Surrogate Endpoint Evaluation: Principal Stratification Criteria and the Prentice Definition

1.1 Setup of randomized trial for assessing clinical efficacy

We consider a single clinical trial that randomizes n participants to active intervention (e.g. treatment or vaccine) versus a control intervention such as placebo, with Z the indicator of assignment to active intervention. Participants are followed for a fixed follow-up period for occurrence of the primary endpoint Y by time τ1 post-randomization, with Y the indicator of endpoint occurrence. For simplicity of exposition we assume no dropout during follow-up, though this could be accommodated straightforwardly under a random censoring assumption. Let S be the candidate surrogate endpoint measured at fixed time τ<τ1 post-randomization, which may be discrete or quantitative and may be multivariate. Let R be the indicator that S is measured; frequently case–cohort, case–control, or two-phase sampling designs are used that only measure S in a judiciously chosen subset. Let Yτ be the indicator of primary endpoint occurrence before the time τ for measuring S. The observed random variables are (Z,R,RS,Yτ,Y). Lastly, let S(z), R(z), Yτ(z), and Y(z) be the potential outcomes if assigned treatment z, for z=0,1, with W the vector of potential outcomes W≡(S(1),S(0),R(1),R(0),Yτ(1),Yτ(0),Y(1),Y(0)). We make the common assumptions for randomized clinical trials of SUTVA, ignorable treatment assignment (Z⊥W), the probability that S is observed in those with Yτ=0 (i.e. P(R=1|Yτ=0)) depends only on observed data (missing at random assumption), and that the (Zi,Wi) are iid, for i=1,…,n. The ignorable treatment assignment assumption will hold by design, and the missing at random assumption will hold by design if all subjects with Yτ=0 contribute a viable sample for potentially measuring S at the visit at τ.

1.2 Background: published definitions and criteria for a principal surrogate endpoint

Joffe and Greene [4] reviewed four frameworks for evaluating surrogate endpoints. The current article focuses on the principal stratification framework in comparison to the Prentice definition of a valid surrogate endpoint [2] (but not to the Prentice criteria). Prentice stated his definition as “a response variable for which a test of the null hypothesis of no relationship to the treatment groups under comparison is also a valid test of the corresponding null hypothesis based on the true endpoint.” As stated in the first paragraph of the Introduction, S is a valid surrogate endpoint if measurement of P(s|Z) alone can inform us about P(y|Z), and the Prentice definition implements this concept in hypothesis testing but not in estimation. In particular, in our notation above a valid Prentice surrogate S satisfies P(Y(1)=1)=P(Y(0)=1) if and only if P(S(1)≤s1)=P(S(0)≤s1) for all s1, or, equivalently based on observable random variables given the trial is randomized, as P(Y=1|Z=1)=P(Y=1|Z=0) if and only if P(S≤s1|Z=1)=P(S≤s1|Z=0) for all s1. This if and only if statement allows the Prentice definition to be divided into two components of Specificity and Sensitivity, where Specificity means that no treatment effect on Y implies no treatment effect on S and Sensitivity means that a treatment effect on Y implies a treatment effect on S. In Section 3 we develop some criteria for separately checking Specificity and Sensitivity.

This article focuses on evaluating a candidate surrogate from a single randomized clinical trial, for evaluating its quality for the same setting as that trial. As such, within the frequency framework of statistics, satisfaction of the Prentice definition means that if the surrogate endpoint is used in an identical trial, then inference of the treatment effect on the surrogate is guaranteed to provide correct inference (in the dichotomous sense of correctly accepting or rejecting the null hypothesis) about the treatment effect on the clinical endpoint. While not directly relevant for answering the important question of whether the surrogate will be valid for a new treatment in the same or new setting, such a result is still useful because it provides indirect evidence that the surrogate would approximately satisfy the Prentice definition for a new treatment if the treatment is similar to the original treatment (e.g. in the same drug class). If two candidate surrogates are assessed in a single efficacy trial and one satisfies the Prentice definition and one does not, then it may be rational to prioritize the Prentice surrogate as a study endpoint in subsequent Phase 1/2 trials of new and similar treatments that will constitute the basis for selecting the most promising new treatment to advance to the next efficacy trial.

Several papers have considered definitions and criteria for a useful principal surrogate endpoint. Within the potential outcomes framework of causal inference, Frangakis and Rubin [3] defined S to be a principal surrogate if every individual with a causal treatment effect on the clinical endpoint Y also has a causal treatment effect on the surrogate S (i.e. “causal necessity”). This definition states that a valid surrogate satisfies P(Yi(1)=1)=P(Yi(0)=1) for all subjects i with Si(1)=Si(0), which departs from the Prentice definition in (1) being required for all individuals and in (2) being unidirectional (instead of if and only if). Gilbert and Hudgens [5], focusing on what could be evaluated from the sampling scheme of a typical randomized trial, modified the causal necessity condition to “average causal necessity” (ACN), i.e. no average causal treatment effect on Y in the subpopulation with S(1)=S(0) and Yτ(1)=Yτ(0)=0; the latter condition was added to ensure that causal treatment effects on S are defined. ACN can be expressed in terms of the “principal effects” or “causal effect predictiveness” (CEP) surface, which is defined in terms of the clinical risks under each treatment assignment,

With h(x,y) a known contrast function satisfying h(x,y)=0 if and only if x=y, for example h(x,y)=x−y, the CEP surface is defined as

and ACN is expressed as CEP(s1,s0)=0 for all s1=s0. Gilbert and Hudgens [5] defined S to be a principal surrogate if ACN and average causal sufficiency (ACS) hold, where a one-sided version of ACS (relevant for active treatment versus control trials considered here) states that there exists a constant C≥0 such that the subgroup of subjects with a sufficient treatment effect on S, those with {S(1)=s1,S(0)=s0 with s1−s0>C}, has a beneficial causal effect on Y, i.e. CEP(s1,s0) has the sign indicating benefit. We refer to one-sided ACS with C=0 as one-sided strong ACS; note that for a biomarker satisfying ACN plus one-sided strong ACS, the subgroup with no individual causal effect on S has zero clinical treatment effect and the subgroup with positive individual causal effect on S has a beneficial clinical treatment effect. Thus Gilbert and Hudgen’s [5] definition of a strong (C=0) one-sided principal surrogate can be stated as P(Y(1)=1|S(1)=S(0),Yτ(1)=Yτ(0)=0)=P(Y(0)=1|S(1)=S(0),Yτ(1)=Yτ(0)=0) and P(Y(1)=1|S(1)>S(0),Yτ(1)=Yτ(0)=0)<P(Y(0)=1|S(1)>S(0),Yτ(1)=Yτ(0)=0).

Moreover, both Frangakis and Rubin [3] and Gilbert and Hudgens [5] expressed the concept that studying the whole CEP surface is important for evaluating the utility of a candidate principal surrogate; the former authors expressed this by stating that a more useful biomarker will have relatively more associative than dissociative effects, whereas the latter authors expressed this by stating that a more useful biomarker will have wide variability in the CEP surface over subgroups defined by (S(1),S(0)), i.e. the biomarker is a strong effect modifier.

The marginal CEP curve causal parameter, closely related to the CEP surface, is also useful for evaluating a principal surrogate, which contrasts the risks averaged over the distribution of S(0): mCEP (s1)≡h(mrisk1(s1),mrisk0(s1)), where

While ACN and ACS are not in general defined for this causal parameter, the “wide variability/strong effect modifier” principal surrogate criterion is operable, and identifiability is achieved under weaker assumptions [6]. Below we consider both the CEP and mCEP full-data causal parameters as useful quantities for evaluating and understanding principal surrogate quality.

Our results below use an additive difference contrast h(x,y)=x−y, with advantage that, under equal early clinical risk (EECR) defined below and a no-harm monotonicity assumption [P(Y(1)=1,Y(0)=0)=0], −CEP(s1,s0) has interpretation as a conditional probability of the disease being averted by assignment to Z=1: −CEP(s1,s0)=P(Y(1)=0,Y(0)=1|S(1)=s1,S(0)=s0,Yτ(1)=Yτ(0)=0); similarly −mCEP(s1)=P(Y(1)=0,Y(0)=1|S(1)=s1,Yτ(1)=Yτ(0)=0). Other principal surrogate evaluation literature has considered the CEP surface [7] or closely related causal parameters. Taylor et al. [8] studied the proportion associative (PA) summary measure of principal surrogate value,

which is the proportion of the study population with a beneficial clinical effect that also has a positive surrogate effect. (This definition assumes no clinical events before τ.) With the additive difference contrast h(x,y)=x−y and no-harm monotonicity defined above, straightforward calculation shows that PA = ∫s1>s0CEP(s1,s0)dp(s1,s0)CE, where P(s1,s0) is the joint cdf of S(1) and S(0) conditional on Yτ(1)=Yτ(0)=0 and CE ≡h(P(Y(1)=1),P(Y(0)=1)) is the overall clinical treatment effect. For the special case of binary S, Li, Taylor, and Elliott [9] studied the 16 causal parameters constituting the joint distribution of (S(1),S(0)) and (Y(1),Y(0)). Under the same assumptions given above, these parameters for Y(1)=0 and Y(0)=1 map to the CEP surface: P(S(1)=s1,S(0)=s0,Y(1)=0,Y(0)=1)=CEP(s1,s0)P(s1,s0)/CE for (s1,s0)∈{0,1}.

Additional work clarified the value or limitations of the above criteria for a biomarker’s utility as a principal surrogate and suggested new criteria. VanderWeele [10] showed that ACN can hold yet the treatment has a causal effect on Y not mediated through S. For example, this situation may occur if there are two independent biological mechanisms of clinical protection, one that operates directly through S and one that does not. On the positive side for ACN, VanderWeele [10] also showed that failure of ACN does imply that the treatment has a causal effect on Y not mediated through S; thus, ACN is a valid criterion to “disprove” full mediation but cannot affirm it. (Thus Frangakis and Rubin’s [3] principal surrogate definition is about a one-way implication, different from the if and only if implications of the Prentice definition.) In addition, Gilbert et al. [11] emphasized that, for the purpose of iteratively developing increasingly efficacious treatments, ACN and ACS may be less important for a useful principal surrogate than the strong effect modifier criterion that CEP(s1,s0) widely varies across subgroups defined by (s1,s0). Strong effect modification may occur in many ways not implying ACN nor ACS, and strong effect modification alone combined with an overall beneficial clinical treatment effect implies that there is at least one subgroup with relatively large clinical efficacy. Strong effect modification “sets the target” for future development of improved treatments, where the goal is to find refined treatments that generate S in the “high efficacy zone” for a greater percentage of active treatment recipients; combining these data results with context-dependent bridging assumptions (e.g. Pearl and Bareinboim [12] initiated a framework for combining data with bridging assumptions) would predict that the refined treatment would confer greater overall clinical efficacy. One way that wide variability could lead to erroneous bridging for improving a treatment would be if new subjects added to the high efficacy zone by the new treatment have a different distribution of clinical effect modifiers than the subgroup in the high efficacy zone in the original trial. Nevertheless, wide variability is a useful criterion for research areas that study a battery of biomarker endpoints as potential surrogates; this criterion may be used for prioritizing/ranking the biomarker endpoints to use in Phase 1/2 trials for comparing refined treatments and for selecting the most promising treatments to advance to the next efficacy trial.

Another criterion for a good surrogate endpoint is the original Prentice [2] definition of a valid replacement endpoint for the clinical endpoint, and below we provide results on the implications of ACN + one-sided strong ACS on the Prentice definition and vice versa. The results show how the implications depend on assumptions about causal treatment effects on the clinical endpoint before and after the biomarker is measured. These implications yield alternative criteria to the original Prentice criteria for checking the Prentice definition of a valid surrogate endpoint.

Many authors including Chen et al. [13], Ju and Geng [14], and VanderWeele [15] rightly assert that a reasonable surrogate endpoint should be assured to avoid the “surrogate paradox” pitfall, defined as the scenario where the effect of the treatment on the surrogate is positive, the surrogate and clinical outcomes are positively correlated, yet the overall clinical treatment effect CE indicates harm by the active treatment. Below we note scenarios, which commonly occur in practice, for which ACN plus one-sided strong ACS guarantee a “consistent surrogate” (defined as the surrogate paradox cannot happen).

The remainder of this article is organized as follows. Section 2 clarifies that principal surrogate analysis is essentially subgroup effect modification analysis. Section 3 provides results on ACN and one-sided ACS as criteria for checking the Prentice surrogate definition. Section 4 provides results on these criteria for checking a consistent surrogate. Section 5 illustrates the relationships with a Zoster vaccine (ZV) efficacy trial, Section 6 provides discussion, and the appendix contains proofs of results.

2.1 Connection to the treatment marker selection problem

Principal surrogate analysis is subgroup analysis (hence suggesting a name such as principal stratification effect modification analysis), with an objective to characterize how clinical treatment efficacy varies over subgroups, where these subgroups are defined by post-randomization principal strata (which by construction may be treated as baseline covariates) and possibly also by actual baseline covariates. The analysis essentially repeats the overall intention-to-treat analysis for each of a range of these subgroups, assessing the effect of treatment assignment on disease risk within each subgroup, and, like the overall analysis, provides little or no direct information about mechanisms or mediators of protection. As such, the principal surrogate problem has a close connection with the “treatment marker selection problem,” which has a goal to determine if and how clinical treatment efficacy varies over subgroups defined by biomarkers measured at baseline (e.g. Huang et al. [16]). While the statistical approaches for these two problems are highly related, the applications are partly overlapping and partly distinct; for instance, both fields seek to rank biomarker endpoints by their strength of effect modification and hence utility for treatment development, but, unlike the treatment marker selection field that often focuses on individual decision making for tailored allocation of therapy, the principal surrogate field has focused on different applications including the prediction of overall treatment efficacy from the biomarker distribution in a similar or new setting [17, 18]. In addition, the treatment marker selection field does not endeavor to identify “perfect” or “valid” treatment selection markers; rather it focuses on characterizing efficacy over subgroups and the ranking of biomarkers by the strength of effect modification. Similarly, principal surrogate evaluation is primarily about comparing candidate surrogates and ranking them by the degree of their utility as effect modifiers, and the field should not be dominated by the objective to identify perfect/valid surrogates. Nevertheless, the joint criterion of ACN together with strong ACS has particular value in checking the Prentice definition or the individual components of the Prentice definition as described below.

The analogy with the treatment marker selection literature also suggests that if very strong baseline effect modifiers exist, then it may be unimportant to develop a biomarker response effect modifier – one can simply predict clinical treatment effects based on actual baseline variables, avoiding the identifiability challenges of the principal stratification framework (Ross Prentice has voiced this point). While true, in practice a response to treatment may be a stronger effect modifier, motivating principal surrogacy assessment, and many such examples exist. The analogy also raises the question as to when the principal strata subgroups are identifiable from the observable data, as an affirmative answer to this question places the principal stratification problem much closer to the treatment marker selection problem where subgroups are obviously directly observable.

2.2 Is the principal stratum for inference observable?

A key issue for the utility of principal stratification research in general is whether the principal stratum for inference is observable versus latent and never observable. Many principal strata of interest in a variety of applications are not observable, rendering the approach unhelpful for decision making for individual patients or for health policy (e.g., Joffe [19]). However, for present application there is a special case where the principal stratum for inference is identifiable from the observable data, the “Constant Biomarker” scenario (i.e. the conditional distribution S(0)|Yτ(0)=0 is degenerate), which has been considered in several papers. In Case CB the CEP surface collapses to the mCEP curve, and ACN and ACS may be assessed based on the mCEP curve. Case CB is important because the principal strata subgroups are specified by {S(1)=s1,Yτ(1)=Yτ(0)=0}, which are identifiable (equating to {S(1)=s1,Yτ(1)=0}) under the no early-protection monotonicity assumption P(Yτ(1)=0,Yτ(0)=1)=0. Where this assumption fails, the principal strata subgroups will be approximately equal to the identifiable subgroups if at most a very small subgroup receives clinical protection by τ, which is especially likely to hold if the rate of disease by τ is much less than the rate of disease after τ.

While Case CB has been motivated by vaccine efficacy trials, it also may occur in general active treatment versus control randomized trials with S defined as the difference in a biomarker readout between time τ and the time of randomization [20]. In this scenario, if τ is reasonably close to baseline and the passage of time from baseline to τ is not expected to alter the biomarker during that time for subjects assigned Z=0, then it may be reasonable to set S(0)=0 for all subjects. While in many applications the measured differences S may have some variability about 0, sometimes this scatter may be assumed to be random measurement error, in which case it is of interest to assess the “de-noised” variable S as a principal surrogate that does have S(0)=0 for all subjects. Another advantage of this “difference biomarker” scenario is that the baseline biomarker may be predictive of S, which aids identifiability and efficiency of estimation of the CEP surface and mCEP curve via the baseline immunogenicity predictor (BIP) augmented trial design [5, 17, 18, 20–23].

In sum, if the principal strata subgroups are identifiable from observable random variables, then principal stratification effect modification assessment has utility similar to baseline covariate subgroup analysis of effect modification, whereas otherwise, the utility is reduced, but still present for the purposes of ranking candidate biomarkers and for providing inputs into bridging formulas for predicting overall treatment efficacy.

3.1 Prentice definition

A criterion for a good principal surrogate endpoint is satisfaction of Prentice’s [2] definition as a valid replacement endpoint for the clinical endpoint. This definition may be expressed as perfect population-level specificity and sensitivity of the surrogate. Henceforth we use h(x,y)=x−y such that CE =P(Y(1)=1)−P(Y(0)=1) and CE <0 indicates clinical benefit, and CEP(s1,s0)=risk1(s1,s0)−risk0(s1,s0).

We define two-sided and one-sided versions of Specificity and Sensitivity as follows:

We use the contrapositive forms of two-sided Specificity and one-sided Specificity to distinguish them: S(1)≠d S(0)⇒CE≠0 and S(1)>st S(0)⇒CE<0, respectively. In the above definitions >st indicates stochastically larger, i.e. S(1)>st S(0) means that P(S(1)>s)≥P(S(0)>s) for all s with > for at least one s. Because CE is an intention-to-treat parameter, in the above definitions we include “undefined” (∗) as one of the values of S, such that S(1)=d S(0) means that P(S(1)≤s)=P(S(0)≤s) for all defined s and P(S(1)=∗)=P(S(0)=∗), where this last equality is equivalent to no early average clinical treatment effect P(Yτ(1)=1)=P(Yτ(0)=1).

Specificity means that rejecting the null hypothesis of no treatment effect on the surrogate implies a treatment effect on the clinical endpoint (CE ≠0 or a one-sided version), whereas Sensitivity means that accepting the null hypothesis of no treatment effect on the surrogate implies no treatment effect on the clinical endpoint (CE=0).

3.2 Overall clinical efficacy averaged over the CEP surface

Criteria for checking Specificity and Sensitivity may be derived solely based on observable random variables, without the need for potential outcomes, following Prentice [2] and subsequent work. However, in this work we study the relationship of Specificity and Sensitivity to ACN and one-sided ACS, which requires potential outcomes notation. In Section 5 we provide an example where principal stratification effect modification analysis supports ACN + one-sided strong ACS for a biomarker endpoint, generating the question of what does this imply about whether Specificity and/or Sensitivity hold? As a preliminary step, we partition CE as a weighted average of the CEP surface across subgroups. The results are developed for the additive difference contrast function h(x,y)=x−y; additional research would be needed for alternative contrasts. For concreteness in the following results we suppose S is discrete with J levels {0,…,J−1} (in addition to the level S=∗). While we present the results for discrete S, they carry over to the case of continuous S by replacing sums with integrals.

Define P(s1,s0)≡P(S(1)=s1,S(0)=s0|Yτ(1)=Yτ(0)=0) for (s1,s0)∈{0,…,J−1}×{0,…,J−1}, P∗(j,k)=P(Yτ(1)=j,Yτ(0)=k) for j,k∈{0,1}×{0,1}, and Pj∗=P(Yτ(j)=1) for j=0,1. We refer to the subgroups defined by {Yτ(1)=0,Yτ(0)=0}, {Yτ(1)=0,Yτ(0)=1} and {Yτ(1)=1,Yτ(0)=0} as the early-always-at-risk, early-protected and early-harmed principal strata, respectively. In addition, define

for j,k∈{0,1}×{0,1}.

Throughout we assume P∗(0,0)>0, which should be safe to assume in almost all meaningful applications for evaluating a surrogate endpoint. We state a result for easy reference in the forthcoming results.

3.3 Results on the relationship of ACN and one-sided ACS to Specificity and Sensitivity

We consider a menu of assumptions that will be selected from to infer results.

Equal Early Clinical Risk (EECR): P(Yτ(1)=Yτ(0))=1, i.e. treatment has no early clinical effect for any individual

Early No-Harm Monotonicity (ENHM): P∗(1,0)≡P(Yτ(1)=1,Yτ(0)=0)=0, i.e. treatment does not cause early harm for any individual (the early-harmed subgroup is empty)

Population Early Monotonicity (PEM): CE(1,0)P∗(1,0)+CE (0,1)P∗(0,1)=P(Y(1)=1,Yτ(1)≠Yτ(0))−P(Y(0)=1,Yτ(1)≠Yτ(0))≤0, i.e. the union of the early-protected and early-harmed subgroups does not have population-level clinical harm

No Negative Marker Effects (NNMEs): P(S(1)≥S(0)|Yτ(1)=Yτ(0)=0)=1, i.e. active treatment versus control does not reduce the biomarker for any individual in the early-always-at-risk subgroup

Monotonicity: CEP(s1,s0)≤0 for all subgroups defined by biomarker levels {S(1)=s1,S(0)=s0}∈{0,…,J−1}×{0,…,J−1}, i.e. treatment does not cause harm for any individual in the early-always-at-risk subgroup

Case CB: P(S(0)=0)=1

The following results attain, with proofs in the appendix. For these results we re-define ACN and one-sided ACS slightly as follows. ACN is CEP(s1,s1)=0 for all s1 with P(s1,s1)>0 and one-sided ACS is CEP(s1,s0)<0 for all s1−s0>C for some C≥0 and P(s1,s0)>0. The results are organized by the strength of the assumption about early clinical treatment effects, from strongest to weakest.

Result 1 (Under EECR): EECR + ACN + Case CB imply Sensitivity. Conversely, EECR + Sensitivity + Case CB imply ACN. Apart from Case CB, EECR + ACN do not imply Sensitivity and EECR + Sensitivity do not imply ACN, even under all four of the extra assumptions PEM + NNMEs + Monotonicity + Case CB.

EECR + ACN + one-sided strong ACS imply Specificity under any of NNMEs, Monotonicity, or Case CB. Conversely, EECR + Specificity + Sensitivity do not imply one-sided ACS for any C≥0, even under all four of the extra assumptions.

Result 2 (Under ENHM): ENHM + ACN do not imply Sensitivity even under all four of the extra assumptions. Conversely, ENHM + Sensitivity + Monotonicity + Case CB imply ACN.

Similar to Result 1, ENHM + ACN + one-sided strong ACS imply Specificity under any of NNMEs, Monotonicity, or Case CB, whereas ENHM + Specificity + Sensitivity do not imply one-sided ACS for any C≥0 even under all four of the extra assumptions.

Result 3 (General): ACN does not imply Sensitivity even under all four of the extra assumptions. Conversely, Sensitivity + PEM + Monotonicity + Case CB imply ACN.

ACN + one-sided strong ACS + PEM imply Specificity under any of NNMEs, Monotonicity, or Case CB. As for Results 1 and 2, Specificity + Sensitivity do not imply one-sided ACS for any C≥0 even under all four of the extra assumptions.

Results 1 and 2 show that the principal surrogate conditions can be used to check the two parts of the Prentice definition. They show that EECR + Case CB are needed for inferring the full Prentice definition from ACN and one-sided strong ACS, where relaxing either one loses the implication. Results 1 and 2 also show the importance of EECR for the principal surrogate criteria to have implications on the Prentice definition (required for inferring Sensitivity), even under all four extra assumptions PEM, NNMEs, Monotonicity, and Case CB. Results 1 and 2 also show that the Prentice definition does not imply ACS even under many possible assumptions; the basic reason is that there are many ways for CE ≠0 with CEP(s1,s0) zero for some s1≠s0 and below zero for other s1≠s0.

A useful application of Result 3 is that in general applications where PEM and NNMEs or Monotonicity hold (which is often plausible), if the estimated vaccine efficacy curve takes the classic shape of being near zero at s1=0 (supporting ACN) and rising above zero for positive values s1>0 (e.g. as in our example illustrated in Figure 2), then one may conclude Specificity. That is, a classic vaccine efficacy curve indicates that an inference of beneficial overall vaccine efficacy follows from the observation that vaccine recipients tend to have higher biomarker responses than placebo recipients.

Next we state Result 1 for the special case that S is binary. The results on implications of ACN and ACS for Sensitivity and Specificity are unchanged, whereas the reverse implications are strengthened. In contrast, Results 2 and 3 are unchanged for S binary compared to S categorical with more than two categories.

Result 1-Binary (Binary S Under EECR): In the special case of S binary and EECR + Case CB, ACN implies Sensitivity and Sensitivity implies ACN. In addition ACN plus one-sided strong ACS imply Specificity and Sensitivity + Specificity imply one-sided strong ACS.

Result 1-Binary shows that EECR + Case CB + S binary constitutes a scenario where both principal surrogate conditions hold if and only if the Prentice definition holds. For a binary S the Prentice definition does not have implications on ACS if EECR is relaxed, however, further highlighting the importance of EECR.

3.4 Results under minor violations of Case CB and EECR

In the example described in Section 5, there may be minor violations of the Case CB and EECR assumptions, raising the question of whether the results are approximately correct under such violations. We state a variant version of Result 1 to address this question with proof in the appendix, and note that the other results have similar properties under minor violations. We use the following extension of the notation.

Define Case CB-ε as P(S(0)>0)=ε for ε a small positive constant, EECR-ε as P(Yτ(1)≠Yτ(0))=ε, ENHM-ε as P(Yτ(1)=1,Yτ(0)=0)=ε, Sensitivity-ε as S(1)=d S(0)⇒CE→0 as ε→0, one-sided Specificity-ε as S(1)>st S(0)⇒CE→c as ε→0 for some negative constant c, ACN-ε as CEP(s1,s1)→0 as ε→0 for all s1∈{0,…,J−1}, and ACN-ε(0,0) as CEP(0,0)→0 as ε→0.

Result 4 (Result 1 Under Minor Violations of Case CB and EECR): EECR + ACN-ε + Case CB-ε imply Sensitivity-ε. Conversely, EECR + Sensitivity-ε + Case CB-ε imply ACN-ε(0,0) but not ACN-ε. EECR + ACN-ε + one-sided strong ACS imply Specificity-ε under any of NNMEs, Monotonicity, or Case CB-ε. The same implications hold replacing EECR with EECR-ε.

Result 4 implies that the principal stratification criteria do correctly check the Prentice definition under minor violations converging to zero in that Sensitivity-ε and Specificity-ε hold when Case CB is relaxed to Case CB-ε and EECR is relaxed to EECR-ε. In addition, while Result 4 shows that the Prentice definition does not imply ACN-ε if Case CB is minorly violated, it shows that the Prentice definition does imply ACN-ε(0,0), which may what matter in practice given that the principal stratum {S(1)=S(0)=0} constitutes the only causal necessity principal stratum containing study subjects as ε→0. See Appendix for a proof of Result 4.

Result 2 extends to a result where ENHM-ε + Sensitivity-ε + Monotonicity + Case CB-ε imply ACN-ε and ENHM-ε + ACN-ε + 1-sided strong ACS imply Specificity-ε under any of NNMEs, Monotonicity, or Case CB-ε. Result 3 extends to a result where Sensitivity-ε + PEM + Monotonicity + Case CB-ε imply ACN-ε. Result 1-Binary extends to a result where, for S binary and assuming EECR-ε + Case CB-ε, ACN-ε implies Sensitivity-ε and Sensitivity-ε implies ACN-ε(0,0) but not ACN-ε; moreover ACN-ε plus one-sided strong ACS imply Specificity-ε and Sensitivity-ε + Specificity-ε imply one-sided strong ACS.

3.5 Interpretation and testability of the assumptions

The first two assumptions EECR and ENHM are about the effect of treatment on Y before the biomarker is measured. The stronger assumption EECR assumes no effect for any individual, and has been used for all but one paper on evaluating a principal surrogate, given the great help it provides toward identifying the CEP surface and the marginal CEP curve. Wolfson and Gilbert [6] considered sensitivity analysis methods that relax EECR to ENHM or to no assumption about early treatment effects. EECR and ENHM are not fully testable but have testable implications, e.g. they can be rejected by finding early clinical treatment effects overall or in subgroups.

PEM is only relevant if EECR fails, as under EECR P∗(1,0)=P∗(0,1)=0, such that CE(1,0) and CE(0,1) are irrelevant, as treatment effects in empty subgroups. There are no obvious testable implications of PEM. It holds under the no-harm monotonicity assumption considered above. Without this monotonicity assumption, it may be relatively plausible in settings where the early-protected subgroup is much larger than the early-harmed subgroup and there is reason to expect that the early protected also receive some later protection. NNMEs will be plausible in many active versus control trials and can be partially checked by comparing the distributions of S(1)|Yτ(1)=0 and S(0)|Yτ(0)=0. Monotonicity will be more plausible in settings with higher overall efficacy and can be partially checked similarly to checking ENHM. Case CB can be checked by examining the distribution of S(0)|Yτ(0)=0.