Multiple Imputation of Missing Composite Outcomes in Longitudinal Data

In longitudinal randomised trials and observational studies within a medical context, a composite outcome—which is a function of several individual patient-specific outcomes—may be felt to best represent the outcome of interest. As in other contexts, missing data on patient outcome, due to patient drop-out or for other reasons, may pose a problem. Multiple imputation is a widely used method for handling missing data, but its use for composite outcomes has been seldom discussed. Whilst standard multiple imputation methodology can be used directly for the composite outcome, the distribution of a composite outcome may be of a complicated form and perhaps not amenable to statistical modelling. We compare direct multiple imputation of a composite outcome with separate imputation of the components of a composite outcome. We consider two imputation approaches. One approach involves modelling each component of a composite outcome using standard likelihood-based models. The other approach is to use linear increments methods. A linear increments approach can provide an appealing alternative as assumptions concerning both the missingness structure within the data and the imputation models are different from the standard likelihood-based approach. We compare both approaches using simulation studies and data from a randomised trial on early rheumatoid arthritis patients. Results suggest that both approaches are comparable and that for each, separate imputation offers some improvement on the direct imputation of a composite outcome.     

Assessing treatment effect heterogeneity in clinical trials with blocked binary outcomes

This paper addresses treatment effect heterogeneity (also referred to, more compactly, as 'treatment heterogeneity') in the context of a controlled clinical trial with binary endpoints. Treatment heterogeneity, variation in the true (causal) individual treatment effects, is explored using the concept of the potential outcome. This framework supposes the existance of latent responses for each subject corresponding to each possible treatment. In the context of a binary endpoint, treatment heterogeniety may be represented by the parameter, pi2, the probability that an individual would have a failure on the experimental treatment, if received, and would have a success on control, if received. Previous research derived bounds for pi2 based on matched pairs data. The present research extends this method to the blocked data context. Estimates (and their variances) and confidence intervals for the bounds are derived. We apply the new method to data from a renal disease clinical trial. In this example, bounds based on the blocked data are narrower than the corresponding bounds based only on the marginal success proportions. Some remaining challenges (including the possibility of further reducing bound widths) are discussed.     

Solutions for surrogacy validation with longitudinal outcomes for a gene therapy

Valid surrogate endpoints S can be used as a substitute for a true outcome of interest T to measure treatment efficacy in a clinical trial. We propose a causal inference approach to validate a surrogate by incorporating longitudinal measurements of the true outcomes using a mixed modeling approach, and we define models and quantities for validation that may vary across the study period using principal surrogacy criteria. We consider a surrogate-dependent treatment efficacy curve that allows us to validate the surrogate at different time points. We extend these methods to accommodate a delayed-start treatment design where all patients eventually receive the treatment. Not all parameters are identified in the general setting. We apply a Bayesian approach for estimation and inference, utilizing more informative prior distributions for selected parameters. We consider the sensitivity of these prior assumptions as well as assumptions of independence among certain counterfactual quantities conditional on pretreatment covariates to improve identifiability. We examine the frequentist properties (bias of point and variance estimates, credible interval coverage) of a Bayesian imputation method. Our work is motivated by a clinical trial of a gene therapy where the functional outcomes are measured repeatedly throughout the trial.     

Estimating treatment effect heterogeneity for binary outcomes via Dirichlet multinomial constraints

In a randomized two-group parallel trial the mean causal effect is typically estimated as the difference in means or proportions for patients receiving, say, either treatment (T) or control (C). Treatment effect heterogeneity (TEH), or unit-treatment interaction, the variability of the causal effect (defined in terms of potential outcomes) across individuals, is often ignored. Since only one of the outcomes, either Y(T) or Y(C), is observed for each unit in such studies, the TEH is not directly estimable. For convenience, it is often assumed to be minimal or zero. We are particularly interested in the 'treatment risk' for binary outcomes, that is, the proportion of individuals who would succeed on C but fail on T. Previous work has shown that the treatment risk can be bounded (Albert, Gadbury and Mascha, 2005), and that the confidence interval width around it can be narrowed using clustered or correlated data (Mascha and Albert, 2006). Without further parameter constraints, treatment risk is unidentifiable. We show, however, that the treatment risk can be directly estimated when the four underlying population counts comprising the joint distribution of the potential outcomes, Y(T) and Y(C), follow constraints consistent with the Dirichlet multinomial. We propose a test of zero treatment risk and show it to have good size and power. Methods are applied to both a randomized as well as a non-randomized study. Implications for medical decision-making at the policy and individual levels are discussed.     

Mixed effects logistic regression models for multiple longitudinal binary functional limitation responses with informative drop-out and confounding by baseline outcomes

In the context of analyzing multiple functional limitation responses collected longitudinally from the Longitudinal Study of Aging (LSOA), we investigate the heterogeneity of these outcomes with respect to their associations with previous functional status and other risk factors in the presence of informative drop-out and confounding by baseline outcomes. We accommodate the longitudinal nature of the multiple outcomes with a unique extension of the nested random effects logistic model with an autoregressive structure to include drop-out and baseline outcome components with shared random effects. Estimation of fixed effects and variance components is by maximum likelihood with numerical integration. This shared parameter selection model assumes that drop-out is conditionally independent of the multiple functional limitation outcomes given the underlying random effect representing an individual's trajectory of functional status across time. Whereas it is not possible to fully assess the adequacy of this assumption, we assess the robustness of this approach by varying the assumptions underlying the proposed model such as the random effects structure, the drop-out component, and omission of baseline functional outcomes as dependent variables in the model. Heterogeneity among the associations between each functional limitation outcome and a set of risk factors for functional limitation, such as previous functional limitation and physical activity, exists for the LSOA data of interest. Less heterogeneity is observed among the estimates of time-level random effects variance components that are allowed to vary across functional outcomes and time. We also note that. under an autoregressive structure, bias results from omitting the baseline outcome component linked to the follow-up outcome component by subject-level random effects.     

Development of a Core Set of Outcomes for Randomized Controlled Trials with Multiple Outcomes – Example of Pulp Treatments of Primary Teeth for Extensive Decay in Children

Objectives

Evidence-based comparisons of interventions can be challenging because of the diversity of outcomes in randomized controlled trials (RCTs). We aimed to describe outcomes in RCTs assessing pulp treatments for primary teeth and to develop a core set of component outcomes to be part of composite outcome defining the failure of a pulp treatment.

Methods

We systematically reviewed articles of RCTs comparing pulp treatments for primary molars published up to February 2012. We abstracted all outcomes assessed in each trial, then used a small-group consensus process to group similar outcomes, which were reduced to a composite outcome of failure of a pulp treatment by a 3-round Delphi process involving expert authors and dentists.

Results

We included 47 reports of RCTs in the review, for 83 reported outcomes (median 11 outcomes per RCT). These outcomes were grouped into 24 overarching outcome categories. We contacted 210 experts for the Delphi process and 25% to 30% participated. The process identified the following 5 component outcomes as part of a composite outcome of failure of a pulp treatment: soft-tissue pathology, pain, pathologic mobility, pathologic radiolucency and pathologic root resorption.

Conclusions

RCTs of pulp treatments for primary teeth investigate diverse outcomes. Our consensus process, involving clinicians but no patient, allowed for compiling a core set of component outcomes to define the composite outcome failure of a pulp treatment for primary teeth.     

Evaluating treatment effects in group sequential multivariate longitudinal studies with covariate adjustment

Jeffries et al. (2018) investigated testing for a treatment difference in the setting of a randomized clinical trial with a single outcome measured longitudinally over a series of common follow-up times while adjusting for covariates. That paper examined the null hypothesis of no difference at any follow-up time versus the alternative of a difference for at least one follow-up time. We extend those results here by considering multivariate outcome measurements, where each individual outcome is examined at common follow-up times. We consider the case where there is interest in first testing for a treatment difference in a global function of the outcomes (e.g., weighted average or sum) with subsequent interest in examining the individual outcomes, should the global function show a treatment difference. Testing is conducted for each follow-up time and may be performed in the setting of a group sequential trial. Testing procedures are developed to determine follow-up times for which a global treatment difference exists and which individual combinations of outcome and follow-up time show evidence of a difference while controlling for multiplicity in outcomes, follow-up, and interim analyses. These approaches are examined in a study evaluating the effects of tissue plasminogen activator on longitudinally obtained stroke severity measurements.     

Analysis of clustered binary outcomes using within-cluster paired resampling

Conditional logistic regression (CLR) is useful for analyzing clustered binary outcome data when interest lies in estimating a cluster-specific exposure parameter while treating the dependency arising from random cluster effects as a nuisance. CLR aggregates unmeasured cluster-specific factors into a cluster-specific baseline risk and is invalid in the presence of unmodeled heterogeneous covariate effects or within-cluster dependency. We propose an alternative, resampling-based method for analyzing clustered binary outcome data, within-cluster paired resampling (WCPR), which allows for within-cluster dependency not solely due to baseline heterogeneity. For example, dependency may be in part caused by heterogeneity in response to an exposure across clusters due to unmeasured cofactors. When both CLR and WCPR are valid, our simulations suggest that the two methods perform comparably. When CLR is invalid, WCPR continues to have good operating characteristics. For illustration, we apply both WCPR and CLR to a periodontal data set where there is heterogeneity in response to exposure across clusters.     

Modulation of aggressive behaviour by fighting experience: mechanisms and contest outcomes

Experience in aggressive contests often affects behaviour during, and the outcome of, later contests. This review discusses evidence for, variations in, and consequences of such effects. Generally, prior winning experiences increase, and prior losing experiences decrease, the probability of winning in later contests, reflecting modifications of expected fighting ability. We examine differences in the methodologies used to study experience effects, and the relative importance and persistence of winning and losing experiences within and across taxa. We review the voluminous, but somewhat disconnected, literature on the neuroendocrine mechanisms that mediate experience effects. Most studies focus on only one of a number of possible mechanisms without providing a comprehensive view of how these mechanisms are integrated into overt behaviour. More carefully controlled work on the mechanisms underlying experience effects is needed before firm conclusions can be drawn.Behavioural changes during contests that relate to prior experience fall into two general categories. Losing experiences decrease willingness to engage in a contest while winning experiences increase willingness to escalate a contest. As expected from the sequential assessment model of contest behaviour, experiences become less important to outcomes of contests that escalate to physical fighting.A limited number of studies indicate that integration of multiple experiences can influence current contest behaviour. Details of multiple experience integration for any species are virtually unknown. We propose a simple additive model for this integration of multiple experiences into an individual's expected fighting ability. The model accounts for different magnitudes of experience effects and the possible decline in experience effects over time.Predicting contest outcomes based on prior experiences requires an algorithm that translates experience differences into contest outcomes. We propose two general types of model, one based solely on individual differences in integrated multiple experiences and the other based on the probability contests reach the escalated phase. The difference models include four algorithms reflecting possible decision rules that convert the perceived fighting abilities of two rivals into their probabilities of winning. The second type of algorithm focuses on how experience influences the probability that a subsequent contest will escalate and the fact that escalated contests may not be influenced by prior experience. Neither type of algorithm has been systematically investigated.Finally, we review models for the formation of dominance hierarchies that assume that prior experience influences contest outcome. Numerous models have reached varied conclusions depending on which factors examined in this review are included. We know relatively little about the importance of and variation in experience effects in nature and how they influence the dynamics of aggressive interactions in social groups and random assemblages of individuals. Researchers should be very active in this area in the next decade. The role of experience must be integrated with other influences on contest outcome, such as prior residency, to arrive at a more complete picture of variations in contest outcomes. We expect that this integrated view will be important in understanding other types of interactions between individuals, such as mating and predator-prey interactions, that also are affected significantly by prior experiences.     

Intra-tumor heterogeneity of MLH1 promoter methylation revealed by deep single molecule bisulfite sequencing

A single tumor may contain cells with different somatic mutations. By characterizing this genetic heterogeneity within tumors, advances have been made in the prognosis, treatment and understanding of tumorigenesis. In contrast, the extent of epigenetic intra-tumor heterogeneity and how it influences tumor biology is under-explored. We have characterized epigenetic heterogeneity within individual tumors using next-generation sequencing. We used deep single molecule bisulfite sequencing and sample-specific DNA barcodes to determine the spectrum of MLH1 promoter methylation across an average of 1000 molecules in each of 33 individual samples in parallel, including endometrial cancer, matched blood and normal endometrium. This first glimpse, deep into each tumor, revealed unexpectedly heterogeneous patterns of methylation at the MLH1 promoter within a subset of endometrial tumors. This high-resolution analysis allowed us to measure the clonality of methylation in individual tumors and gain insight into the accumulation of aberrant promoter methylation on both alleles during tumorigenesis.     

Covariation between the physiological and behavioral components of pathogen transmission: host heterogeneity determines epidemic outcomes

Although heterogeneity in contact rate, physiology, and behavioral response to infection have all been empirically demonstrated in host–pathogen systems, little is known about how interactions between individual variation in behavior and physiology scale‐up to affect pathogen transmission at a population level. The objective of this study is to evaluate how covariation between the behavioral and physiological components of transmission might affect epidemic outcomes in host populations. We tested the consequences of contact rate covarying with susceptibility, infectiousness, and infection status using an individual‐based, dynamic network model where individuals initiate and terminate contacts with conspecifics based on their behavioral predispositions and their infection status. Our results suggest that both heterogeneity in physiology and subsequent covariation of physiology with contact rate could powerfully influence epidemic dynamics. Overall, we found that 1) individual variability in susceptibility and infectiousness can reduce the expected maximum prevalence and increase epidemic variability; 2) when contact rate and susceptibility or infectiousness negatively covary, it takes substantially longer for epidemics to spread throughout the population, and rates of epidemic spread remained suppressed even for highly transmissible pathogens; and 3) reductions in contact rate resulting from infection‐induced behavioral changes can prevent the pathogen from reaching most of the population. These effects were strongest for theoretical pathogens with lower transmissibility and for populations where the observed variation in contact rate was higher, suggesting that such heterogeneity may be most important for less infectious, more chronic diseases in wildlife. Understanding when and how variability in pathogen transmission should be modelled is a crucial next step for disease ecology.     

Looking for a needle in a haystack: inference about individual fitness components in a heterogeneous population

Studies of wild vertebrates have provided evidence of substantial differences in lifetime reproduction among individuals and the sequences of life history ‘states’ during life (breeding, nonbreeding, etc.). Such differences may reflect ‘fixed’ differences in fitness components among individuals determined before, or at the onset of reproductive life. Many retrospective life history studies have translated this idea by assuming a ‘latent’ unobserved heterogeneity resulting in a fixed hierarchy among individuals in fitness components. Alternatively, fixed differences among individuals are not necessarily needed to account for observed levels of individual heterogeneity in life histories. Individuals with identical fitness traits may stochastically experience different outcomes for breeding and survival through life that lead to a diversity of ‘state’ sequences with some individuals living longer and being more productive than others, by chance alone. The question is whether individuals differ in their underlying fitness components in ways that cannot be explained by observable ‘states’ such as age, previous breeding success, etc. Here, we compare statistical models that represent these opposing hypotheses, and mixtures of them, using data from kittiwakes. We constructed models that accounted for observed covariates, individual random effects (unobserved heterogeneity), first‐order Markovian transitions between observed states, or combinations of these features. We show that individual sequences of states are better accounted for by models incorporating unobserved heterogeneity than by models including first‐order Markov processes alone, or a combination of both. If we had not considered individual heterogeneity, models including Markovian transitions would have been the best performing ones. We also show that inference about age‐related changes in fitness components is sensitive to incorporation of underlying individual heterogeneity in models. Our approach provides insight into the sources of individual heterogeneity in life histories, and can be applied to other data sets to examine the ubiquity of our results across the tree of life.     

Testing for heterogeneity in the utility of a surrogate marker

In studies that require long-term and/or costly follow-up of participants to evaluate a treatment, there is often interest in identifying and using a surrogate marker to evaluate the treatment effect. While several statistical methods have been proposed to evaluate potential surrogate markers, available methods generally do not account for or address the potential for a surrogate to vary in utility or strength by patient characteristics. Previous work examining surrogate markers has indicated that there may be such heterogeneity, that is, that a surrogate marker may be useful (with respect to capturing the treatment effect on the primary outcome) for some subgroups, but not for others. This heterogeneity is important to understand, particularly if the surrogate is to be used in a future trial to replace the primary outcome. In this paper, we propose an approach and estimation procedures to measure the surrogate strength as a function of a baseline covariate W and thus examine potential heterogeneity in the utility of the surrogate marker with respect to W. Within a potential outcome framework, we quantify the surrogate strength/utility using the proportion of treatment effect on the primary outcome that is explained by the treatment effect on the surrogate. We propose testing procedures to test for evidence of heterogeneity, examine finite sample performance of these methods via simulation, and illustrate the methods using AIDS clinical trial data.     

Temperature-specific outcomes of cytoplasmic-nuclear interactions on egg-to-adult development time in seed beetles

The integration of the mitochondrial and nuclear genomes coordinates cellular energy production and is fundamental to life among eukaryotes. Therefore, there is potential for strong selection to shape the interactions between the two genomes. Several studies have now demonstrated that epistatic interactions between cytoplasmic and nuclear genes for fitness can occur both at a "within" and "across" population level. Genotype-by-environment interactions are common for traits that are encoded by nuclear genes, but the effects of environmental heterogeneity on traits that are partly encoded by cytoplasmic genes have received little attention despite the fact that there are reasons to believe that phenotypic effects of cytoplasmic genetic variation may often be environment specific. Consequently, the importance of environmental heterogeneity to the outcomes of cyto-nuclear fitness interactions and to the maintenance of mitochondrial polymorphism is unclear. Here, we assess the influence of temperature on cyto-nuclear effects on egg-to-adult development time in seed beetles (Callosobruchus maculatus). We employed an "across-population" design, sourcing beetles from five distinct populations and using backcrossing to create orthogonal combinations of distinct introgression lines, fixed for their cytoplasmic and nuclear lineages. We then assayed development times at two different temperatures and found sizeable cyto-nuclear effects in general, as well as temperature- and block-specific cyto-nuclear effects. These results demonstrate that environmental factors such as temperature do exert selection on cytoplasmic genes by favoring specific cyto-nuclear genetic combinations, and are consistent with the suggestion that complex genotype-by-environment interactions may promote the maintenance of polymorphism in mitochondrial genes.     

Evaluating Outcomes Used in Cardiothoracic Surgery Interventional Research: A Systematic Review of Reviews to Develop a Core Outcome Set

Background

When planning clinical trials, it is a key element to choose appropriate outcomes that ensure the comparability of effects of interventions in ways that minimise bias. We hypothesise that outcome measures in cardiothoracic surgical trials are inconsistent and without standard. Therefore, comparing the relative effectiveness of interventions across studies is problematic. We surmise that cardiothoracic research has focused habitually on the identification of risk factors and on the reduction of adverse outcomes with less consideration of factors that contribute to well being and positive health outcomes (salutogenesis).

Methods and Findings

We conducted a systematic review of reviews to determine both the type and number of outcomes reported in current cardiothoracic surgery interventional research, in order to identify a list of potential outcomes for a minimum core outcome set (COS). Special focus was placed on outcomes that emphasise salutogenesis. We interpreted salutogenic outcomes as those relating to optimum and/or positive health and well being. We searched Issue 7 (July 2014) of the Cochrane Database of Systematic Reviews. Systematic reviews of randomised trials on non-minimal-invasive off- or on-pump cardiothoracic surgery (elective and emergency, excluding transplants) investigating pre-, intra- or postsurgical interventions related to the outcome of the procedure were eligible for inclusion. We excluded protocols and withdrawn systematic reviews. Two review authors extracted outcome data independently. Unique lists of salutogenically and non-salutogenically focused outcomes were established. 15 systematic reviews involving 371 randomized trials and 58,253 patients were included in this review. Applied definitions of single and composite endpoints varied significantly, and patient-centred, salutogenically focused outcomes were seldom reported. One third of included reviews did not assess patient-centred outcomes at all; all other reviews were unable to perform meta-analyses due to an absence of data or heterogeneity in outcome measures. This compares to 36 non-salutogenically focused outcome domains representing 121 individual non-salutogenically focused outcomes, whereof 50% were assessed only once. Measures of mortality, cerebrovascular complications and hospitalisation were reported most frequently. Two reviews chose a composite endpoint as primary outcome. Pooled analysis of composite endpoints was not possible, as the required data was not reported per patient in all components.

Conclusion

In cardiothoracic surgical trials, choice and definition of non-salutogenically focused single and composite outcomes are inconsistent. There is an absence of patient centred, salutogenically focused outcome parameters in cardiac trials. We recommend the development of a core outcome set of salutogenically focused and non-salutogenically focused outcomes for cardiothoracic surgical research.     

Assessing exposure-time treatment effect heterogeneity in stepped-wedge cluster randomized trials

A stepped-wedge cluster randomized trial (CRT) is a unidirectional crossover study in which timings of treatment initiation for clusters are randomized. Because the timing of treatment initiation is different for each cluster, an emerging question is whether the treatment effect depends on the exposure time, namely, the time duration since the initiation of treatment. Existing approaches for assessing exposure-time treatment effect heterogeneity either assume a parametric functional form of exposure time or model the exposure time as a categorical variable, in which case the number of parameters increases with the number of exposure-time periods, leading to a potential loss in efficiency. In this article, we propose a new model formulation for assessing treatment effect heterogeneity over exposure time. Rather than a categorical term for each level of exposure time, the proposed model includes a random effect to represent varying treatment effects by exposure time. This allows for pooling information across exposure-time periods and may result in more precise average and exposure-time-specific treatment effect estimates. In addition, we develop an accompanying permutation test for the variance component of the heterogeneous treatment effect parameters. We conduct simulation studies to compare the proposed model and permutation test to alternative methods to elucidate their finite-sample operating characteristics, and to generate practical guidance on model choices for assessing exposure-time treatment effect heterogeneity in stepped-wedge CRTs.     

SIMEX variance component tests in generalized linear mixed measurement error models

In the analysis of clustered data with covariates measured with error, a problem of common interest is to test for correlation within clusters and heterogeneity across clusters. We examined this problem in the framework of generalized linear mixed measurement error models. We propose using the simulation extrapolation (SIMEX) method to construct a score test for the null hypothesis that all variance components are zero. A key feature of this SIMEX score test is that no assumptions need to be made regarding the distributions of the random effects and the unobserved covariates. We illustrate this test by analyzing Framingham heart disease data and evaluate its performance by simulation. We also propose individual SIMEX score tests for testing the variance components separately. Both tests can be easily implemented using existing statistical software.     

Random main effects of treatment: A case study with a network meta‐analysis

If the number of treatments in a network meta‐analysis is large, it may be possible and useful to model the main effect of treatment as random, that is to say as random realizations from a normal distribution of possible treatment effects. This then constitutes a third sort of random effect that may be considered in connection with such analyses. The first and most common models treatment‐by‐trial interaction as being random and the second, rather rarer, models the main effects of trial as being random and thus permits the recovery of intertrial information. Taking the example of a network meta‐analysis of 44 similar treatments in 10 trials, we illustrate how a hierarchical approach to modeling a random main effect of treatment can be used to produce shrunk (toward the overall mean) estimates of effects for individual treatments. As a related problem, we also consider the issue of using a random‐effect model for the within‐trial variances from trial to trial. We provide a number of possible graphical representations of the results and discuss the advantages and disadvantages of such an approach.     

The Use of Two‐Way Linear Mixed Models in Multitreatment Meta‐Analysis

Summary Meta‐analysis summarizes the results of a series of trials. When more than two treatments are included in the trials and when the set of treatments tested differs between trials, the combination of results across trials requires some care. Several methods have been proposed for this purpose, which feature under different labels, such as network meta‐analysis or mixed treatment comparisons. Two types of linear mixed model can be used for meta‐analysis. The one expresses the expected outcome of treatments as a contrast to a baseline treatment. The other uses a classical two‐way linear predictor with main effects for treatment and trial. In this article, we compare both types of model and explore under which conditions they give equivalent results. We illustrate practical advantages of the two‐way model using two published datasets. In particular, it is shown that between‐trial heterogeneity as well as inconsistency between different types of trial is straightforward to account for.     

A potential outcomes approach to developmental toxicity analyses

Estimating the effects of a toxin on fetal development in animal models such as mice can be problematic, because the number of pups that develop and survive until birth may simultaneously affect developmental outcomes such as birth weight and be affected by the introduction of a toxin into the fetal environment. Also, comparing pups that survived until birth at a high dose of the toxin with pups that survived at low doses may underestimate the effect of the toxin, because the lower dose means include the less healthy pups that would not survive if exposed to a higher level of toxin. We consider this problem in a potential outcomes framework that defines the effect of the dose on the outcome as the difference between what the outcome would have been for a pup had the dam in which the pup develops been exposed to dose level Z=z* rather than dose level Z=z. To disentangle the direct effect of dose from the effect of litter size, we focus on effects defined within principal strata that are a function of the survival status of the pups at each of the possible dose levels. A unique contribution to the potential outcomes literature is that we allow the outcome for a subject to be dependent on the principal stratum to which other subjects within a cluster belong.