	$(-0.21,-0.01)$	$(-0.20,0.00)$
p-value	0.0003				$0.0053$	$0.9972$	$0.9941$

Table 6. Table 6: Results Monotonicity S 1 [ 2 ] ​ ( ω , t ) + S 0 [ 1 ] ​ ( ω , t ) ≤ 1 superscript subscript 𝑆 1 delimited-[] 2 𝜔 𝑡 superscript subscript 𝑆 0 delimited-[] 1 𝜔 𝑡 1 S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1 for all ω ∈ Ω 𝜔 Ω \omega\in\Omega

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
Estimate	$0.09$	$0.16$	$0.18$	$0.13$	$0.02$	$-0.34$	$-0.48$
95% CI	$(0.05,0.13)$	$(0.10,0.22)$	$(0.11,0.25)$	$(0.05,0.21)$	$(-0.05,0.10)$	$(-0.41,-0.27)$	$(-0.54,0.41)$
99% CI	$(0.04,0.15)$	$(0.08,0.24)$	$(0.08,0.28)$	$(0.03,0.23)$	$(-0.08,0.13)$	$(-0.43,-0.25)$	$(-0.57,-0.39)$
p-value				$0.0003$	0.30	1	1

Table 7. Table 7: Results Monotonicity S 1 [ 2 ] ​ ( ω , t ) + S 0 [ 1 ] ​ ( ω , t ) ≤ 1 superscript subscript 𝑆 1 delimited-[] 2 𝜔 𝑡 superscript subscript 𝑆 0 delimited-[] 1 𝜔 𝑡 1 S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1 and S 1 [ 0 ] ​ ( ω , t ) + S 0 [ 1 ] ​ ( ω , t ) ≤ 1 superscript subscript 𝑆 1 delimited-[] 0 𝜔 𝑡 superscript subscript 𝑆 0 delimited-[] 1 𝜔 𝑡 1 S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1 for all ω ∈ Ω 𝜔 Ω \omega\in\Omega

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
Estimate	$0.10$	$0.19$	$0.22$	0.19	0.14	$-0.08$	-0.07
95% CI	$(0.06,0.14)$	$(0.13,0.25)$	$(0.15,0.29)$	$(0.12,0.27)$	$(0.07,0.22)$	$(-0.16,0.00)$	$(-0.14,0.01)$
99% CI	$(0.05,0.15)$	$(0.11,0.27)$	$(0.13,0.31)$	$(0.10,0.29)$	$(0.05,0.24)$	$(-0.18,0.02)$	$(-0.17,0.03)$
p-value						$0.978$	$0.953$

Table 8. Table 8: Quality of Life at 12 weeks

	$Q>70,S=1,R=1$	$Q\le 70,S=1,R=1$	$S=0$	$R=0$	Total
$X=1$	73	136	13	$23+93=116$	338
$X=0$	89	89	11	$30+117=147$	336
Total	162	225	24	263	674

Table 9. Table 9: Quality of Life at 12 weeks

	$Q>75,S=1,R=1$	$Q\le 75,S=1,R=1$	$S=0$	$R=0$	Total
$X=1$	63	146	13	$23+93=116$	338
$X=0$	71	107	11	$30+117=147$	336
Total	134	253	24	263	674

Table 10. Table 10: Counterfactual Distribution

$\omega$	$Y_1$	$Y_0$	$S_1$	$S_0$	Probability
1	1	1	1	1	$P\left(Y_1=1,Y_0=1,S_1=1,S_0=1\right)$
2	0	1	1	1	$P\left(Y_1=0,Y_0=1,S_1=1,S_0=1\right)$
3	1	0	1	1	$P\left(Y_1=1,Y_0=0,S_1=1,S_0=1\right)$
4	0	0	1	1	$P\left(Y_1=0,Y_0=0,S_1=1,S_0=1\right)$
5	2	1	0	1	$P\left(Y_1=2,Y_0=1,S_1=0,S_0=1\right)$
6	2	0	0	1	$P\left(Y_1=2,Y_0=0,S_1=0,S_0=1\right)$
7	1	2	1	0	$P\left(Y_1=1,Y_0=2,S_1=1,S_0=0\right)$
8	0	2	1	0	$P\left(Y_1=0,Y_0=2,S_1=1,S_0=0\right)$
9	2	2	0	0	$P\left(Y_1=2,Y_0=2,S_1=0,S_0=0\right)$
10	3	3	2	2	$P\left(Y_1=3,Y_0=3,S_1=2,S_0=2\right)$
11	3	1	2	1	$P\left(Y_1=3,Y_0=1,S_1=2,S_0=1\right)$
12	3	0	2	1	$P\left(Y_1=3,Y_0=0,S_1=2,S_0=1\right)$
13	3	2	2	0	$P\left(Y_1=3,Y_0=2,S_1=2,S_0=0\right)$
14	1	3	1	2	$P\left(Y_1=1,Y_0=3,S_1=1,S_0=2\right)$
15	0	3	1	2	$P\left(Y_1=0,Y_0=3,S_1=1,S_0=2\right)$
16	2	3	0	2	$P\left(Y_1=2,Y_0=3,S_1=0,S_0=2\right)$

Table 11. Table 11: Survive and cancer progression at time t 𝑡 t ( Y ​ ( t ) = 1 , S ​ ( t ) = 1 ) formulae-sequence 𝑌 𝑡 1 𝑆 𝑡 1 (Y(t)=1,S(t)=1)

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
$X=1$	6	30	72	86	116	175	144
$X=0$	40	94	146	150	164	147	121

Table 12. Table 12: Survive and cancer does not progress at time t , 𝑡 t, ( Y ​ ( t ) = 0 , S ​ ( t ) = 1 ) formulae-sequence 𝑌 𝑡 0 𝑆 𝑡 1 (Y(t)=0,S(t)=1)

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
$X=1$	320	289	243	221	172	65	17
$X=0$	278	219	160	146	109	58	14

Table 13. Table 13: Did not survive until t , 𝑡 t, ( Y ​ ( t ) = 2 , S ​ ( t ) = 0 ) formulae-sequence 𝑌 𝑡 2 𝑆 𝑡 0 (Y(t)=2,S(t)=0)

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
$X=1$	3	10	14	22	41	88	166
$X=0$	3	8	15	25	47	114	182

Table 14. Table 14: Censored at time t , 𝑡 t, ( Y ​ ( t ) = 3 , S ​ ( t ) = 2 ) formulae-sequence 𝑌 𝑡 3 𝑆 𝑡 2 (Y(t)=3,S(t)=2)

	1 month	2 months	3 months	4 months	6 months	12 months	18 months
$X=1$	9	9	9	9	9	10	11
$X=0$	15	15	15	15	16	17	19

Equations

$$P^{c(t)}(y,1-y,1,1)$$

$$P^{c(t)}(y,1-y,1,1)-[P^{c(t)}(1-y,y,1,1)+P^{c(t)}(3,3,2,2)+P^{c(t)}(3,y,2,1)+P^{c(t)}(1-y,3,1,2)],$$

$$P^{c(t)}(y,1-y,1,1)>P^{c(t)}(1-y,y,1,1)+P^{c(t)}(3,3,2,2)+P^{c(t)}(3,y,2,1)+P^{c(t)}(1-y,3,1,2).$$

$$P^{c(t)}(y,1-y,1,1)-P^{c(t)}(1-y,y,1,1),$$

$$R(\omega ,t)=\{\begin{array}{ll}1& \text{no missingness in Y and S for unit ω at time t}\\ 0& \text{missingness in tuple Y and S for unit ω at time t }\end{array}$$

$$RS(\omega ,t)=\{\begin{array}{ll}1& \text{no missingness in outcome S for unit ω}\text{ at time t}\\ 0& \text{missingness if any element in outcome S for unit ω at time t, }\end{array}$$

$$RY(\omega ,t)=\{\begin{array}{ll}1& \text{no missingness in tuples Y for unit ω at time t}\\ 0& \text{missingness if any element in tuples Y for unit ω at time t }\end{array}$$

$$P\left(Y(t)\in y_a,S(t)=1,R(t)=1\mid X=1\right)+P(Y(t)\notin y_b,S(t)=1,R(t)=1\mid X=0)-1$$

$$P\left(Y_1(t)\in y_a,Y_0(t)\notin y_b,S_1(t)=1,R_1(t)=1,S_0(t)=1,R_0(t)=1\right)$$

$$-\sum _{(r_1,r_0)\in \{0,1{\}}^2}P\left(Y_1(t)\notin y_a,Y_0(t)\in y_b,S_1(t)=1,R_1(t)=r_1,S_0(t)=1,R_0(t)=r_0\right),$$

$$P\left(Y_1(t)\in y_a,Y_0(t)\notin y_b,S_1(t)=1,S_0(t)=1\right)$$

$$-P\left(Y_1(t)\notin y_a,Y_0(t)\in y_b,S_1(t)=1,S_0(t)=1\right).$$

$$\frac{P(Y(t)\in y_a,S(t)=1,R(t)=1\mid X=1)+P(Y(t)\notin y_b,S(t)=1,R(t)=1\mid X=0)-1}{P\left(S(t)=1,R(t)=1\mid X=1\right)-P\left(R(t)=0\mid X=0\right)-P\left(S(t)=0,R(t)=1\mid X=0\right)}$$

$$\frac{\left(\begin{array}[]{c}P\left(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b},S_{1}(t)=1,R_{1}(t)=1,S_{0}(t)=1,R_{0}(t)=1\right)\\ -\sum_{(r_{1},r_{0})\in\{0,1\}^{2}}P\left(Y_{1}(t)\not\in y_{a},Y_{0}(t)\in y_{b},S_{1}(t)=1,R_{1}(t)=r_{1},S_{0}(t)=1,R_{0}(t)=r_{0}\right)\end{array}\right)}{P(S_{1}(t)=1,S_{0}(t)=1,R_{1}(t)=1,R_{0}(t)=1)},$$

$$P\left(Y_1(t)\in y_a,Y_0(t)\notin y_b\mid S_1(t)=1,R_1(t)=1,S_0(t)=1,R_0(t)=1\right)$$

$$-P\left(Y_1(t)\notin y_a,Y_0(t)\in y_b\mid S_1(t)=1,R_1(t)=1,S_0(t)=1,R_0(t)=1\right),$$

$$\frac{P(Y(t)\in y_a,S(t)=1,R(t)=1\mid X=1)+P(Y(t)\notin y_b,S(t)=1,R(t)=1\mid X=0)-1}{P\left(S(t)=1,R(t)=1\mid X=1\right)+P(R(t)=0\mid X=1)-P\left(S(t)=0,R(t)=1\mid X=0\right)}$$

$$\frac{\left(\begin{array}[]{c}P\left(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b},S_{1}(t)=1,S_{0}(t)=1\right)\\ -P\left(Y_{1}(t)\not\in y_{a},Y_{0}(t)\in y_{b},S_{1}(t)=1,S_{0}(t)=1\right)\end{array}\right)}{P(S_{1}(t)=1,S_{0}(t)=1)},$$

$$P(Y_1(t)\in y_a,Y_0(t)\notin y_b\mid S_1(t)=S_0(t)=1)$$

$$-P\left(Y_1(t)\notin y_a,Y_0(t)\in y_b\mid S_1(t)=1,S_0(t)=1\right).$$

$$P(Y(t)\notin y_b,(S(t),R(t))\in \{(1,1)\}\mid X=0)-P(Y(t)\notin y_a,(S(t),R(t))\in \{(1,1)\}\mid X=1)$$

$$P(Y_1(t)\in y_a,Y_0(t)\notin y_b,(S_1(t),R_1(t))\in \{(1,1)\},(S_0(t),R_0(t))\in \{(1,1)\})$$

$$-P(Y_1(t)\notin y_a,Y_0(t)\in y_b,(S_1(t),R_1(t))\in \{(1,1)\},(S_0(t),R_0(t))\in \{(1,1)\})$$

$$-P(Y_1(t)\notin y_a,Y_0(t)\in y_b,(S_1(t),R_1(t))\in \{(1,1)\},(S_0(t),R_0(t))\in \{(1,0)\}).$$

$$\frac{P(Y(t)\notin y_b,(S(t),R(t))\in \{(1,1)\}\mid X=0)-P(Y(t)\notin y_a,(S(t),R(t))\in \{(1,1)\}\mid X=1)}{P(S(t)=1,R(t)=1\mid X=0)}$$

$$\displaystyle\frac{\left(\begin{array}[]{c}P(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b},S_{1}(t)=1,S_{0}(t)=1,R{}_{1}(t)=R_{0}(t)=1)\\ -P(Y_{1}(t)\not\in y_{a},Y_{0}(t)\in y_{b},S_{1}(t)=1,S_{0}(t)=1,R_{1}(t)=R{}_{0}(t)=1)\end{array}\right)}{P(S_{1}(t)=1,S_{0}(t)=1,R{}_{1}(t)=R_{0}(t)=1)},$$

$$\begin{array}[]{c}\begin{array}[]{c}P(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b}\mid R_{1}(t)=R_{0}(t)=1,S_{1}(t)=S_{0}(t)=1)\\ -P(Y(t)\not\in y_{a},Y_{0}(t)\in y_{b}\mid R_{1}(t)=R_{0}(t)=1,S_{1}(t)=S_{0}(t)=1).\end{array}\end{array}$$

$$P(Y(t)\in y_a,S(t)=1,R(t)=1\mid X=1)+P(Y(t)\notin y_b,S(t)=1,R(t)=1\mid X=0)$$

$$+P(Y(t)\in y_a,S(t)=1,R(t)=0\mid X=1)+P(Y(t)\notin y_b,S(t)=1,R(t)=0\mid X=0)$$

$$-1$$

Full text

Quantifying and Detecting Individual Level ‘Always Survivor’ Causal

Effects Under ‘Truncation by Death’ and Censoring Through Time

Jaffer M. Zaidi, Eric J. Tchetgen Tchetgen, Tyler J. VanderWeele

Abstract

The analysis of causal effects when the outcome of interest is possibly truncated by death has a long history in statistics and causal inference. The survivor average causal effect is commonly identified with more assumptions than those guaranteed by the design of a randomized clinical trial or using sensitivity analysis. This paper demonstrates that individual level causal effects in the ‘always survivor’ principal stratum can be identified with no stronger identification assumptions than randomization even in the presence of informative censoring. We illustrate the practical utility of our methods using data from a randomized clinical trial on patients with prostate cancer. Our methodology is the first and, as of yet, only proposed procedure that enables detecting causal effects in the presence of truncation by death using only the assumptions that are guaranteed by design of the clinical trial. This methodology is applicable to all distributions concerning the outcome variable.

1 Introduction

Researchers would often like to evaluate the effect of a treatment or exposure on an outcome that could be truncated by or missing due to death [3, 14, 6]. For individuals who die before the end of study, the outcome of interest remains undefined. Such issues occur both in clinical trials and in observational studies. Many researchers note that a comparison of risk differences of those populations that survive at least until the end of the study do not offer causal conclusions [8, 10, 6]. Studies based upon principal stratification, in which comparisons are made of average causal effects in the ‘always survivor’ group, attempt to provide causal conclusions in the presence of truncation by death.

We refer the interested reader to Wang et al [13], who provide a detailed description of the different approaches with regard to survivor average causal effects. Our approach differs from these previous approaches [2, 13, 3, 14, 11] in that our causal estimand is not the survivor average causal effect estimand used in these previous papers. Instead, we derive a null hypothesis that when falsified informs us that the ‘always survivor’ principal stratum must exist and that there are individuals in this principal stratum of ‘always survivors’ for whom the treatment has an effect of the post-survival outcome of interest and allowing us to construct lower bounds on the size of this subgroup. Our approach embeds testing for these causal effects firmly within the Neyman-Pearson paradigm. The identification assumptions used to detect such effects are exactly those assumptions made to identify the average treatment effect, that is, we make no identification assumption other than randomization of treatment.

2 Notation and assumptions

Suppose a binary treatment $X\in\{0,1\}$ is randomized at baseline. We assume data has been collected over time on survival and on a binary outcome of interest defined only among those who survive. We assume that censoring may take place either for survival or for the binary outcome. The individuals of our study, denoted by symbol $\omega,$ compose a finite population $\Omega.$ At pre-specified times $T=\{1,\ldots,t_{f}\}$ over the course of the study duration $t_{f}$, we define variable $S(\omega,t)\in\{0,1,2\}$ of an individual $\omega$ for each $t\in T.$ Here, $S(\omega,t)=2$ denotes that the individual’s survival response was censored or missing at time $t.$ If the individual’s response is not censored at time $t,$ then $S(\omega,t)=1$ denotes that the individual survived at least until time $t,$ and $S(\omega,t)=0$ denotes that the individual did not survive until time $t.$ A binary outcome of primary interest is also measured at each of the pre-specified times $t\in T,$ though at each $t\in T,$ the outcome might be truncated through death, that is, undefined because the individual died.

If the individual’s response is censored by time $t,$ then let $Y(\omega,t)=3,$ and if the individual was not censored but did not survive until $t\in T,$ then let $Y(\omega,t)=2.$ If the individual was not censored and did not die before time $t$ then let $Y(\omega,t)=1$ if the binary outcome is present and $Y(\omega,t)=0$ if the binary outcome is absent. Consequently the ordinarily binary outcome is transformed to a categorical variable. For this paper, it is important to note that censoring by time $t$, denoted as $Y(\omega,t)=3,$ refers to either drop-out of the individual from the trial or missing data, and that death of an uncensored individual is treated as a different event and is denoted as $Y(\omega,t)=2.$ If the individual $\omega$ survives at least until $t\in T$, then $Y(\omega,t)=0$ denotes that the primary outcome of interest is 0 at follow up period $t$, and $Y(\omega,t)=1$ denotes that the primary outcome of interest is 1 at follow up period $t$.

We use this re-coding of the outcome variables $Y$ and $S$ for notational simplicity. However, we also offer an alternative proof of our results in the Generalizations and Extensions section that uses a different coding of the variables that avoids specifying a new level for the missing responses. The reason we first provide this first coding of the outcome variables is to draw the connection between the statistical units in the randomized experiment and the counterfactual question under investigation. We first use deterministic counterfactuals or potential outcomes in order for readers to understand how an adaptation of Neyman’s seminal contribution to the use of potential outcomes in agricultural field experiments [9, 7] provides a design based understanding of causal effects in the presence of truncation by death and missing outcome data. The framework can be extended to use stochastic counterfactuals for continuous outcomes as developed in the Generalizations and Extensions section.

Define potential outcomes $Y_{x}(\omega,t)$ and $S_{x}(\omega,t)$ to be the value of $Y(\omega,t)$ and $S(\omega,t)$ respectively at follow up time $t$ had we set the value of the treatment $X$ to $x$ for individual $\omega.$ Similarly, should the individual’s counterfactual $S_{x}(\omega,t)=0$ for $t\in T,$ then $Y_{x}(\omega,t)=2,$ and if $S_{x}(\omega,t)=1$ for $t\in T,$ then $Y_{x}(\omega,t)$ is the counterfactual $Y(\omega,t)$ had we forced individual to take treatment $x\in\{0,1\}.$ Also, for some $t\in T,$ let $Y_{x}(\omega,t)=3$ denote that had we forced individual to take treatment $x\in\{0,1\}$ their counterfactual of the primary outcome $Y(\omega,t)$ would be censored or missing to the experimenter in the counterfactual world. Similarly, let $S_{x}(\omega,t)=2$ denote that had we forced individual to take treatment $x\in\{0,1\}$ their counterfactual of the survival status would be censored or missing to the experimenter. This setting of potential outcomes or counterfactuals allowing for the possibility of censoring or missing data at a given time $t$ in the counterfactual world provides us with a theoretical framework to examine causal effects in the presence of censoring and truncation by death.

Consider now the setting of a randomized trial. With treatment randomized at baseline, we can make the ‘weak ignorability’ assumption $(Y_{x}(\omega,t),S_{x}(\omega,t))\amalg X$ for all $t\in T.$ We require the consistency assumption for both $Y_{x}(\omega,t)$ and $S_{x}(\omega,t),$ which means that when $X(\omega)=x,$ then $Y_{x}(\omega,t)=Y(\omega,t)$ and $S_{x}(\omega,t)=S(\omega,t).$ This assumption states that the value of $Y(\omega,t)$ and $S(\omega,t)$ that would be observed if $X$ had been set to what in fact they were observed to be is equal respectively to the value of $Y(\omega,t)$ and $S(\omega,t)$ that was observed. The randomization and consistency assumption are assumed throughout this paper. Additional assumptions, whenever needed, are explicitly detailed in the relevant section, theorem and proposition.

For ease of notation, we drop the $\omega$ in $Y(\omega,t)$ and $S(\omega,t)$ whenever the meaning is clear. Denote for all $t\in T,$ $P^{c(t)}(y_{1},y_{0},s_{1},s_{0})=P(Y_{1}(t)=y_{1},Y_{0}(t)=y_{0},S_{1}(t)=s_{1},S_{0}(t)=s_{0})$, where $y_{1}$ and $y_{0}$ take values in the set $\{0,1,2,3\}$, and $s_{1}$ and $s_{0}$ that take values in the set $\{0,1,2\}.$ Also, for all $t\in T,$ we shall use the notation $P_{y,s.x}^{r(t)}=P(Y(t)=y,S(t)=s\mid X=x)$ where $y\in\{0,1,2,3\},$ $s\in\{0,1,2\},$ and $x\in\{0,1\}.$ The counterfactual probability $P^{c(t)}(y_{1},y_{0},s_{1},s_{0})$ is an important quantity for understanding the magnitude of different causal effects, and is used throughout this paper. The observed probability $P_{y,s.x}^{r(t)}$ is used in empirical conditions throughout this paper. Under the assumption of randomization and consistency of counterfactuals, the empirical conditions that only use the observed probability $P_{y,x.s}^{r(t)}$ imply resulting constraints placed upon $P^{c(t)}(y_{1},y_{0},s_{1},s_{0}),$ which, as we will show, enables investigators to evaluate the magnitude of causal effects for always survivors using only empirical conditions from the observed data.

The superscripts are meant to distinguish these probabilities from one another. The symbol $c(t)$ denotes that this is probability concerns the counterfactual world, and similarly the symbol $r(t)$ denotes that this probability concerns our observed data or real world. All proofs are provided in Online Supplement 1. Let $I(\cdot)$ denote the usual indicator function. Define $Y_{x}^{[y]}(\omega,t)=I(Y_{x}(\omega,t)=y),$ and $S_{x}^{[s]}(\omega,t)=I(S_{x}(\omega,t)=s)$ for $y\in\{0,1,2,3\},$ $s\in\{0,1,2\},$ $\omega\in\Omega$ and $t\in T.$ We say an individual is an ‘always survivor’ at time $t$ if $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ that is the individual survives at least until time $t$ regardless of treatment. We say that an individual displays an individual level ‘always survivor causal effect’ at time $t,$ if for such an individual $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y,$ $S_{1}(\omega,t)=1,$ and $S_{0}(\omega,t)=1$ for $y\in\{0,1\},$ which means that this individual would have survived at least until time $t$ regardless of treatment and there is a causal effect on $Y.$

Joint counterfactuals of the form $Y_{x}(\omega,t)=1$ and $S_{x}(\omega,t)=0$ are never considered, as the individual $\omega\in\Omega$ has not survived until time $t\in T.$ Consequently, any counterfactual probability $P^{c(t)}(y_{1},y_{0},s_{1},s_{0})$ for which $s_{1}=0$ and $y_{1}\in\{0,1\}$ or $s_{0}=0$ and $y_{0}\in\{0,1\}$ is identically zero. Also $P^{c(t)}(y_{1},y_{0},s_{1},s_{0})=0$ whenever $s_{1}=2$ and $y_{1}\neq 3$ or $s_{0}=2$ and $y_{0}\neq 3.$ Similarly, $P_{y,s.x}^{r(t)}$ is equal to zero whenever any of the following conditions hold: (1) $s=2$ and $y\neq 3;$ (2) or $y=3$ and $s\neq 2$ (3) or $s=0$ and $y\neq 2;$ (4) or $y=2$ and $s\neq 0.$ Situations where, at time $t,$ the realized survival value $s$ is not censored or missing but the realized value $y$ is censored or missing are possible in practice. However, in our application below, these situations do not occur in our data analysis by design of the clinical trial. The theoretical framework to accommodate such situations follows the arguments presented here, and Online Supplement 2 fully explicates the framework and results in the setting where there exists individuals $\omega\in\Omega$ for whom $Y(\omega,t)$ is censored but $S(\omega,t)$ is not censored for any specified time $t$.

We also provide a section that generalizes our work to tuple valued $Y(\omega,t)$ that has support on $\mathbf{R}^{n_{y}},$ where $n_{y}$ denotes the number of elements of tuple $Y(\omega,t),$ as well as multivariate $S(\omega,t).$ This generalization also allows for more extensive sensitivity analyses for the role of missing data on outcomes. The simpler setting presented below is provided to make the individual level interpretation transparent.

3 Identification of individual level always survivor causal effects

Here, we only require randomization and consistency for Theorem 1, Proposition 1, and Corollary 1. In the next section, we will examine the derivation of similar results in the presence of monotonicity assumptions.

Theorem 1

Suppose $X$ is randomized at baseline. If for some $t\in T$ and $y\in\{0,1\}$ $P_{y,1.1}^{r(t)}+P_{1-y,1.0}^{r(t)}>1$, then there exists a non-empty subpopulation $\Omega_{s}\subseteq\Omega$ such that for every $\omega\in\Omega_{s},$ $S_{1}(\omega,t)=S_{0}(\omega,t)=1$ and $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y.$

Theorem 1 allows for the empirical detection of individual level total effects within the ‘always survivor’ principal stratum. A difference between our result and previous methods [2, 13, 3, 14, 11] is that Theorem 1 effectively only requires randomization and yet still provides conditions for which the ‘always survivor’ principal stratum exists and has individuals for whom the treatment changes the outcome. To the best of our knowledge, this is the first time an empirical condition to detect causal effects is formulated in the presence of truncation by death and censoring without any additional assumptions other than those guaranteed by design of the clinical trial. Proposition 1 provides further context, comparing the proportion of individuals randomized at baseline that are always survivors and who display a positive (or negative) effect of treatment $X$ on an outcome $Y$ to the proportion of individuals randomized at baseline that are always survivors and who display a negative (or positive) effect of treatment $X$ on outcome $Y.$ Corollary 1 provides further results for this difference for all individuals in the always survivor principal stratum (observed or censored). Proofs of all results are given in Online Supplement 1.

Proposition 1

Suppose $X$ is randomized at baseline. For some $t\in T$ and $y\in\{0,1\},$ the contrast $P_{y,1.1}^{r(t)}+P_{1-y,1.0}^{r(t)}-1$ is equal to

[TABLE]

Corollary 1

Suppose $X$ is randomized at baseline. The expression $P_{1,1.1}^{r(t)}+P_{0,1.0}^{r(t)}-1$ is a lower bound on the risk difference between the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment causes the outcome and the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment prevents the outcome of interest.

In counterfactual notation, the expression $P_{1,1.1}^{r(t)}+P_{0,1.0}^{r(t)}-1$ is a lower bound on $P^{c(t)}(1,0,1,1)-[P^{c(t)}(0,1,1,1)+P^{c(t)}(3,3,2,2)+P^{c(t)}(3,1,2,1)+P^{c(t)}(0,3,1,2)].$ Note $P^{c(t)}(1,0,1,1)$ is a lower bound on the proportion of individuals randomized at baseline that are always survivors for whom the treatment causes the outcome $Y$ as it does not count any censored individual that could be an always survivor for whom the treatment causes the outcome $Y.$ The term in the square brackets is an upper bound on the proportion of individuals randomized at baseline that are always survivors for whom the treatment prevents the outcome $Y,$ because any censored individual that could possibly be an always survivor at time $t$ for whom the treatment prevents the outcome $Y$ is represented in the square brackets. Note, from Theorem 1 and Proposition 1, if $P_{1,1.1}^{r(t)}+P_{0,1.0}^{r(t)}-1>0,$ then we have established that treatment efficacy at time $t$ of population randomized at baseline. Our result establishes that the proportion of individuals randomized at baseline that are always survivors at time $t$ and treatment causes the outcome at time $t$ is necessarily greater by at least $P_{1,1.1}^{r(t)}+P_{0,1.0}^{r(t)}-1$ than the proportion of individuals that are always survivors at time $t$ and the treatment prevents the outcome at time $t.$

It follows from Corollary 1, the expression $\max\{P_{1,1.1}^{r(t)}+P_{0,1.0}^{r(t)}-1,0\}$ is thus a lower bound on the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment causes the outcome: $P^{c(t)}(1,0,1,1)$. Similarly, the expression $P_{0,1.1}^{r(t)}+P_{1,1.0}^{r(t)}-1$ is a lower bound on the risk difference between the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment prevents the outcome and the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment causes the outcome of interest. In counterfactual notation, the expression $P_{0,1.1}^{r(t)}+P_{1,1.0}^{r(t)}-1$ is a lower bound on $P^{c(t)}(0,1,1,1)-[P^{c(t)}(1,0,1,1)+P^{c(t)}(3,3,2,2)+P^{c(t)}(3,0,2,1)+P^{c(t)}(1,3,1,2)]$. The expression $\max\{P_{0,1.1}^{r(t)}+P_{1,1.0}^{r(t)}-1,0\}$ is a lower bound on the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment prevents the outcome: $P^{c(t)}(0,1,1,1).$

Thus, without any assumption on the censoring mechanism, we can formulate lower bounds on the proportion of individuals randomized at baseline that are always survivors (censored or not) at time $t$ and for whom the treatment causes (prevents) the outcome minus the proportion of individuals randomized at baseline that are always survivors (censored or not) at time $t$ and for whom the treatment prevents (causes respectively) the outcome. To the best of our knowledge, no other paper provides empirical conditions to detect causal effects in the presence of censoring without assumptions on the censoring mechanism. Corollary 1 also gives us a contrast only involving real-world probabilities corresponding to the minimum proportion of of individuals randomized at baseline that are always survivors (censored or not) at time $t$ for whom treatment has an effect.

Theorem 1, Proposition 1 and Corollary 1 provide scientists with interpretations, under weaker assumptions, that are not possible in any of the previous approaches [4, 13, 2] to deal with truncation by death. Previous approaches that examine the survivor average causal effect make more assumptions than randomization and consistency [2, 4, 13]. Bounds on the survivor average causal effect and other causal estimands are discussed in the Online Supplement 2 as well as the Generalization and Extensions Section. Other approaches rely on either principal ignorability or a sufficient covariate to essentially choose the set of individuals that are always survivors at time $t,$ and then characterize the average treatment effect for this set of always survivors [2, 13].

Our approach is substantially different in terms of assumptions and interpretation. Previous approaches, including principal ignorability or sufficient covariate methods, make assumptions that are not guaranteed by design, and therefore cannot guarantee the causal interpretation of their results [10, 11, 13, 14, 2, 3]. For example, to identify survivor average causal effects two recent papers propose a substitution variable denoted $A$ that essentially amounts to providing the statistician with the relevant units that form the always survivor principal strata. Some of the assumptions, for example principal ignorability or sufficient covariate assumptions, that are used to get these relevant units are not only assumptions which cannot be guaranteed to hold in any experimental design, but are also unfalsifiable [2, 11] from the observed data as these assumptions involve conditioning on joint potential outcomes. Scientists that use these previous methods [2, 12] will rely on assumptions that are not only unverifiable, or even unfalsifiable, but also offer little in the way of formal scientific or statistical methodology for the statistician to assess whether they are even plausible given that the observed data distribution cannot be used to assess assumptions where joint counterfactuals (e.g. $S_{1}(t)=1$ and $S_{0}(t)=1$) are both in the conditioning statement.

4 Identification of individual level always survivor causal effects

under monotonicity

Monotonicity assumptions are often employed in methods to evaluate principal stratum direct effects. Recall, we denote $Y_{x}^{[y]}(\omega,t)=I(Y_{x}(\omega,t)=y),$ and $S_{x}^{[s]}(\omega,t)=I(S_{x}(\omega,t)=s)$ for $y\in\{0,1,2,3\},$ $s\in\{0,1,2\},$ and $t\in T.$ Here we will consider additional results under two different monotonicity assumptions. In the first monotonicity assumption, we will assume $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega$ and for a pre-specified $t\in T.$ This assumption translates to it being the case that no individual exists who would be alive and not censored at least until time $t$ when forced to take control, but would be dead and not censored by time $t$ when forced to take treatment. This means that we do not observe any individual for whom $S_{1}(\omega,t)=0$ and $S_{0}(\omega,t)=1$ for the specified $t\in T.$ In the absence of censoring, this assumption is simply that for no individual does the treatment itself cause death. If treatment is considered to be non-smoking and the control is smoking, scientific literature might lead us to believe that such an assumption is reasonable to make. Subject matter expertise could provide guidance for which $t\in T,$ $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ If for all $\omega\in\Omega,$ $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ holds for only for a subset of $T_{s}\subset T,$ then Theorem 2A and Proposition 2A, provided below, applies only for those $t\in T_{s}.$ While such monotonicity constraints are never verifiable, they are falsifiable, and again can sometimes be justified with subject matter knowledge.

For the second monotonicity assumption, we shall assume that $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega$ and for a pre-specified $t\in T.$ Heuristically, this assumption means that there is no individual who would drop out of the study by time $t$ when forced to take treatment but who would not drop out of the study and is alive when forced to take control condition (e.g. a prior or older treatment). This means that we do not observe any individual for whom $S_{1}(\omega,t)=2$ and $S_{0}(\omega,t)=1$ for the specified $t\in T.$ Again, such monotonicity constraints are falsifiable and could be judged using subject matter knowledge. For instance, if clinicians believe that the new treatment is less toxic than the old treatment, then this could provide some evidence that patients might be less disposed to drop out if they are forced to take the less toxic new treatment in comparison to the more toxic old treatment.

The monotonicity assumption on censoring is weaker than the non-informative censoring that is commonly used in statistical analysis. Non-informative censoring would have that if an individual $\omega\in\Omega$ is censored under treatment at any fixed time $t\in T,$ denoted $S_{1}(\omega,t)=2$, then the same individual would be censored under the control condition at the same fixed time $t\in T$, denoted $S_{0}(\omega,t)=2.$ In contrast, the monotonicity assumption on censoring, enables statisticians to examine situations where censoring is likely to occur sooner in one of the two arms than in the other.

We will consider analogous results to Theorem 1 and Proposition 1 when one or both of these monotonicity assumptions hold. If such monotonicity assumptions are untenable for the population in question, investigators can still use the empirical conditions provided earlier that do not need any monotonicity assumptions. Causally interpretable sensitivity analyses are also presented in Section 6 when one or both of the monotonicity assumptions do not hold. Such sensitivity analysis provide researchers tools to quantify the consequences of violations of these assumptions, and also directly consider how each individual in their sample could contribute to the causal conclusions.

Theorem 2A

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ For some $t\in T$ and $y\in\{0,1\},$ if the empirical condition $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{3,2.1}^{r(t)}>0$ holds, then there exists a non-empty subpopulation $\Omega_{s}\subseteq\Omega$ such that for every $\omega\in\Omega_{s},$ $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ and $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y.$

Proposition 2A

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ For some $t\in T$ and $y\in\{0,1\},$ the contrast $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{3,2.1}^{r(t)}$ is equal to

[TABLE]

Corollary 2A

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ The expression $P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)}-P_{3,2.1}^{r(t)}$ is a lower bound on the risk difference between the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment causes the outcome and the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment prevents the outcome of interest.

The expression $\max\{P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)}-P_{3,2.1}^{r(t)},0\}$ is thus also a lower bound on the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom treatment causes the outcome. Similarly, the expression $P_{1,1.0}^{r(t)}-P_{1,1.1}^{r(t)}-P_{3,2.1}^{r(t)}$ is a lower bound on the risk difference between the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment prevents the outcome and the proportion of individuals randomized at baseline that always survivors (observed or censored) at time $t$ for whom the treatment causes the outcome of interest. The expression $\max\{P_{1,1.0}^{r(t)}-P_{1,1.1}^{r(t)}-P_{3,2.1}^{r(t)},0\}$ is a lower bound on the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom treatment prevents the outcome.

The interpretations for Theorem 2A and Proposition 2A are similar to those provided for Theorem 1 and Proposition 1. Specifically, if we believe that no individual exists that would die and not be censored by time $t$ when forced to take treatment but would not die and not be censored by time $t$ when forced to take control and if $P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)}-P_{3,2.1}^{r(t)}>0,$ then there exists a set of individuals that would live at least until time $t$ and for whom the treatment causes outcome. From Corollary 2A, we learn that the proportion of individuals randomized at baseline that are always survivors (observed or censored) until time $t$ and for whom the treatment causes (prevents) outcome of interest is at least greater the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ regardless of treatment assignment and for whom the treatment prevents (causes respectively) outcome of interest. Also, Corollary 2A provides a contrast involving only real-world probabilities that is the minimum proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom the treatment causes (or prevents) the outcome. The other monotonicity assumption also provides useful results and is presented next.

Theorem 2B

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ For some $t\in T$ and $y\in\{0,1\},$ if the empirical condition $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{2,0.1}^{r(t)}>0$ holds, then there exists a non-empty subpopulation $\Omega_{s}\subseteq\Omega$ such that for every $\omega\in\Omega_{s},$ $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ and $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y.$

Proposition 2B

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ For some for some $t\in T$ and $y\in\{0,1\},$ the empirical condition $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{2,0.1}^{r(t)}$ is equal to

[TABLE]

Corollary 2B

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ The expression $\max\{P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)}-P_{2,0.1}^{r(t)},0\}$ is a lower bound on the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom treatment causes the outcome.

Similarly, the expression $\max\{P_{1,1.0}^{r(t)}-P_{1,1.1}^{r(t)}-P_{2,0.1}^{r(t)},0\}$ is a lower bound on the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom treatment prevents the outcome. The interpretations for Theorem 2B and Proposition 2B are similar to those provided for Theorem 2A and Proposition 2A. Finally, if we believe that the two monotonicity assumptions hold, then we have the following results.

Theorem 2C

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ and $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ For some $t\in T$ and $y\in\{0,1\},$ if the empirical condition $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}>0$ holds, then there exists a non-empty subpopulation $\Omega_{s}\subseteq\Omega$ such that for every $\omega\in\Omega_{s},$ $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ and $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y.$

Proposition 2C

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ and $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ For some $t\in T$ and $y\in\{0,1\},$ the contrast $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}$ is equal to

[TABLE]

Corollary 2C

Suppose $X$ is randomized at baseline. In addition, suppose that for some $t\in T,$ $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ and $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega.$ The expression $\max\{P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)},0\}$ is a lower bound on the proportion of individuals at randomized at baseline that are always survivors (observed or censored) at time $t$ for whom treatment causes the outcome.

Similarly, the expression $\max\{P_{1,1.0}^{r(t)}-P_{1,1.1}^{r(t)},0\}$ is a lower bound on the proportion of individuals randomized at baseline that are always survivors (observed or censored) at time $t$ for whom treatment prevents the outcome. The interpretations of Theorem, Proposition, and Corollary 2C are similar to those provided for Theorem, Proposition and Corollary 2B respectively. The conditions are to detect individual level always survivor causal effects are weaker the more monotonicity assumptions are made. Again, if such monotonicity assumptions cannot be justified on scientific grounds, investigators can still use the results that do not require such assumptions. The sensitivity analysis that is presented in Section 6 enables researchers to assess the consequences of violations of such monotonicity assumptions and also check how violations of these monotonicity assumptions impact the causal findings.

5 Inference for individual level always survivor causal effects

To investigate individual level always survivor causal effects, null hypotheses can be formulated that when falsified produce the inequalities associated with Theorem 1, 2A, 2B, or 2C. For Theorem 1, if we reject $P_{y,1.1}^{r(t)}+P_{1-y,1.0}^{r(t)}\leq 1$ for $y\in\{0,1\},$ then we conclude with a fixed type one error rate that there exists a non-empty subpopulation $\Omega_{s}\subseteq\Omega$ such that for every $\omega\in\Omega_{s},$ $S_{1}(\omega,t)=S_{0}(\omega,t)=1$ and $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y.$ Testing $P_{y,1.1}^{r(t)}+P_{1-y,1.0}^{r(t)}\leq 1$ is equivalent to testing $P_{1-y,1.0}^{r(t)}\leq P_{1-y,1.1}^{r(t)}+P_{2,0.1}^{r(t)}+P_{3,2.1}^{r(t)},$ which can also be converted into a one-sided difference of proportions. Note testing $P_{1-y,1.0}^{r(t)}\leq P_{y,1.1.}^{r(t)}+P_{2,0.1}^{r(t)}+P_{3,2.1}^{r(t)}$ is equivalent to testing $P(Y(t)=1-y,S(t)=1\mid X=0)\leq P(\{Y(t)=1-y,S(t)=1\}\cup\{Y(t)=2,S(t)=0\}\cup\{Y(t)=3,S(t)=2\}\mid X=1)$.

To use Theorem 2A, if assume $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega$ and we reject, $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{3,2.1}^{r(t)}\leq 0$ for some $y\in{0,1}$, then we conclude with a fixed type one error rate that there exists a non-empty subpopulation $\Omega_{s}$ such that for every $\omega\in\Omega_{s},$ $S_{1}(\omega,t)=S_{0}(\omega,t)=1$ and $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y.$ Note testing $P_{1-y,1.0}^{r(t)}\leq P_{y,1.1.}^{r(t)}+P_{3,2.1}^{r(t)}$ is equivalent to testing $P(Y(t)=1-y,S=1\mid X=0)\leq P(\{Y(t)=1-y,S(t)=1\}\cup\{Y(t)=3,S(t)=2\}\mid X=1)$. Similarly, to use Theorem 2B, if assume $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega$ and we reject, $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{2,0.1}^{r(t)}\leq 0$ for some $y\in\{0,1\}$, then we conclude with a fixed type one error rate that there exists a non-empty subpopulation $\Omega_{s}\subseteq\Omega$ such that for every $\omega\in\Omega_{s},$ $S_{1}(\omega,t)=1$ $S_{0}(\omega,t)=1$ and $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y.$ Note testing $P_{1-y,1.0}^{r(t)}\leq P_{y,1.1.}^{r(t)}+P_{2,0.1}^{r(t)}$ is equivalent to testing $P(Y(t)=1-y,S=1\mid X=0)\leq P(\{Y(t)=1-y,S(t)=1\}\cup\{Y(t)=2,S(t)=0\}\mid X=1)$. Finally, to use Theorem 2C, if assume and $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ and $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega$ and we reject, $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}\leq 0$ for some $y\in{0,1}$, then we conclude with a fixed type one error rate that there exists a non-empty subpopulation $\Omega_{s}$ such that for every $\omega\in\Omega_{s},$ $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ and $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y.$

In our data application below we have seven fixed time periods of interest and so we use Theorem 1 seven times and apply a Bonferroni correction. We also apply Theorem 2A, 2B, 2C arguing that the relevant monotonicity assumptions likely hold, and demonstrate similar conclusions. Note for the data application, we use the equivalent test expressed in a one-sided difference of proportions instead of the complement of the inequality associated with Theorems 1, 2A, 2B, 2C.

The Online Supplement 2 provides Bayesian and randomization based inferential methods to assess results associated with Theorem 1, 2A, 2B, 2C. For our data application, we use the standard t-test for a one-sided difference of proportions to assess always survivor causal effects, because of the moderately large sample size and ease of implementation for readers. In the setting of smaller sample sizes, the randomization or Bayesian approach might be more appropriate.

6 Sensitivity analysis for monotonicity assumptions

As demonstrated in Section 3, we provide methods for identification of individual level always survivor causal effects with only the assumptions guaranteed by design of the clinical trial. Theorem 1, Proposition 1 and Corollary 1 effectively only require randomization at baseline to first detect always survivor causal effects and secondly provide a population level characterization of such causal effects. Monotonicity assumptions on survival or censoring enable scientists to detect always survivor causal effects under assumptions that are justified using subject matter knowledge, and can assist researchers to detect always survivor causal effects when the tests without such monotonicity assumptions are too stringent. Sensitivity analysis can provide scientists and statisticians with methods to interpret their conclusions when a set of assumptions is not guaranteed by design.

Theorem 3A

Suppose $X$ is randomized at baseline. For some $t\in T$ and $y\in\{0,1\},$ the expression $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{3,2.1}^{r(t)}-d_{m}(t)$ for $d_{m}(t)=P^{c(t)}(2,1-y,0,1)-P^{c(t)}(1-y,2,1,0)-P^{c(t)}(3,2,2,0)$ is equal to

[TABLE]

and consequently if $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{3,2.1}^{r(t)}>d_{m}(t),$ then

[TABLE]

Theorem 3B

Suppose $X$ is randomized at baseline. If for some $t\in T$ and $y\in\{0,1\}$ the expression $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{2,0.1}^{r(t)}-a_{m}(t)$ for $a_{m}(t)=P^{c(t)}(3,1-y,2,1)-P^{c(t)}(1-y,2,1,0)-P^{c(t)}(1-y,3,1,2)-P^{c(t)}(2,y,0,1)-P^{c(t)}(2,2,0,0)-P^{c(t)}(2,3,0,2)$ is equal to

[TABLE]

and consequently if $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{2,0.1}^{r(t)}>a_{m}(t),$ then $P^{c(t)}(y,1-y,1,1)>P^{c(t)}(1-y,y,1,1).$

Theorem 3C

Suppose $X$ is randomized at baseline. If for some $t\in T$ and $y\in\{0,1\}$ the expression $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-k_{m}(t)$ for $k_{m}(t)=P^{c(t)}(2,1-y,0,1)+P^{c(t)}(3,1-y,2,1)-P^{c(t)}(1-y,2,1,0)-P^{c(t)}(1-y,3,1,2)$ is equal to

[TABLE]

and consequently if $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}>k_{m}(t),$ then $P^{c(t)}(y,1-y,1,1)>P^{c(t)}(1-y,y,1,1).$

The quantities $d_{m}(t),$ $a_{m}(t)$ and $k_{m}(t)$ are not point identified in randomized studies. A simple, yet effective, approach for sensitivity analysis would be to first estimate the left hand side of the inequalities associated with Theorem 3A, Theorem 3B, and Theorem 3C, and then crudely interpret $d_{m}(t),$ $a_{m}(t),$ $k_{m}(t)$ as the proportion of individuals of a specific counterfactual form to render the respective inferences associated with Theorem 3A, 3B, 3C to be invalid. As an example, suppose the researcher estimates $P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)}$ for a specific $t\in T$ as 0.2. Using Theorem 2C, the scientist would conclude that at least 20% of her population randomized at baseline would follow the counterfactual response pattern $Y_{1}(\omega,t)=1,$ $Y_{0}(\omega,t)=0,$ $S_{1}(\omega,t)=1,$ and $S_{0}(\omega,t)=1.$ For the scientist’s conclusion to be possibly incorrect on the existence of such an individual level effect, the exact 20% of individuals in her population that she believed to follow counterfactual response pattern $Y_{1}(\omega,t)=1,$ $Y_{0}(\omega,t)=0,$ and $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ would need to follow either one of two counterfactual response types $\{\omega_{1},\omega_{2}\}\in\Omega:$ $Y_{1}(\omega_{1},t)=2,$ $Y_{0}(\omega_{1},t)=1,$ $S_{1}(\omega_{1},t)=0,$ $S_{0}(\omega_{1},t)=1$ or $Y(\omega_{2},t)=3,$ $Y_{0}(\omega_{2},t)=0,$ $S_{1}(\omega_{2},t)=2,$ $S_{0}(\omega_{2},t)=1.$ Additionally, there has to be at least 20% (possibly significantly greater than 20%, if $P^{c(t)}(0,2,1,0)$ and $P^{c(t)}(0,3,1,2)$ are non-zero) of individuals that follow counterfactual response types $Y_{1}(\omega_{1},t)=2,$ $Y_{0}(\omega_{1},t)=1,$ $S_{1}(\omega_{1},t)=0,$ $S_{0}(\omega_{1},t)=1$ or $Y(\omega_{2},t)=3,$ $Y_{0}(\omega_{2},t)=0,$ $S_{1}(\omega_{2},t)=2,$ $S_{0}(\omega_{2},t)=1$ at time $t$ for there to be no individual of counterfactual response type $Y_{1}(\omega,t)=1,$ $Y_{0}(\omega,t)=0,$ $S_{1}(\omega,t)=1,$ and $S_{0}(\omega,t)=1$ in our population.

Theorem 3A, 3B, and 3C provide further context on how violations of individual level monotonicity assumptions impact the relevant counterfactual proportions. As an example, consider a researcher that estimates $P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)}-P_{3,2.1}^{r(t)}$ as 0.25 at time $t=1.$ If the scientist believes that at most five percent of her clinical trial subject’s randomized at baseline satisfy the following three conditions, (1) not survive when given the new treatment by time $t=1$, (2) survive when given the control condition at least until time $t=1$, and (3) not develop the outcome of interest under the control condition (denoted $P^{c(t)}(2,0,0,1)\leq 0.05)$, then from Theorem 3A, we know that the proportion of individuals randomized at baseline that are always survivors at least until time $t=1$ that would develop outcome $Y$ under the treatment condition but when prescribed the control condition is at least $0.25-0.05=0.20$.

Such sensitivity analysis enables researchers to assess either how many individuals of a specific counterfactual response type need to exist in a population for the detection of individual principal stratum causal effects to be invalid if a particular monotonicity assumption fails, or how given individual level counterfactual response types attenuate the always survivor causal effect of interest. Additionally, researchers will develop an understanding of the individual reasons for trial censoring or death at each particular time in the trial. If a trial subject dies in a random car crash at time $t=1$, then a researcher might have strong arguments to support the belief that the subjects death had nothing to do with treatment assignment. Such information can be used to choose and refine suitable values of $d_{m}(t),$ $a_{m}(t)$ and $k_{m}(t)$ for different times in the study.

Similarly, if the control condition arm experiences a greater number of adverse events (for instance higher toxicities) for the duration the trial and the rate of censoring is at least as large in the control arm as in the treatment arm, then similarly a researcher might have grounds to believe that at particular times in the trial that $P^{c(t)}(3,1-y,2,1)$ is less than $P^{c(t)}(1-y,3,1,2).$ In such situations, the researcher might have solid grounds to believe that $P_{1-y,1.0}^{r(t)}-P_{1-y,1.1}^{r(t)}-P_{2,0.1}^{r(t)}$ is fairly close to $P^{c(t)}(y,1-y,1,1)-P^{c(t)}(1-y,y,1,1).$ After unblinding the trial, researchers might discover that some toxicities only occur in one of the two arms, and that such toxicities cause the patients to drop out of the study. A careful sensitivity analysis that keeps track of the each individual’s reasons for drop out or death could provide researchers with greater assurance in detecting always survivor causal effects and also enable them to further refine the constants $k_{m}(t)$ and $a_{m}(t)$ that are associated with Theorems and Corollaries 3B and 3C.

The data analysis presented below illustrates how such sensitivity analysis for monotonicity assumptions can reassure researchers about the veracity of their scientific findings. We present a second more generalized version of sensitivity analysis in the section titled Generalizations and Extensions as well as some notes on sensitivity analysis in Online Supplement 2.

7 Application to Southwest Oncology Group Trial

Our results are applied to the data from the Southwest Oncology Group (SWOG) Trial, which was a phase III trial to compare docetaxel plus estramustine against mitoxantrone plus prednisone in men with metastatic, hormone-independent prostate cancer [5]. For the trial, the primary end-point was overall survival [5]. A total of 338 patients were randomized to the docetaxel plus estramustine (henceforth referred to as docetaxel), and a total of 336 were randomized to the mitoxantrone plus prednisone (henceforth referred to as mitoxantrone). The trial found that docetaxel improved median survival in comparison to mitoxantrone [5]. Progression of cancer is an important secondary outcome of interest and may be subject to truncation by death. Our hypothesis is that for some pre-specified times $t\in T_{\text{SWOG}}$, the proportion of individuals randomized at baseline who would remain alive regardless of treatment at least until the pre-specified time $t$ and whose cancer would progress under mitoxantrone but not under docetaxel at time $t$ is greater than the proportion of individuals randomized at baseline who would remain alive regardless of treatment at least until the pre-specified time $t$ and whose cancer would progress under docetaxel but not under mitoxantrone at time $t$. Such a hypothesis can be evaluated using our results, but no previous approach could answer such a question using only the set of assumptions guaranteed by design.

The time periods are 1 month $(t=1)$, 2 months $(t=2)$, 3 months $(t=3)$, 4 months $(t=4)$, 6 months $(t=5)$, 12 months $(t=6)$, and 18 months $(t=7)$ post treatment initiation. Here, $X=1$ denotes that the individual was randomized to the docetaxel arm, and $X=0$ denotes that the individual was randomized to the mitoxantrone arm. Outcome $Y(t)$ is defined as a categorical variable that takes values in the set $\{0,1,2,3\}$ for each $t\in T_{\text{SWOG}}=\{1,\ldots,7\},$ where $Y(t)=0$ means that the individual’s cancer did not progress at time point $t,$ $Y(t)=1$ means that the individual’s cancer progressed at time point $t,$ $Y(t)=2$ means that the individual did not survive until time $t,$ and finally $Y(t)=3$ means that the individual’s observation was censored or missing at time $t.$ Survival status at time $t,$ denoted $S(t)$ is also defined as a categorical variable that takes value in the set $\{0,1,2\}$ for each $t\in T_{\text{SWOG}},$ where $S(t)=0$ means that the individual did not survive at time period $t,$ $S(t)=1$ means that the individual survived to at least time period $t,$ and $S(t)=2$ means that the individual’s survival status is censored or missing at time period $t.$ We provide the relevant contingency tables in the appendix and sample R code that is used to produce all our results in Online Supplement 2.

7.1 Data Analysis without monotonicity assumptions

Without any monotonicity assumptions, we need to test $P_{0,1.1}^{r(t)}+P_{1,1.0}^{r(t)}\leq 1$ for each $t\in\{1,\ldots,7\}$ to evaluate at each time point whether there exist individuals for whom $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ $Y_{1}(\omega,t)=0,$ $Y_{0}(\omega,t)=1,$ that is these, individuals would survive at least until time $t$ regardless of treatment, but treatment with docetaxel (vs. mitoxantrone) would prevent their cancer from progressing. This hypothesis test is equivalent to testing $P_{1,1.0}^{r(t)}\leq P_{1,1.1}^{r(t)}+P_{2,0.1}^{r(t)}+P_{3,2.1}^{r(t)}.$ To conduct this hypothesis test, we provide the following three contingency tables:

Two-sided confidence intervals for $P(Y(t)=1,S(t)=1\mid X=0)-P(\{Y(t)=1,S(t)=1\}\cup\{Y(t)=3,S(t)=2\}\cup\{Y(t)=2,S(t)=0\}\mid X=1)$ and one-sided p-values for Wald test of the null hypothesis $P(Y(t)=1,S(t)=1\mid X=0)-P(\{Y(t)=1,S(t)=1\}\cup\{Y(t)=3,S(t)=2\}\cup\{Y(t)=2,S(t)=0\}\mid X=1)\leq 0$ are provided below:

The results from Table 4 demonstrate by Corollary 1 for the Southwest Oncology Group Trial, that there is statistical evidence suggesting that the proportion of individuals randomized at baseline that are always survivors (observed or censored) at one month post treatment initiation whose cancer would not progress on docetaxel but would progress on mitoxantrone is greater by 0.07 with a 95 percent confidence interval of $(0.02,0.11)$ than the proportion of individuals randomized at baseline that are always survivors (observed or censored) at one month post treatment initiation whose cancer would progress on docetaxel but would not progress on mitoxantrone. Our estimate of the number of individuals randomized at baseline that are always survivors (observed or censored) at one month post treatment initiation whose cancer would not progress on docetaxel but would progress on mitoxantrone is greater by $0.07\cdot 674=47$ than the number of individuals randomized at baseline that are always survivors (observed or censored) at one month post treatment initiation whose cancer would progress on docetaxel but would not progress on mitoxantrone. We do not find evidence for such individual level always survivor causal effects from 6 months onwards. At three months, statistical evidence (even after Bonferonni correction for multiple testing with a threshold of $0.05/7=0.007$) suggests that the proportion individuals randomized at baseline that are always survivors three months post treatment initiation whose cancer would not progress on docetaxel but would on mitoxantrone is at least $0.15$ with a 95 percent confidence interval of $(0.08,0.23)$. Note that this is a lower bound of such individuals, and with randomization alone this proportion is not point identified. In terms of the $336+338=674$ individuals that were enrolled in this clinical trial, our estimate is that there are at least $0.15\cdot 674=81$ more individuals randomized at baseline who would survive at least until 3 months regardless of which treatment arm they were randomized and their cancer would progress under mitoxantrone but not docetaxel than the individuals randomized at baseline who would survive at at least until three months regardless of which treatment arm they were randomized and their cancer would progress under docetaxel but not mitoxantrone.

The conclusions drawn in this data analysis could not be made with any of the previous methods that used principal stratification to deal with truncation by death [2, 13]. Two papers [2, 13] have used similar data and were unable to draw such conclusions [2, 13]. Note also that these other papers, in contrast to ours, made more assumptions than those guaranteed by randomization alone. We are able to draw conclusions here even with weaker assumption because of the focus, not on the survivor average causal effect, but rather detecting and quantifying individuals who were always survivors for whom treatment had a causal effect on cancer progression.

7.2 Data analysis with monotonicity assumption $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$

For each $t\in\{1,\ldots,7\},$ we test the null hypothesis $P_{1,1.0}^{r(t)}\leq P_{1,1.1}^{r(t)}+P_{3,2.1}^{r(t)}$ to evaluate for a fixed $t$ whether the proportion of individuals for whom $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ and $Y_{1}(\omega,t)=0,$ $Y_{0}(\omega,t)=1$ is greater than the proportion of individuals for whom $S_{1}(\omega,t)=S_{0}(\omega,t)=1$ and $Y_{1}(\omega,t)=1,$ $Y_{0}(\omega,t)=0$ under the assumption that $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega$ and $t\in T_{\text{SWOG}}.$ The following table provides $95\%$ and $99\%$ confidence intervals for $P_{1,1.0}^{r(t)}-P_{1,1.1}^{r(t)}-P_{3,2.1}^{r(t)}$ for each $t\in T_{\text{SWOG}}$. Notice $P(S(t)=0\mid X=1)+P(S(t)=1\mid X=0)\leq 1$ for all $t\in T_{\text{SWOG}}$ and therefore we fail to falsify the monotonicity assumption, though of course this does not guarantee that the assumption holds.

Two-sided confidence intervals of $P(Y(t)=1,S(t)=1\mid X=0)-P(\{Y(t)=1,S(t)=1\}\cup\{Y(t)=3,S(t)=2\}\mid X=1)$ and p-values for Wald test of the null hypothesis $P(Y(t)=1,S(t)=1\mid X=0)-P(\{Y(t)=1,S(t)=1\}\cup\{Y(t)=3,S(t)=2\}\mid X=1)\leq 0$ are provided below:

For this section, we have assumed that $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $t\in T_{\text{SWOG}}.$ Notice with this assumption, we have the same conclusions as before regarding always survivors between one to four months, and now also have statistical evidence for individual level always survivor causal effects at six months.

By sensitivity analysis Theorem 3A, in order for the inference of detecting always survivors six months post treatment initiation whose cancer would not progress on docetaxel but would on mitoxantrone to be invalid, we would need at least 12 percent of our study population at baseline to follow the counterfactual response $S_{1}(\omega,t)=0,$ $S_{0}(\omega,t)=1,$ and $Y_{1}(\omega,t)=2,$ $Y_{0}(\omega,t)=0$ for $t=5,$ that is at least 12 percent of our population randomized at baseline would need to die within six months if they were prescribed docetaxel but would survive with a cancer that has not progressed at least until six months post treatment initiation if prescribed mitoxantrone.

7.3 Data analysis with monotonicity assumption $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$

For each $t\in\{1,\ldots,7\},$ we test the null hypothesis $P_{1,1.0}^{r(t)}\leq P_{1,1.1}^{r(t)}+P_{2,0.1}^{r(t)}$ to evaluate for a fixed $t$ whether the proportion of individuals for whom $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ $Y_{1}(\omega,t)=0,$ and $Y_{0}(t,\omega)=1$ is greater than the proportion of individuals for whom $S_{1}(\omega,t)=S_{0}(\omega,t)=1,$ and $Y_{1}(\omega,t)=1,$ $Y_{0}(\omega,t)=0$ under the assumption that $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega$ and $t\in T_{\text{SWOG}}.$ The following table provides $95\%$ and $99\%$ confidence intervals for $P_{1,1.0}^{r(t)}-P_{1,1.1}^{r(t)}-P_{2,0.1}^{r(t)}$ for each $t\in T_{\text{SWOG}}$. Notice that $P(S(t)=2\mid X=1)+P(S(t)=1\mid X=0)\leq 1$ for all $t\in T_{\text{SWOG}},$ though of course this does not guarantee that the assumption holds.

Two-sided confidence intervals of $P(Y(t)=1,S(t)=1\mid X=0)-P(\{Y(t)=1,S(t)=1\}\cup\{Y(t)=2,S(t)=0\}\mid X=1)$ and one-sided p-values for Wald test of the null hypothesis $P(Y(t)=1,S(t)=1\mid X=0)-P(\{Y(t)=1,S(t)=1\}\cup\{Y(t)=2,S(t)=0\}\mid X=1)\leq 0$ are provided below:

Under the assumption of monotonicity for censoring, we find evidence for individual principal stratum causal effects up to 4 months but not beyond.

7.4 Data analysis with monotonicity assumptions $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$

and $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$

For each $t\in\{1,\ldots,7\},$ we test the null hypothesis $P_{1,1.0}^{r(t)}\leq P_{1,1.1}^{r(t)}$ to evaluate for a different fixed $t$ whether the proportion of individuals for whom $Y_{1}(\omega,t)=0,$ $Y_{0}(t,\omega)=1,$ $S_{1}(t,\omega)=1,$ and $S_{0}(t,\omega)=1$ is greater than the proportion of individuals for whom $Y_{1}(\omega,t)=1,$ $Y_{0}(\omega,t)=0,$ $S_{1}(\omega,t)=1,$ and $S_{0}(\omega,t)=1$ under the assumption that $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ and $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all $\omega\in\Omega$ and $t\in T_{\text{SWOG}}.$ The following table provides $95\%$ and $99\%$ confidence intervals for $P_{1,1.0}^{r(t)}-P_{1,1.1}^{r(t)}$ for each $t\in T_{\text{SWOG}}$. As before, notice $P(S(t)=0\mid X=1)+P(S(t)=1\mid X=0)\leq 1$ and $P(S(t)=2\mid X=1)+P(S(t)=1\mid X=0)\leq 1$ for all $t\in T_{\text{SWOG}}$ and therefore we fail to falsify the monotonicity assumption, though of course this does not guarantee that the assumption holds.

Two-sided confidence intervals of $P(Y(t)=1,S(t)=1\mid X=0)-P(Y(t)=1,S(t)=1\mid X=1)$ and one-sided p-values for Wald test of the null hypothesis $P(Y(t)=1,S(t)=1\mid X=0)-P(Y(t)=1,S(t)=1\mid X=1)\leq 0$ are provided below:

Under the assumption of monotonicity for censoring and survival, statistical evidence suggests that the proportion of individuals randomized at baseline that are always survivors three months post treatment initiation whose cancer would not progress on docetaxel but would on mitoxantrone is at least 0.22 with a 95 percent confidence interval of $(0.15,0.29)$. Additionally, we again have statistical evidence for individual principal stratum causal effects up to 6 months but not beyond. Sensitivity analysis could also be applied to the results using Theorem 3C and Corollary 3C.

8 Generalizations and Extensions

This section generalizes results to allow for outcomes in $\mathbf{R}^{n}$ and extends notation further. Let $T$ the range of the study period from baseline $t=0$ to the end of study $t_{f}.$ Let $Y_{x}(\omega,t),$ and $S_{x}(\omega,t)$ denote the potential outcomes at time $t$ for individual $\omega,$ and correspondingly $Y(\omega,t)$ and $S(\omega,t)$ are the observed variables at time $t$ for individual $\omega.$ The variables $Y(\omega,t)$ and $Y_{x}(\omega,t)$ have support on $\mathbf{R}^{n_{y}}$ where $n_{y}$ is the number of elements corresponding to the tuple $Y(\omega,t)$. The potential outcomes $Y_{x}(\omega,t)$ and $S_{x}(\omega,t),$ and, correspondingly, the observed variables, $Y(\omega,t)$ and $S(\omega,t),$ could be missing. Instead of denoting a new level indicating that these observations are missing as presented in the previous sections, we account for the missingness in a different way that is presented below. The coarsest handling of missing data will stem from using variables $R(\omega,t)$ that are defined as follows:

[TABLE]

For more refined analysis of censoring or missing data, consider the following variables:

[TABLE]

and

[TABLE]

Clearly, $R(\omega,t)=RS(\omega,t)\cdot RY(\omega,t).$ We will use $(S_{x}(t),RY_{x}(t),RS_{x}(t))\in\{(s,a,b)\}$ to denote that $S_{x}(\omega,t)=s$, $RY_{x}(\omega,t)=a,$ $RM_{x}(\omega,t)=b.$ Let $Y(\omega,t)=\star$ denote that the individual has died by time $t$ which is denoted $S(\omega,t)=0.$ Similarly, for the potential outcomes if the individual $\omega$ has died by time $t,$ then $S_{x}(\omega,t)=0$ and $Y_{x}(\omega,t)=\star.$ The random tuple $(Y_{1}(t),Y_{0}(t),S_{1}(t),S_{0}(t),RY_{1}(t),RY_{0}(t),RS_{1}(t),RS_{0}(t))$ is defined for each unit of the randomized experiment, and all the results provided below retain the individual level interpretations that are provided in the previous sections. We allow $(Y_{1}(t),Y_{0}(t),S_{1}(t),S_{0}(t),RY_{1}(t),RY_{0}(t),RS_{1}(t),RS_{0}(t))$ to have a distribution function $P(Y_{1}(t)\in y_{1},Y_{0}(t)\in y_{0},S_{1}(t)=s_{1},S_{0}(t)=s_{0},RY_{1}(t)=ry_{1},RY_{0}(t)=ry_{0},RS_{1}(t)=rs_{1},RS_{0}(t)=rs_{0}),$ where $y_{1}$ and $y_{0}$ are subsets of $\mathbf{R}^{n_{y}}$ (or $y_{1}$ and $y_{0}$ are equal to $\star$ whenever $s_{1}=0$ or $s_{0}=0$ respectively) and $s_{1},s_{0},ry_{1},ry_{0}$ all take values in the set $\{0,1\}.$ An important consideration here is that the distribution function $P(Y_{1}(t)\in y_{1},Y_{0}(t)\in y_{0},S_{1}(t)=s_{1},S_{0}(t)=s_{0},RY_{1}(t)=ry_{1},RY_{0}(t)=ry_{0},RS_{1}(t)=rs_{1},RS_{0}(t)=rs_{0})$ retains all the axiomatic properties of probability theory as provided by Kolmogorov. In contrast, an causal estimand of the form $P\left(Y_{x}(\omega,t)\in y_{a}\right)$ that is restricted to some $y_{a}$ in $\mathbf{R}^{n_{y}}$ might not be defined on $y_{a}$ and is undefined when $S_{x}(\omega,t)=0$ at time $t$.

For our second data application, $S$ still denotes survival, but a scientist could take $S$ to denote survival and any other outcome of interest to get at a more refined principal strata. For example, $S$ could denote whether the individual has survived is not HIV infected. In this setting $S_{1}(\omega,t)=S_{0}(\omega,t)=1$ denotes that the individual is an always uninfected, always survivor. Depending on the scientific question under investigation, these more refined individual level causal effects might be important. We will also look at quantifying individual level causal effects at different time points in the sensitivity analysis section provided below and further in Online Supplement 2. For instance we will examine whether we can detect an individual that remains alive at least until time $t_{U}$ with treatment, $t_{L}$ with control, and will have the outcome in a range $y_{a}$ at time $t_{U}$ (denoted $Y_{1}(\omega,t_{U})\in y_{a}$ ,$S_{1}(\omega,t_{U})=1$) and will not have the outcome in a range $y_{b}$ at time $t_{L}$ (denoted $Y_{1}(\omega,t_{L})\notin y_{b},S_{1}(\omega,t)=1$). We will see that these type of causal questions are useful to understand the life course of disease trajectory.

Theorem Generalized Always Survivor Causal Effect

Assume $X$ is randomized at baseline. For some $t\in T,$ $y_{a}\subset\mathbf{R}^{n_{y}},$ and $y_{b}\subset\mathbf{R}^{n_{y}},$ the contrast

[TABLE]

is a lower bound on

[TABLE]

and is therefore also a lower bound on

[TABLE]

For some $t\in T,$ $y_{a}\subset\mathbf{R}^{n_{y}}$ and $y_{b}\subset\mathbf{R}^{n_{y}},$ the expression

[TABLE]

is a lower bound on

[TABLE]

and therefore is also a lower bound on

[TABLE]

For some $t\in T,$ $y_{a}\subset\mathbf{R}^{n_{y}}$ and $y_{b}\subset\mathbf{R}^{n_{y}},$ when $P(Y(t)\in y_{a},S(t)=1,R(t)=1\mid X=1)+P(Y(t)\not\in y_{b},S(t)=1,R(t)=1\mid X=0)>1,$ the expression

[TABLE]

is a lower bound on

[TABLE]

which is equivalent to

[TABLE]

The result above is a generalization of Proposition 1 and Corollary

1. With baseline randomization, we enable statisticians to quantify always survivor causal effects. In many instances, the scientist will take $y_{a}=y_{b},$ but as we will see in our data analysis below, scientists also might want to have $y_{a}\neq y_{b}$ to quantify a specific form of individual level effect. As noted earlier, without monotonicity assumptions, the empirical conditions to detect individual level always survivor causal effects might be too stringent. In this setting there are many different monotonicity conditions that a scientist might be willing to make. We provide one result that makes the full set of monotonicity assumptions.

Theorem Generalized Always Survivor Causal Effect Monotonicity

Assume $X$ is randomized at baseline. If we have the monotonicity assumption at time $t\in T$ that there is no individual $\omega\in\Omega$ of response type $(S_{1}(t),R_{1}(t))\not\in\{(1,1)\}$ and $(S_{0}(t),R_{0}(t))\in\{(1,1)\}$ , then for some for some $t\in T,$ $y_{a}\subset\mathbf{R}^{n_{y}}$ and $y_{b}\subset\mathbf{R}^{n_{y}}$ contrast

[TABLE]

is a lower bound on

[TABLE]

When there is no individual $\omega\in\Omega$ of response type $(S_{1}(\omega,t),R_{1}(\omega,t))\not\in\{(1,1)\}$ and $(S_{0}(\omega,t),R{}_{0}(\omega,t))\in\{(1,1)\}$, then the expression

[TABLE]

and also is a lower bound on

[TABLE]

which is equal to

[TABLE]

The results to detect and quantify individual level always survivor causal effects under truncation by death with missing data under partial monotonicity assumptions are provided in Online Supplement 2. All estimands provided above can be estimated from the observed data the associated parameters do not involve the joint of $R(t)=0$ and $S(t)$ or $Y(t)$.

Generalized Sensitivity Analysis

We now provided methods to conduct sensitivity analysis with missing data. The sensitivity analysis provided herein differs from the sensitivity analysis provided in Section 4 for monotonicity assumptions. The sensitivity analysis in Section 4 is substantially easier to implement than the sensitivity analysis provided in this section. However, the methods provided in this section enable statisticians the ability to understand how missing responses as well as monotonicity assumptions impact the quantification of individual level always survivor causal effects. Statisticians should use both sensitivity analysis methods to assess the impact of violations of their assumptions on the resulting conclusions. They should first apply the sensitivity analysis methods provided in Section 4, because those methods are immediate. The following results require a little more deliberation from both the statistician and scientist, but such sensitivity analyses offers different insights on what needs to be assumed from the observed data to quantify or point identify $P(Y_{1}(t)\in y_{a},Y_{0}(t)\notin y_{b},S_{1}(t)=1,S_{0}(t)=1)-P(Y_{1}(t)\notin y_{a},Y_{0}(t)\in y_{b},S_{0}(t)=1,S_{0}(t)=1).$ In contrast, the simpler sensitivity analysis will not enable statisticians to quantify or point identify $P(Y_{1}(t)\in y_{a},Y_{0}(t)\notin y_{b},S_{1}(t)=1,S_{0}(t)=1)-P(Y_{1}(t)\notin y_{a},Y_{0}(t)\in y_{b},S_{0}(t)=1,S_{0}(t)=1).$

Theorem Generalized Missing Data Sensitivity Analysis

Suppose $X$ is randomized at baseline. For some $t\in T,$ $y_{a}\subset\mathbf{R}^{n_{y}}$ and $y_{b}\subset\mathbf{R}^{n_{y}}$ the expression

[TABLE]

is a lower bound on

[TABLE]

which is equivalent to

[TABLE]

When

[TABLE]

is greater than equal to zero, the expression

[TABLE]

is a lower bound on

[TABLE]

The parameters $P(Y(t)\in y_{a},S(t)=1,R(t)=0\mid X=1),$ $P(Y(t)\notin y_{b},S(t)=1,R(t)=0\mid X=0)$ and $P\left(S(t)=0,R(t)=1\mid X=1\right)-P\left(S(t)=0,R(t)=0\mid X=0\right)$ cannot be estimated from the observed data, because when the data is missing the realized values of $Y(t)$ and $S(t)$ are no longer known at time $t.$ Therefore, the statistician will need to make some assumption about the values of these parameters given the observed data as well as subject matter knowledge about each individual’s reason for dropping out of the study. A second sensitivity analysis method is provided below. For the second sensitivity analysis, we will have two time points, $t_{U}$ indexing all $X=1$ (treatment variables) and $t_{L}$ indexing all $X=0$ (control variables). In most applications, we will take $t_{L}=t_{U}=t,$ but in one very important example below $t_{L}$ and $t_{U}$ are distinct time points. If the statistician is interested in always survivor causal effects at time $t$ and wishes to use the sensitivity analysis approach from the next theorem, they should replace all instances where $t_{L}$ and $t_{U}$ appear with $t.$

Theorem Generalized Missing Data and Monotonicity Sensitivity Analysis

Assume $X$ is randomized at baseline. For $y_{a}\subset\mathbf{R}^{n_{y}},$ $y_{b}\subset\mathbf{R}^{n_{y}},$ $t_{L}\in T$ and $t_{U}\in T,$ the expression

[TABLE]

where

[TABLE]

is equal to

[TABLE]

or equivalently

[TABLE]

Consequently, the expression

[TABLE]

is equal to the normalized principal stratum direct effect

[TABLE]

which is equivalent to

[TABLE]

Sensitivity analysis parameters $r(t_{L},t_{U}),$ $P\left(S(t_{L})=1,R(t_{L})=0\mid X=0\right)$ and $P\left(S_{1}(t_{U})=1,S_{0}(t_{L})\neq 1\right)$ cannot be estimated from the observed data. However, researchers can deliberate on reasonable values for these sensitivity analysis parameters to detect and quantify individual level effects. Also, note $r(t_{L},t_{U})$ is equal $P(Y_{1}(t_{U})\in\star,Y_{0}(t_{L})\notin y_{b},S_{1}(t_{U})=0,S_{0}(t_{L})=1)-P\left(Y_{1}(t_{U})\notin y_{a},Y_{0}(t_{L})\in\star,S_{1}(t_{U})=1,S_{0}(t_{L})=0\right),$ and depending on the scientific question under investigation, these sensitivity analysis parameters can be meaningfully intuited. The advantage of such type of sensitivity analyses is that they are tied intrinsically to the actual units, which enables statisticians to choose these parameters carefully with conversations with the study investigators. In the presence of truncation by death, statisticians can fairly attempt to target

[TABLE]

and not $P(Y_{1}(t)\in y_{a},Y_{0}(t)\notin y_{b})-P(Y_{0}(t)\notin y_{a},Y_{0}(t)\in y_{b})$ as the counterfactuals $Y_{x}(\omega,t)$ are not meaningfully defined on $\mathbf{R}^{n_{y}}$ for individuals for whom $S_{x}(\omega,t)=0.$ To identify the counterfactual risk difference $P(Y_{1}(t)\in y_{a},Y_{0}(t)\notin y_{b},S_{1}(t)=1,S_{0}(t)=1)-P\left(Y_{1}(t)\notin y_{a},Y_{0}(t)\in y_{b},S_{1}(t)=1,S_{0}(t)=1\right)$ all causal models are making assumptions of baseline randomization and an implicit assumption regarding sensitivity analysis parameters $c(t_{L}=t,t_{U}=t),$ $P\left(Y(t_{L})\notin y_{b},S(t_{L})=1,R(t_{L})=0\mid X=0\right)$ and $P\left(Y(t_{U})\notin y_{a},S(t_{U})=1,R(t_{U})=0\mid X=1\right).$

A Bayesian approach to using the above sensitivity analysis would place a Dirichlet prior $\alpha$ on the Multinomial counts associated on the joint $Z(t)=(Y(t),S(t),R(t)).$ Counts associated with $(Y(t),S(t),R(t)=0)$ are not observed, and the statistician can choose a range of suitable values in collaboration with the scientific investigator.

9 Second Application to Southwest Oncology Group Trial

Consider variable $Q,$ which is the quality of life index at 12 weeks. Given that we have a single time point, we drop the indexing on $t,$ and, similarly, sensitivity analysis parameter $c(t_{L}=t,t_{U}=t)$ is measured at a single time point $t_{L}=t_{U}=t$ of 12 weeks. We have the following contingency table

Using Theorem Generalized Always Survivor Causal Effect Monotonicity, to test the hypothesis that there is an individual that would have survived regardless of treatment and their quality of life is below $70$ units with docetaxel and above 70 units with mitoxantrone, the scientist could examine the null hypothesis $P(Q>70,S=1,R=1\mid X=0)\leq P(DQ>70,S=1,R=1\mid X=1).$ The required monotonicity assumptions ($S_{1}(\omega)\geq S_{0}(\omega)$ and $R_{1}(\omega)\geq R_{0}(\omega)$ for all units in our population) appear reasonable in this setting. Using Theorem Generalized Always Survivor Causal Effect Monotonicity, rejecting $P(Q>70,S=1,R=1\mid X=0)\leq P(Q>70,S=1,R=1\mid X=1)$ tells the scientist that there must exist at least one individual $\omega\in\Omega$ such that $Q_{1}(\omega)\leq 70,$ $Q_{0}(\omega)>70,$ $S_{1}(\omega)=S_{0}(\omega)=R_{1}(\omega)=R_{0}(\omega)=1,$ and the same theorem also tells us that $P(Q>70,S=1,R=1\mid X=0)\leq P(Q>70,S=1,R=1\mid X=1)$ is a lower bound on

[TABLE]

The risk difference $P(Q>70,S=1,R=1\mid X=0)-P(Q>70,S=1,R=1\mid X=1)$ is estimated as $\frac{89}{336}-\frac{73}{338}=0.05$ with a 95% confidence interval of $(-0.02,0.12).$ Assuming the monotonicity assumptions hold true, the counterfactual interpretation of $P(Q>70,S=1,R=1\mid X=0)-P(Q>70,S=1,R=1\mid X=1)=0.05$ is that on average we should expect at least $674\cdot 0.05=34$ more individuals to survive regardless of treatment to twelve weeks and their quality of life measurement is above 70 with mitoxantrone and below 70 with docetaxel than the individuals who also survive regardless of treatment to twelve weeks and whose quality of life is below 70 with mitoxantrone and above 70 with docetaxel. Using the second sensitivity analysis in the Generalization and Extensions section, a statistician might entertain values for $P(Q>70,S=1,R=0\mid X=0)-P(Q>70,S=1,R=0\mid X=1).$ Given that $P(Q>70,S=1,R=1\mid X=0)-P(Q>70,S=1,R=1\mid X=1)$ is positive, a scientist might speculate that $P(Q>70,S=1,R=0\mid X=0)-P(Q>70,S=1,R=0\mid X=1)$ is also positive and similarly around 0.05. Given the results of the clinical trial where it was established that docetaxel was protective for survival in comparison to mitoxantrone, a statistician might reasonable assume that $r(t_{L},t_{U})$ is negative (the number of failures should be higher for the mitoxantrone arm in comparison to the docetaxel arm). In such a setting, the counterfactual interpretation of

[TABLE]

enables scientists to make the following conclusion: on average we should expect at least 10% of our population ($647\cdot 0.10\approx 65)$ randomized at baseline to survive regardless of treatment and their quality of life is above 70 with mitoxantrone but below 70 with docetaxel. While such sensitivity analysis require careful consideration, the advantage of such analyses is that they are intrinsically tied to the actual units of the statistical experiment, which enables scientists and statisticians deliberate together to come to a consensus decision regarding the results of their actual experiment. Quantifying that 10% of the population displays such individual level effects is informative, but it might not be that useful from a decision-theoretic or regulatory perspective if all 65 of such individuals are of the counterfactual response type $Q_{1}(\omega)=69,$ $Q_{0}(\omega)=71,$ $S_{1}(\omega)=S_{0}(\omega)=1.$ Instead, we might be interested in a meaningful change in quality of life.

In such a setting, we might be more interested to discover whether there are a large number of individuals that follow the counterfactual response type $Y_{1}(\omega)\leq 70,$ $Y_{0}(\omega)>75,$ $S_{1}(\omega)=S_{0}(\omega)=R_{1}(\omega)=R_{0}(\omega)=1.$ To answer this scientific question, we would look at rejecting the null hypothesis $P\left(Y>75,S=1,R=1\mid X=0\right)\leq P\left(Y>70,S=1,R=1\mid X=1\right)$. Use the following contingency table, along the the previous contingency table:

The estimated risk difference $P\left(Y>75,S=1,R=1\mid X=0\right)-P\left(Y>70,S=1,R=1\mid X=1\right)=\frac{71}{336}-\frac{73}{388}$ is approximately zero with a 95 percent confidence interval of $(-0.07,0.06).$ We therefore find no statistical evidence for these counterfactual response types of individuals in our population. Our analysis reveals that scientists can deliberate on continuous outcomes, and decide on meaningful hypothesis testing or estimation for informed decision making. In cancer clinical trials where quality of life is an extremely important consideration in end of life decisions, individuals might be willing to sacrifice some gain in individual level life expectancy for an improved quality of life. A difference of one point on a scale from 0 to 100 might not be meaningful enough for the individual to contemplate such a sacrifice, but if the difference is 10 points, then the doctor and patient might well wish to deliberate.

10 Survival Average Causal Effects

Online Supplement 1 provides a fully Bayesian approach using the Dirichlet-Multinomial distribution to provide lower bounds on $P(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b}\mid S_{1}(t)=S_{0}(t)=1)-P(Y_{1}(t)\not\in y_{a},Y_{0}(t)\in y_{b}\mid S_{1}(t)=S_{0}(t)=1).$ This approach is applied to the first and second data analysis. The causal estimand $P(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b}\mid S_{1}(t)=S_{0}(t)=1)-P(Y_{1}(t)\not\in y_{a},Y_{0}(t)\in y_{b}\mid S_{1}(t)=S_{0}(t)=1)$ is arguably less important than $P(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b},S_{1}(t)=1,S_{0}(t)=1)-P(Y_{1}(t)\not\in y_{a},Y_{0}(t)\in y_{b},S_{1}(t)=1,S_{0}(t)=1)$ given that the more traditional survivor average causal effect will never inform us how many people are in the always-survivor principal stratum. In contrast, $P(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b},S_{1}(t)=1,S_{0}(t)=1)-P(Y_{1}(t)\not\in y_{a},Y_{0}(t)\in y_{b},S_{1}(t)=1,S_{0}(t)=1)$ provides information on how many people remain in the always survivor principal strata as well as the treatment effect on $Y(t).$ Our Theorems demonstrated that bounds or senstivity analysis for the survivor average causal effect is just rescaling of$P(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b},S_{1}(t)=1,S_{0}(t)=1)-P(Y_{1}(t)\not\in y_{a},Y_{0}(t)\in y_{b},S_{1}(t)=1,S_{0}(t)=1).$

If scientists want life course trajectories, take $Y_{x}(\omega,t),$ and $S_{x}(\omega,t)$ (also factuals $Y(\omega,t),$ and $S(\omega,t)$) to be multivariate tuples each defined on $\mathbf{R}^{n_{v}\times n_{y}}\cup\{\star,\ldots,\star\}_{n_{v}\times n_{y}}$ and $\mathbf{R}^{n_{v}\times n_{s}}\cup\{\star,\ldots,\star\}_{n_{v}\times n_{s}}$ respectively, where $n_{v}$ denotes the number of time points of interest, $n_{y}$ denotes the number of elements associated with tuple $Y_{x}(\omega,t_{i})$ and $n_{s}$ denote the number of elements associated with tuple $S_{x}(\omega,t_{i})$. The tuple denoted as $\{\star,\ldots,\star\}_{n_{v}\times n}$ denotes a tuple with elements $\star$ of length $n_{v}\times n$ that is used to notationally allow for undefined variable status whenever needed. In such a setting, use variables $Y_{x}(\omega,t)=(Y_{x}(\omega,t_{1}),\ldots,Y_{x}(\omega,t_{n_{v}}))$ and $S_{x}(\omega,t)=(S_{x}(\omega,t_{1}),\ldots,S_{x}(\omega,t_{n_{v}}))$ in all the Theorems below. Each $Y_{x}(\omega,t_{i})$ and $S_{x}(\omega,t_{i})$ is defined on $\mathbf{R}^{n_{y}}\cup\{\star,\ldots,\star\}_{n_{y}}$ and $\mathbf{R}^{n_{s}}\cup\{\star,\ldots,\star\}_{n_{s}}$ respectively. The full generality of defining $Y_{x}(\omega,t)$ and $S_{x}(\omega,t)$ to be multivariate tuples each defined on $\mathbf{R}^{n_{v}\times n_{y}}\{\star,\ldots,\star\}_{n_{v}\times n_{y}}$ and $\mathbf{R}^{n_{v}\times n_{s}}\cup\{\star,\ldots,\star\}_{n_{v}\times n_{s}}$ respectively was not necessary for our applciations, but we provide this notation in order or scientists to consider as refined scientific questions they deem important.

11 Conclusion

This work demonstrates that testing and estimating individual causal effects within ‘always survivor’ principal stratum can be conducted with simply the identification assumptions for a non-zero total effect. The identifiability assumptions we use to identify these individual level always survivor causal effects are no stronger than randomization. Consequently, our results embeds testing for individual level always survivor causal effects firmly within the Neyman-Pearson paradigm. We also enable statistician to assess the existence of such individual level always survivor causal effects using randomization based and Bayesian inference as described in the Online Supplement 2. Previous literature required stronger assumptions to identify or provide bounds for survivor average causal effect, which is a different causal estimand of interest [1, 4]. Our results are generally applicable to clinical trials in the situation where an outcome of interest is potentially not observed when death of the individual occurs prior to the end of study. Our identifiability assumptions are guaranteed by design in a randomized trial.

Our proposed methodology provides a procedure to evaluate whether a new treatment would delay cancer progression. The effect of a new treatment on overall survival is a different scientific question than whether new treatment delays cancer progression. Our data analysis demonstrates the usefulness of our methods in this setting. In oncology trials, determining treatment effectiveness sooner rather than later can extend and improve patient lives as the progression to mortality for many type III cancers can be rapid. Our methodology is the first and, as of yet, only proposed procedure that enables detecting and quantifying causal effects in the presence of truncation by death and censoring using only the assumptions that are guaranteed by design of the clinical trial.

Acknowledgements

The authors thank Linbo Wang and Peng Ding for providing them with the actual data from the Southwest Oncology Group Trial that is used in all the analysis presented in this article. This research was funded by the National Institutes of Health.

Online Supplement 1

Let $I(\cdot)$ denote the usual indicator function. Define $Y_{x}^{[y]}(\omega,t)=I(Y_{x}(\omega,t)=y),$ and $S_{x}^{[s]}(\omega,t)=I(S_{x}(\omega,t)=s)$ for $y\in\{0,1,2,3\},$ $s\in\{0,1,2\},$ and $t\in T.$ Also note $Y_{x}^{[y]}(\omega,t)S_{x}^{[s]}(\omega,t)=I(Y_{x}(\omega,t)=y,S_{x}(\omega,t)=s)$ as a property of indicator functions. For ease of notation and space considerations, we drop the $\omega$ in $Y(\omega,t)$, $S(\omega,t)$, $Y_{x}(\omega,t)$, $S_{x}(\omega,t)$, whenever the meaning is clear. The main paper and Online Supplement 1 assumes deterministic counterfactuals. The Online Supplement 2 provides the same results with stochastic counterfactuals and derives the same results. The Online Supplement 2 also provides results for the situation where there exists individuals $\omega$ whose variable $S(\omega,t)$ observed at time $t,$ but the outcome $Y(\omega,t)$ is censored. Table 8 presented at the end of this paper is used for all our proofs.

Proofs of identification of individual level always survivor causal

effects

We first present the proofs of identification of individual level always survivor causal effects.

Proof of Theorem 1

We prove the contrapositive. Assume that no individual $\omega$ of response type $Y_{1}(\omega,t)=y,$ $Y_{0}(\omega,t)=1-y,$ $S_{1}(\omega,t)=1$ and $S_{0}(\omega,t)=1$ exists in our population for a fixed $t\in T$ and $y\in\{0,1\}.$ Then for all individuals $\omega$ in our population $\Omega,$ $I(Y_{1}(\omega,t)=y,S_{1}(\omega,t)=1)+I(Y_{0}(\omega,t)=1-y,S_{0}(\omega,t)=1)\leq 1.$ Taking expectations,

[TABLE]

The first to second line follows from $(Y_{x}(t),S_{x}(t))\amalg X$ for all $t$ and the second to third line follows from consistency of counterfactuals. The same proof applies for any arbitrary $t\in T.$

Proof of Proposition 1

For any fixed $t\in T,$ taking expectation of $Y_{1}^{[1]}(\omega,t)S_{1}^{[1]}(\omega,t)+Y_{0}^{[0]}(\omega,t)S_{0}^{[1]}(\omega,t)-1$ and $Y_{1}^{[0]}(\omega,t)S_{1}^{[1]}(\omega,t)+Y_{0}^{[1]}(\omega,t)S_{1}^{[1]}(\omega,t)-1$ gives the result. A table with the relevant frequencies of the counterfactuals is provided below in Table 8 (see also Tables 1.3s in Online Supplement 2 to avoid computation). The same proof applies for any arbitrary $t\in T.$

Proof of Corollary 1

For any fixed $t\in T,$ consider from arguments presented in Theorem 1 and Proposition 1 (see Tables 1.3s and 1.4s in Online Supplement 2),

[TABLE]

Therefore,

[TABLE]

Consequently,

[TABLE]

This demonstrates $P_{1,1.1}^{r(t)}+P_{0,1.0}^{r(t)}-1$ serves as a lower bound for the counterfactual contrast

[TABLE]

The first term in this counterfactual contrast, $P^{c(t)}(1,0,1,1),$ is a lower bound of the proportion of individuals randomized at baseline that are always survivors (observed or censored) and for whom the treatment causes the outcome, because it does not include any censored individuals that are also always survivors and for whom the treatment causes the outcome. The term in the square brackets is an upper bound on the proportion of individual randomized at baseline that are always survivors (observed or censored) for whom the treatment prevents the outcome, because it effectively treats any censored individual that could possibly be an always survivor and for whom the treatment prevents the outcome is actually an always survivor and for whom the treatment prevents the outcome. To see that $P_{1,1.1}^{r(t)}+P_{0,1.0}^{r(t)}-1$ serves as a lower bound to $P^{c(t)}(1,0,1,1),$ note

[TABLE]

The proofs of the other bounds are similar. The same proof applies for any arbitrary $t\in T.$ This completes the proof.

Proof of Theorem 2A

We prove the contrapositive. For a fixed $t\in T,$ assume that no individual $\omega$ of response type $Y_{1}(\omega,t)=1,$ $Y_{0}(\omega,t)=0,$ $S_{1}(\omega,t)=1$ and $S_{0}(\omega,t)=1$ exists in our population. Then for all individuals $\omega$ in our population $\Omega,$ $I(Y_{0}(\omega,t)=0,S_{0}(\omega,t)=1)-I(Y_{1}(\omega,t)=0,S_{1}(\omega,t)=1)-I(Y_{1}(\omega,t)=3,S_{1}(\omega,t)=2)\leq 0.$ This last assertion is true after examining the counterfactual table provided in tables 2.1s-2.3s in Online Supplement 2. Taking expectations, we have

[TABLE]

The first to second line follows from $(Y_{x}(t),S_{x}(t))\amalg X$ and the second to third line follows from consistency of counterfactuals. The same proof applies for any arbitrary $t\in T.$ The other inequality associated with Theorem 2A can be derived similarly. This completes the proof.

The proofs of Theorem 2B and 2C can be derived similarly.

Proof of Proposition 2A

For a fixed integer $t\in T,$ taking expectation of $Y_{0}^{[1]}(\omega,t)S_{0}^{[1]}(\omega,t)-Y_{1}^{[1]}(\omega,t)S_{1}^{[1]}(\omega,t)-Y_{1}^{[3]}(\omega,t)S_{1}^{[2]}(\omega,t)$ and $Y_{0}^{[0]}(\omega,t)S_{0}^{[1]}(\omega,t)-Y_{1}^{[0]}(\omega,t)S_{1}^{[1]}(\omega,t)-Y_{1}^{[3]}(\omega,t)S_{1}^{[2]}(\omega,t)$ gives the result. A table with the relevant frequencies of the counterfactuals is provided below. In Table 8, set any probability that does not satisfy the monotonicity assumption $S_{1}^{[1]}(\omega,t)+S_{0}^{[0]}(\omega,t)\leq 1$ to zero. The same proof applies for any arbitrary $t\in T.$ This completes the proof.

The proofs of Proposition 2B and 2C can be derived similarly.

Proof of Corollary 2A

For a fixed $t\in T,$ from Table 8 (see also Table 2.3s in Online Supplement 2 to avoid computation), baseline randomization and consistency of counterfactuals, we have

[TABLE]

Therefore,

[TABLE]

Consequently,

[TABLE]

This demonstrates $P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)}-P_{3,2.1}^{r(t)}$ serves as a lower bound for the counterfactual contrast

[TABLE]

The first term in this counterfactual contrast, $P^{c(t)}(1,0,1,1),$ is a lower bound of the proportion of individuals that are always survivors (observed or censored) and for whom the treatment causes the outcome, because it does not include any censored individuals that are also always survivors and for whom the treatment causes the outcome. The term in the square brackets is an upper bound on the always survivors (observed or censored) for whom the treatment prevents the outcome, because it effectively treats any censored individual that could possibly be an always survivor and for whom the treatment prevents the outcome is actually an always survivor and for whom the treatment prevents the outcome. To see that $P_{0,1.0}^{r(t)}-P_{0,1.1}^{r(t)}-P_{3,2.1}^{r(t)}$ serves as a lower bound to $P^{c(t)}(1,0,1,1),$ note

[TABLE]

The proofs of the other bounds are similar. The same proof applies for any arbitrary $t\in T.$ This completes the proof.

The proofs of Corollary 2B and 2C can be derived similarly.

Proofs involving sensitivity analysis

We now present the proofs for the sensitivity analysis of monotonicity assumptions.

Proof of Theorem 3A

For a fixed $t\in T,$ from Table 8 (see also Table 2.3s in Online Supplement 2 to avoid computation), we have under consistency of counterfactuals and baseline randomization,

[TABLE]

Therefore,

[TABLE]

Take $d_{m}(t)=P^{c(t)}(2,1-y,0,1)-P^{c(t)}(1-y,2,1,0)-P^{c(t)}(3,2,2,0),$ then, we have

[TABLE]

Consequently, if

[TABLE]

where

[TABLE]

then,

[TABLE]

The same proof applies for any arbitrary $t\in T.$ This completes the proof.

An alternative proof would examine

[TABLE]

and then expand $P(Y_{0}(\omega,t)=1-y,S_{0}(\omega,t)=1),$ $P(Y_{1}(\omega,t)=1-y,S_{1}(\omega,t)=1),$ and $P(Y_{1}(\omega,t)=3,S_{1}(\omega,t)=2)$ over the remaining counterfactual variables. For example,

[TABLE]

Here, $\{(Y_{1},S_{1})\}$ denotes the joint compatible state space of the joint variables $Y_{1}(\omega,t)$ and $S_{1}(\omega,t).$ Explicitly, in our context, $\{(Y_{1},S_{1})\}=\{(0,1),(1,1),(2,0),(3,2)\}$ is the state of compatible realizations of the joint counterfactual random variable $(Y_{1},S_{1}).$ Section 5 and 8 of Online Supplement 2 provides this proof using stochastic counterfactuals. The proofs of Theorems 3B and 3C are provided in Section 12 of Online Supplement 2.

Proof of Theorem Generalized Always Survivor Causal Effect

Begin with the equality

[TABLE]

Under randomization and consistency of counterfactuals, the last expression in the above equality is equal to

[TABLE]

Now, use the law of total probability $P(A)=P(A,B)+P(A,B^{c}),$ where $B^{c}$ denotes the complement of $B.$ Consequently, this last expression is equal to

[TABLE]

Again, use the law of total probability $P(A)=P(A,B)+P(A,B^{c}),$ where $B^{c}$ denotes the complement of $B.$ Consequently, this last expression is equal to

[TABLE]

Simplifying this last expression, we have that $P\left(Y(t)\in y_{a},S(t)=1\mid X=1\right)+P(Y(t)\not\in y_{b},S(t)=1\mid X=0)-1$ is equal to

[TABLE]

We will label this last expression involving counterfactuals as expression D. Given that probability is greater than equal to zero and less than equal to 1, $P\left(Y(t)\in y_{a},S(t)=1,R(t)=1\mid X=1\right)+P(Y(t)\not\in y_{b},S(t)=1,R(t)=1\mid X=0)-1$ serves as a lower bound on

[TABLE]

Consequently, the expression $P\left(Y(t)\in y_{a},S(t)=1,R(t)=1\mid X=1\right)+P(Y(t)\not\in y_{b},S(t)=1,R(t)=1\mid X=0)-1$ also serves as a lower bound on

[TABLE]

from an application of the theorem of total probability.

This completes the proof for the the first part of the Theorem.

From the first part of the Theorem

[TABLE]

is equal to expression D, and to simplify expression D, we will use identities:

Identity one:

[TABLE]

Identity two :

[TABLE]

Identity three:

[TABLE]

Denote

[TABLE]

Now, the contrast

[TABLE]

is equal to expression D if and only if the following equality holds

[TABLE]

Denote

[TABLE]

out of space considerations. Note, $y\in[0,1).$

Note, the normalized principal stratum direct effect is

[TABLE]

Instead, we target

[TABLE]

Now, from

[TABLE]

We want to show that the next expression, which we denote expression A,

[TABLE]

is greater than or equal to one to show that

[TABLE]

is a lower bound to

[TABLE]

Through inspection, expression A is greater than or equal to one if and only if

[TABLE]

We denote

[TABLE]

as expression B.

Now, from the earlier part of the Theorem, we had the denominator of expression B is equal to expression D. The numerator of expression B is

[TABLE]

We have $P(S_{1}(t)R_{1}(t)=1,S_{0}(t)R_{0}(t)=1)\geq P\left(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{a},S_{1}(t)\cdot R_{1}(t)=1,S_{0}(t)\cdot R_{0}(t)=1\right)$ as $P(A)\geq P(A,B).$

Now,

[TABLE]

This in turn is equal to

[TABLE]

If we examine

[TABLE]

then the numerator is greater than the denominator as every term that is negative in the numerator is represented in the counterfactual representation of the denominator, and the only positive term in the numerator is greater than equal to the only positive term in the counterfactual representation of the denominator. Since expression B is greater than equal to 1, expression A is also greater than equal to 1.

This implies that

[TABLE]

serves a a lower bound on

[TABLE]

Now,

[TABLE]

is equal to

[TABLE]

This holds as

[TABLE]

This implies that

[TABLE]

is a lower bound on

[TABLE]

The expression

[TABLE]

is a function only of the observed data distribution, and has no terms that involve counterfactuals or potential outcomes.

We could only make this bound sharper if we could find a function of the observed data that was greater than 1 but less than expression A. There is no immediate way to find such a function of the observed data, because expression A is not point identified from randomization at baseline and no function of the observed data can be found that is guaranteed to be less than expression A and also greater than one for any observed data on $(S(t),Y(t),X).$

For the last part of the Theorem, we provide a sketch proof. The complete proof of this final result of the Theorem is provided in the proof of Theorem Generalized Missing Data Sensitivity Analysis.

For some $t\in T,$ $y_{a}\subset\mathbf{R}^{n_{y}}$ and $y_{b}\subset\mathbf{R}^{n_{y}}$ the numerator of the expression

[TABLE]

is a lower bound on $P(Y_{1}(t)\in y_{a},Y_{0}(t)\not\in y_{b},S_{1}(t)=1,S_{0}(t)=1)-P(Y_{1}(t)\notin y_{a},Y_{0}(t)\in y_{a},S_{1}(t)=1,S_{0}(t)=1).$ The expression

[TABLE]

is equal to

[TABLE]

The above expression is bounded from below with the following expression

[TABLE]

Conseqently, when $P(Y(t)\in y_{a},S(t)=1,R(t)=1\mid X=1)+P(Y(t)\not\in y_{b},S(t)=1,R(t)=1\mid X=0)>1,$ the expression

[TABLE]

also serves as a lower bound to

[TABLE]

which is equivalent to

[TABLE]

This completes the proof.

Proof of Generalized Always Survivor Causal Effect Monotonicity

We begin with the expression

[TABLE]

Under consistency of counterfactuals and randomization, this expression is equal to

[TABLE]

Now apply the law of total probability, $P(A)=\sum P(A,B_{i}).$

[TABLE]

Now again apply DeMorgan’s laws and $P(A\cup B)=P(A)+P(B)$ for disjoint $A$ and $B.$

[TABLE]

Now, again apply the law of total probability. Also, apply $Y_{1}(t)\in\star$ when $S_{1}(t)=0.$

[TABLE]

Simplifying and rearranging terms

[TABLE]

Under the monotonicity condition that there is no individual $\omega\in\Omega$ such that $(S_{1}(\omega,t),R_{1}(\omega,t))\not\in\{(1,1)\}$ and $(S_{0}(\omega,t),R_{0}(\omega,t))\in\{(1,1)\}.$

[TABLE]

is a lower bound on

[TABLE]

When $P(Y(t)\not\in y_{a},(S(t),R(t))\in\{(1,1)\}\mid X=0)>P(Y(t)\not\in y_{a},(S(t),R(t))\in\{(1,1)\}\mid X=1),$ then

[TABLE]

serves as a lower on

[TABLE]

and also

[TABLE]

which is equivalent to

[TABLE]

Proof of Theorem Generalized Missing Data Sensitivity Analysis

From the law of total probability, we have that

[TABLE]

is equal to

[TABLE]

and therefore,

[TABLE]

is a lower bound on

[TABLE]

We label this equality A. Now from the law of total probability, we have the following identities

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Therefore, we have that

[TABLE]

is equal to

[TABLE]

We label this equality as equality $B.$

Provided $P(S_{1}(t)=1,S_{0}(t)=1)>0,$ Equality B only holds if and only if

[TABLE]

Using the identity,

[TABLE]

this equation for the normalized principal stratum direct effect simplifies to

[TABLE]

Denote

[TABLE]

out of space considerations. Note, $x\in[0,1).$ The normalized principal stratum direct effect can be formulated to equal:

[TABLE]

We want to show that the next expression, which we denote expression A,

[TABLE]

is greater than or equal to one to show that

[TABLE]

is a lower bound to

[TABLE]

Through inspection, expression A is greater than or equal to one if and only if

[TABLE]

Now use the identity that $P(Y_{1}(t)=\star,Y_{0}(t)=\star,S_{1}(t)=0,S_{0}(t)=0)=P(S_{1}(t)=0,S_{0}(t)=0),$ and therefore,

[TABLE]

is equal to

[TABLE]

If we examine

[TABLE]

then the numerator is greater equal to the denominator as

[TABLE]

from the counterfactual representation of

[TABLE]

Also, both the numerator and denominator contain $-P(S_{1}(t)=0,S_{0}(t)=0),$ which means that

[TABLE]

This means that expression A is greater than equal to 1. This implies finally that

[TABLE]

Therefore,

[TABLE]

is a lower bound on

[TABLE]

provided $P(Y(t)\in y_{a},S(t)=1\mid X=1)+P(Y(t)\not\in y_{b},S(t)=1\mid X=0)-1>0.$

Now, note that

[TABLE]

Consequently, when $P(Y(t)\in y_{a},S(t)=1\mid X=1)+P(Y(t)\not\in y_{b},S(t)=1\mid X=0)-1>0$, the expression

[TABLE]

is a lower bound to

[TABLE]

which is equivalent to

[TABLE]

Note, the expression

[TABLE]

is a function only of the observed data distribution, and has no terms that involve counterfactuals or potential outcomes. However,

[TABLE]

cannot be calculated from the observed data if there is missing responses. As before,

[TABLE]

is equal to

[TABLE]

The parameters $P(Y(t)\in y_{a},S(t)=1,R(t)=0\mid X=1)+P(Y(t)\notin y_{b},S(t)=1,R(t)=0\mid X=0)$ and $P(S(t)=1,R(t)=0\mid X=1)-P\left(S(t)=1,R(t)=0\mid X=0\right)$ are sensitivity analysis parameters, because we cannot calculate any probability that involves joints of $(R(t)=0,S(t)=s),$ $(R(t)=0,Y(t)=y),$ or $(R(t)=0,S(t)=s,Y(t)=y)$ for $s\in\{0,1\}$ and $y\in\{0,1\}.$ The moment the individual drops out of the clinical trial, $Y(t)$ and $S(t)$ become undefined. We could only make this bound sharper if we could find a function of the observed data that was greater than 1 but less than expression A. There is no immediate way to find such a function of the observed data, because expression A is not point identified from randomization at baseline and no function of the observed data can be found that is guaranteed to be less than expression A and also greater than one for any observed data on $(S(t),Y(t),R(t),X).$ This completes the proof.

The expression

[TABLE]

is bounded from below from the expression

[TABLE]

Consequently, when $P(Y(t)\in y_{a},S(t)=1,R(t)=1\mid X=1)+P(Y(t)\notin y_{b},S(t)=1,R(t)=1\mid X=0)>1,$ the expression

[TABLE]

is a lower bound on

[TABLE]

which is equivalent to

[TABLE]

Proof of Theorem Generalized Missing Data and Monotonicity Sensitivity

Analysis

Begin with the expression

[TABLE]

Under randomization and consistency of counterfactuals this expression is equal to

[TABLE]

We now will use the law of total probability. The above counterfactual contrast is equal to

[TABLE]

We again apply the law of total probability. The above counterfactual contrast is equal to

[TABLE]

We again apply the law of total probability. The above counterfactual contrast is equal to

[TABLE]

Simplifying this expression we have the above expression is equal to

[TABLE]

and therefore the expression

[TABLE]

where

[TABLE]

is equal to

[TABLE]

Consequently, the expression

[TABLE]

is equal to the normalized principal stratum direct effect

[TABLE]

which is equivalent to

[TABLE]

This last part falls out by realizing that

[TABLE]

is equal to $P(S_{1}(t_{U})=1,S_{0}(t_{L})=1)$ under randomization and consistency of counterfactuals. This completes the proof.

Associated Tables

The tables drop the explicit dependence on $\omega$ and integer $t\in T,$ and use the shorthand $Y_{x}(\omega,t)=Y_{x}$ and $S_{x}(\omega,t)=S_{x}$ for $x\in\{0,1\}$ out of space considerations. Recall that we denote $Y_{x}^{[y]}(\omega,t)=I(Y_{x}(\omega,t)=y),$ and $S_{x}^{[s]}(\omega,t)=I(S_{x}(\omega,t)=s)$ for $y\in\{0,1,2,3\}$ and $s\in\{0,1,2\}.$ Likewise, we drop the dependence on $\omega$ and $t,$ and use shorthand notation $S_{x}^{[s]}$ and $Y_{x}^{[y]}$ instead of $S_{x}^{[s]}(\omega,t)$ and $Y_{x}^{[y]}(\omega,t)$ for $y\in\{0,1\}$ out of space considerations. The reader should always assume that all counterfactuals presented in the tables below are implicitly dependent on $\omega$ and $t$ even if the shorthand notation does not explicitly indicate such a dependence. The tables below provide a complete enumeration of the different counterfactual response types. Without monotonicity assumptions there are 16 different counterfactual response types, and then with the monotonicity assumption that $S_{1}^{[0]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ for all individuals in our population, the response types $\omega\in\{5,6\}$ are no longer possible. Under monotonicity assumption that $S_{1}^{[2]}(\omega,t)+S_{0}^{[1]}(\omega,t)\leq 1$ response types $\omega\in\{11,12\}$ in Table 8 are no longer possible.

All of our results can be derived using Table 8 only, but we provide further tables in Online Supplement 2 that could aid readers. The table below is for any fixed $t\in T.$ The counterfactual table presented below is the joint distribution of $(Y_{1}(\omega,t),Y_{0}(\omega,t),S_{1}(\omega,t),S_{0}(\omega,t)),$ where all joint counterfactuals $(Y_{1}(\omega,t),Y_{0}(\omega,t),S_{1}(\omega,t),S_{0}(\omega,t))$ are measured at the same fixed $t\in T.$

Data

A total of 338 individuals were randomized to Docetaxel and 336 individuals were randomized to Mitoxantrone. The full dataset is provided here to enable reproducibility of our results.

Survive and Cancer Progression

Survive and Cancer Did not Progress

Did not Survive

Censored