On estimation of the survivor average causal effect in observational studies when important confounders are missing due to death

Testing weak nulls in matched observational studies

We develop sensitivity analyses for the sample average treatment effect in matched observational studies while allowing unit-level treatment effects to vary. The methods may be applied to studies using any optimal without-replacement matching algorithm. In contrast to randomized experiments and to paired observational studies, we show for general matched designs that over a large class of test statistics, any procedure bounding the worst-case expectation while allowing for arbitrary effect heterogeneity must be unnecessarily conservative if treatment effects are actually constant across individuals. We present a sensitivity analysis which bounds the worst-case expectation while allowing for effect heterogeneity, and illustrate why it is generally conservative if effects are constant. An alternative procedure is presented that is asymptotically sharp if treatment effects are constant, and that is valid for testing the sample average effect under additional restrictions which may be deemed benign by practitioners. Simulations demonstrate that this alternative procedure results in a valid sensitivity analysis for the weak null hypothesis under a host of reasonable data-generating processes. The procedures allow practitioners to assess robustness of estimated sample average treatment effects to hidden bias while allowing for effect heterogeneity in matched observational studies.

Translating questions to estimands in randomized clinical trials with intercurrent events

Intercurrent (post‐treatment) events occur frequently in randomized trials, and investigators often express interest in treatment effects that suitably take account of these events. Contrasts that naively condition on intercurrent events do not have a straight‐forward causal interpretation, and the practical relevance of other commonly used approaches is debated. In this work, we discuss how to formulate and choose an estimand, beyond the marginal intention‐to‐treat effect, from the point of view of a decision maker and drug developer. In particular, we argue that careful articulation of a practically useful research question should either reflect decision making at this point in time or future drug development. Indeed, a substantially interesting estimand is simply a formalization of the (plain English) description of a research question. A common feature of estimands that are practically useful is that they correspond to possibly hypothetical but well‐defined interventions in identifiable (sub)populations. To illustrate our points, we consider five examples that were recently used to motivate consideration of principal stratum estimands in clinical trials. In all of these examples, we propose alternative causal estimands, such as conditional effects, sequential regime effects, and separable effects, that correspond to explicit research questions of substantial interest.

Causal Inference in Studying the Long-Term Health Effects of Disasters: Challenges and Potential Solutions

Two frequently encountered but underrecognized challenges for causal inference in studying the long-term health effects of disasters among survivors include 1) time-varying effects of disasters on a time-to-event outcome and 2) selection bias due to selective attrition. In this paper, we review approaches for overcoming these challenges and demonstrate application of the approaches to a real-world longitudinal data set of older adults who were directly affected by the 2011 Great East Japan Earthquake and Tsunami (n = 4,857). To illustrate the problem of time-varying effects of disasters, we examined the association between degree of damage due to the tsunami and all-cause mortality. We compared results from Cox regression analysis assuming proportional hazards with those derived using adjusted parametric survival curves allowing for time-varying hazard ratios. To illustrate the problem of selection bias, we examined the association between proximity to the coast (a proxy for housing damage from the tsunami) and depressive symptoms. We corrected for selection bias due to attrition in the 2 postdisaster follow-up surveys (conducted in 2013 and 2016) using multivariable adjustment, inverse probability of censoring weighting, and survivor average causal effect estimation. Our results demonstrate that analytical approaches which ignore time-varying effects on mortality and selection bias due to selective attrition may underestimate the long-term health effects of disasters.

An Investigation of Selection Bias in Estimating Racial Disparity in Stroke Risk Factors

Selection due to survival or attrition might bias estimates of racial disparities in health, but few studies quantify the likely magnitude of such bias. In a large national cohort with moderate loss to follow-up, we contrasted racial differences in 2 stroke risk factors, incident hypertension and incident left ventricular hypertrophy, estimated by complete-case analyses, inverse probability of attrition weighting, and the survivor average causal effect. We used data on 12,497 black and 17,660 white participants enrolled in the United States (2003-2007) and collected incident risk factor data approximately 10 years after baseline. At follow-up, 21.0% of white participants and 23.0% of black participants had died; additionally 22.0% of white participants and 28.4% of black participants had withdrawn. Individual probabilities of completing the follow-up visit were estimated using baseline demographic and health characteristics. Adjusted risk ratio estimates of racial disparities from complete-case analyses in both incident hypertension (1.11, 95% confidence interval: 1.02, 1.21) and incident left ventricular hypertrophy (1.02, 95% confidence interval: 0.84, 1.24) were virtually identical to estimates from inverse probability of attrition weighting and survivor average causal effect. Despite racial differences in mortality and attrition, we found little evidence of selection bias in the estimation of racial differences for these incident risk factors.

A powerful approach to the study of moderate effect modification in observational studies

Effect modification means the magnitude or stability of a treatment effect varies as a function of an observed covariate. Generally, larger and more stable treatment effects are insensitive to larger biases from unmeasured covariates, so a causal conclusion may be considerably firmer if this pattern is noted if it occurs. We propose a new strategy, called the submax-method, that combines exploratory, and confirmatory efforts to determine whether there is stronger evidence of causality-that is, greater insensitivity to unmeasured confounding-in some subgroups of individuals. It uses the joint distribution of test statistics that split the data in various ways based on certain observed covariates. For L binary covariates, the method splits the population L times into two subpopulations, perhaps first men and women, perhaps then smokers and nonsmokers, computing a test statistic from each subpopulation, and appends the test statistic for the whole population, making <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn> <mml:mi>L</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn></mml:math> test statistics in total. Although L binary covariates define <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>2</mml:mn> <mml:mi>L</mml:mi></mml:msup> </mml:math> interaction groups, only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn> <mml:mi>L</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn></mml:math> tests are performed, and at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn></mml:math> of these tests use at least half of the data. The submax-method achieves the highest design sensitivity and the highest Bahadur efficiency of its component tests. Moreover, the form of the test is sufficiently tractable that its large sample power may be studied analytically. The simulation suggests that the submax method exhibits superior performance, in comparison with an approach using CART, when there is effect modification of moderate size. Using data from the NHANES I epidemiologic follow-up survey, an observational study of the effects of physical activity on survival is used to illustrate the method. The method is implemented in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> package <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>submax</mml:mi></mml:math> which contains the NHANES example. An online Appendix provides simulation results and further analysis of the example.

Latent Class Survival Models Linked by Principal Stratification to Investigate Heterogenous Survival Subgroups Among Individuals With Early-Stage Kidney Cancer

Rates of kidney cancer have been increasing, with small incidental tumors experiencing the fastest growth rates. Much of the increase could be due to increased use of CT scans, MRIs, and ultrasounds for unrelated conditions. Many tumors might never have been detected or become symptomatic in the past. This suggests that many patients might benefit from less aggressive therapy, such as active surveillance by which tumors are surgically removed only if they become sufficiently large. However, it has been difficult for clinicians to identify subgroups of patients for whom treatment might be especially beneficial or harmful. In this work, we use a principal stratification framework to estimate the proportion and characteristics of individuals who have large or small hazard rates of death in two treatment arms. This allows us to assess who might be helped or harmed by aggressive treatment. We also use Weibull mixture models. This work differs from much previous work in that the survival classes upon which principal stratification is based are latent variables. That is, survival class is not an observed variable. We apply this work using Surveillance Epidemiology and End Results-Medicare claims data. Clinicians can use our methods for investigating treatments with heterogeneous effects.

An adaptive Mantel-Haenszel test for sensitivity analysis in observational studies

In a sensitivity analysis in an observational study with a binary outcome, is it better to use all of the data or to focus on subgroups that are expected to experience the largest treatment effects? The answer depends on features of the data that may be difficult to anticipate, a trade-off between unknown effect-sizes and known sample sizes. We propose a sensitivity analysis for an adaptive test similar to the Mantel-Haenszel test. The adaptive test performs two highly correlated analyses, one focused analysis using a subgroup, one combined analysis using all of the data, correcting for multiple testing using the joint distribution of the two test statistics. Because the two component tests are highly correlated, this correction for multiple testing is small compared with, for instance, the Bonferroni inequality. The test has the maximum design sensitivity of two component tests. A simulation evaluates the power of a sensitivity analysis using the adaptive test. Two examples are presented. An R package, sensitivity2x2xk, implements the procedure.

The cross-cut statistic and its sensitivity to bias in observational studies with ordered doses of treatment

A common practice with ordered doses of treatment and ordered responses, perhaps recorded in a contingency table with ordered rows and columns, is to cut or remove a cross from the table, leaving the outer corners--that is, the high-versus-low dose, high-versus-low response corners--and from these corners to compute a risk or odds ratio. This little remarked but common practice seems to be motivated by the oldest and most familiar method of sensitivity analysis in observational studies, proposed by Cornfield et al. (1959), which says that to explain a population risk ratio purely as bias from an unobserved binary covariate, the prevalence ratio of the covariate must exceed the risk ratio. Quite often, the largest risk ratio, hence the one least sensitive to bias by this standard, is derived from the corners of the ordered table with the central cross removed. Obviously, the corners use only a portion of the data, so a focus on the corners has consequences for the standard error as well as for bias, but sampling variability was not a consideration in this early and familiar form of sensitivity analysis, where point estimates replaced population parameters. Here, this cross-cut analysis is examined with the aid of design sensitivity and the power of a sensitivity analysis.

Identifiability of subgroup causal effects in randomized experiments with nonignorable missing covariates

Although randomized experiments are widely regarded as the gold standard for estimating causal effects, missing data of the pretreatment covariates makes it challenging to estimate the subgroup causal effects. When the missing data mechanism of the covariates is nonignorable, the parameters of interest are generally not pointly identifiable, and we can only get bounds for the parameters of interest, which may be too wide for practical use. In some real cases, we have prior knowledge that some restrictions may be plausible. We show the identifiability of the causal effects and joint distributions for four interpretable missing data mechanisms and evaluate the performance of the statistical inference via simulation studies. One application of our methods to a real data set from a randomized clinical trial shows that one of the nonignorable missing data mechanisms fits better than the ignorable missing data mechanism, and the results conform to the study's original expert opinions. We also illustrate the potential applications of our methods to observational studies using a data set from a job-training program.

Case Definition and Design Sensitivity

In a case-referent study, cases of disease are compared to non-cases with respect to their antecedent exposure to a treatment in an effort to determine whether exposure causes some cases of the disease. Because exposure is not randomly assigned in the population, as it would be if the population were a vast randomized trial, exposed and unexposed subjects may differ prior to exposure with respect to covariates that may or may not have been measured. After controlling for measured pre-exposure differences, for instance by matching, a sensitivity analysis asks about the magnitude of bias from unmeasured covariates that would need to be present to alter the conclusions of a study that presumed matching for observed covariates removes all bias. The definition of a case of disease affects sensitivity to unmeasured bias. We explore this issue using: (i) an asymptotic tool, the design sensitivity, (ii) a simulation for finite samples, and (iii) an example. Under favorable circumstances, a narrower case definition can yield an increase in the design sensitivity, and hence an increase in the power of a sensitivity analysis. Also, we discuss an adaptive method that seeks to discover the best case definition from the data at hand while controlling for multiple testing. An implementation in R is available as SensitivityCaseControl.

Causal inference for bivariate longitudinal quality of life data in presence of death by using global odds ratios

In longitudinal clinical trials, if a subject drops out due to death, certain responses, such as those measuring quality of life (QoL), will not be defined after the time of death. Thus, standard missing data analyses, e.g., under ignorable dropout, are problematic because these approaches implicitly 'impute' values of the response after death. In this paper we define a new survivor average causal effect for a bivariate response in a longitudinal quality of life study that had a high dropout rate with the dropout often due to death (or tumor progression). We show how principal stratification, with a few sensitivity parameters, can be used to draw causal inferences about the joint distribution of these two ordinal quality of life measures.

The balanced survivor average causal effect

Statistical analysis of longitudinal outcomes is often complicated by the absence of observable values in patients who die prior to their scheduled measurement. In such cases, the longitudinal data are said to be "truncated by death" to emphasize that the longitudinal measurements are not simply missing, but are undefined after death. Recently, the truncation by death problem has been investigated using the framework of principal stratification to define the target estimand as the survivor average causal effect (SACE), which in the context of a two-group randomized clinical trial is the mean difference in the longitudinal outcome between the treatment and control groups for the principal stratum of always-survivors. The SACE is not identified without untestable assumptions. These assumptions have often been formulated in terms of a monotonicity constraint requiring that the treatment does not reduce survival in any patient, in conjunction with assumed values for mean differences in the longitudinal outcome between certain principal strata. In this paper, we introduce an alternative estimand, the balanced-SACE, which is defined as the average causal effect on the longitudinal outcome in a particular subset of the always-survivors that is balanced with respect to the potential survival times under the treatment and control. We propose a simple estimator of the balanced-SACE that compares the longitudinal outcomes between equivalent fractions of the longest surviving patients between the treatment and control groups and does not require a monotonicity assumption. We provide expressions for the large sample bias of the estimator, along with sensitivity analyses and strategies to minimize this bias. We consider statistical inference under a bootstrap resampling procedure.

Impact of multiple matched controls on design sensitivity in observational studies

In an observational study, one treated subject may be matched for observed covariates to either one or several untreated controls. The common motivation for using several controls rather than one is to increase the power of a test of no effect under the doubtful assumption that matching for observed covariates suffices to remove bias from nonrandom treatment assignment. Does the choice between one or several matched controls affect the sensitivity of conclusions to violations of this doubtful assumption? With continuous responses, it is known that reducing the heterogeneity of matched pair differences reduces sensitivity to unmeasured biases, but increasing the sample size has a highly circumscribed effect on sensitivity to bias. Is the use of several controls rather than one analogous to a reduction in heterogeneity or to an increase in sample size? The issue is examined for Huber's m-statistics, including the t-test, the examination having three components: an example, asymptotic calculations using design sensitivity, and a simulation. Use of multiple controls with continuous responses yields a nontrivial reduction in sensitivity to unmeasured biases. An example looks at lead and cadmium in the blood of smokers from the 2008 National Health and Nutrition Examination Survey. A by-product of the discussion is a new result giving the design sensitivity for the permutation distribution of m-statistics.

Level-I trauma center effects on return-to-work outcomes

BACKGROUND

Injury is the leading cause of death for persons aged 1-44 years in the United States. Injuries have a substantial economic cost. For that reason, regional systems of trauma care in which the more acutely injured patients are transported to Level-I (L-I) trauma centers (TCs) has been widely advocated. However, the cost of TC care is high, raising questions about the value of such an approach.

OBJECTIVES

To study L-I TC effectiveness and study return-to-work (RTW) outcomes.

RESEARCH DESIGN

Using data from National Study on the Costs and Outcomes of Trauma, the authors address the issue of selection bias by comparing naive estimates to matching techniques, as well as to nonlinear instrumental variable models (2SRI) and bivariate probit estimators.

SUBJECTS

Individuals ages 18-64 who were mainly working before traumatic injury. Patients selected for the study were treated at 69 hospitals located in 12 states in the United States. N = 1790.

MEASURES

Treatment is binary indicator on whether treated at L-I TC. Outcome is binary indicator on whether returned to work within 3 months after injury. Covariates include: demographics, pre-injury characteristics (job, health and insurance status), injury descriptors, other income sources, etc.

RESULTS

Across all models that control for unobserved factors, the authors find that L-I TC treatment is positively associated with RTW within 3 months after injury. The estimated average marginal effect of treatment on the probability of RTW ranges from 23 to 38 percentage points.

CONCLUSIONS

Benefits of L-I TC care extend beyond mortality and morbidity.

Sensitivity analysis for unmeasured confounding in principal stratification settings with binary variables

Within causal inference, principal stratification (PS) is a popular approach for dealing with intermediate variables, that is, variables affected by treatment that also potentially affect the response. However, when there exists unmeasured confounding in the treatment arms--as can happen in observational studies--causal estimands resulting from PS analyses can be biased. We identify the various pathways of confounding present in PS contexts and their effects for PS inference. We present model-based approaches for assessing the sensitivity of complier average causal effect estimates to unmeasured confounding in the setting of binary treatments, binary intermediate variables, and binary outcomes. These same approaches can be used to assess sensitivity to unknown direct effects of treatments on outcomes because, as we show, direct effects are operationally equivalent to one of the pathways of unmeasured confounding. We illustrate the methodology using a randomized study with artificially introduced confounding and a sensitivity analysis for an observational study of the effects of physical activity and body mass index on cardiovascular disease.

Identifiability of causal effects for binary variables with baseline data missing due to death

We discuss identifiability and estimation of causal effects of a treatment in subgroups defined by a covariate that is sometimes missing due to death, which is different from a problem with outcomes censored by death. Frangakis et al. (2007, Biometrics 63, 641-662) proposed an approach for estimating the causal effects under a strong monotonicity (SM) assumption. In this article, we focus on identifiability of the joint distribution of the covariate, treatment and potential outcomes, show sufficient conditions for identifiability, and relax the SM assumption to monotonicity (M) and no-interaction (NI) assumptions. We derive expectation-maximization algorithms for finding the maximum likelihood estimates of parameters of the joint distribution under different assumptions. Further we remove the M and NI assumptions, and prove that signs of the causal effects of a treatment in the subgroups are identifiable, which means that their bounds do not cover zero. We perform simulations and a sensitivity analysis to evaluate our approaches. Finally, we apply the approaches to the National Study on the Costs and Outcomes of Trauma Centers data, which are also analyzed by Frangakis et al. (2007) and Xie and Murphy (2007, Biometrics 63, 655-658).

Principal stratification--uses and limitations

Pearl (2011) asked for the causal inference community to clarify the role of the principal stratification framework in the analysis of causal effects. Here, I argue that the notion of principal stratification has shed light on problems of non-compliance, censoring-by-death, and the analysis of post-infection outcomes; that it may be of use in considering problems of surrogacy but further development is needed; that it is of some use in assessing "direct effects"; but that it is not the appropriate tool for assessing "mediation." There is nothing within the principal stratification framework that corresponds to a measure of an "indirect" or "mediated" effect.

Semiparametric regression models for repeated measures of mortal cohorts with non-monotone missing outcomes and time-dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non-response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time-dependent covariates that are missing not at random with non-monotone missingness patterns via inverse-probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse-probability-weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age.

CAUSAL EFFECTS OF TREATMENTS FOR INFORMATIVE MISSING DATA DUE TO PROGRESSION/DEATH

In longitudinal clinical trials, when outcome variables at later time points are only defined for patients who survive to those times, the evaluation of the causal effect of treatment is complicated. In this paper, we describe an approach that can be used to obtain the causal effect of three treatment arms with ordinal outcomes in the presence of death using a principal stratification approach. We introduce a set of flexible assumptions to identify the causal effect and implement a sensitivity analysis for non-identifiable assumptions which we parameterize parsimoniously. Methods are illustrated on quality of life data from a recent colorectal cancer clinical trial.

Extent of gastric resection impacts patient quality of life: the Dysfunction After Upper Gastrointestinal Surgery for Cancer (DAUGS32) scoring system

BACKGROUND

Quality of life is an important outcome measure in the care of patients with cancer. We developed a new scoring system specifically for the evaluation of patients with upper gastrointestinal cancer and postoperative gastrointestinal dysfunction. This study was undertaken to evaluate the scoring system's validity in comparing outcomes after gastric resection.

MATERIALS AND METHODS

Patients with gastric cancer, 3 months to 3 years postoperatively, were surveyed using the survey instrument. Postoperative dysfunction scores and the status of resuming activities of daily living were compared with the surgical procedure performed by analysis of variance and multiple-comparison techniques.

RESULTS

Of 211 patients surveyed, 165 (119 men, 46 women; mean age, 65.1 ± 10.5 years) responded. Procedures included distal gastrectomy in 100, total gastrectomy in 57, and pylorus-preserving gastrectomy in 8. The overall dysfunction score was 61.8 ± 15.5. The dysfunction score was 58.9 ± 15.0 after distal gastrectomy, 66.8 ± 14.1 after total gastrectomy, and 62.4 ± 21.6 after pylorus-preserving gastrectomy. These values differed significantly among the groups (P = .007). Dysfunction scores according to postoperative activity status were 49.1 ± 15.6 in 71 patients who resumed their activities, 56.9 ± 15.7 in 39 patients with reduced activities, 57.3 ± 8.8 in 15 patients with minimal activities, and 63.3 ± 11.8 (P < .05) in 16 patients who did not resume activities because of poor physical condition.

CONCLUSIONS

This scoring system for postoperative gastrointestinal dysfunction provides an objective measure of dysfunction related to specific surgical procedures and correlates with activities of daily living in the postoperative period.

A tutorial on principal stratification-based sensitivity analysis: application to smoking cessation studies

BACKGROUND

One problem with assessing effects of smoking cessation interventions on withdrawal symptoms is that symptoms are affected by whether participants abstain from smoking during trials. Those who enter a randomized trial but do not change smoking behavior might not experience withdrawal-related symptoms.

PURPOSE

We present a tutorial of how one can use a principal stratification sensitivity analysis to account for abstinence in the estimation of smoking cessation intervention effects. The article is intended to introduce researchers to principal stratification and describe how they might implement the methods.

METHODS

We provide a hypothetical example that demonstrates why estimating effects within observed abstention groups is problematic. We demonstrate how estimation of effects within groups defined by potential abstention that an individual would have in either arm of a study can provide meaningful inferences. We describe a sensitivity analysis method to estimate such effects, and use it to investigate effects of a combined behavioral and nicotine replacement therapy intervention on withdrawal symptoms in a female prisoner population.

RESULTS

Overall, the intervention was found to reduce withdrawal symptoms but the effect was not statistically significant in the group that was observed to abstain. More importantly, the intervention was found to be highly effective in the group that would abstain regardless of intervention assignment. The effectiveness of the intervention in other potential abstinence strata depends on the sensitivity analysis assumptions.

LIMITATIONS

We make assumptions to narrow the range of our sensitivity analysis estimates. While appropriate in this situation, such assumptions might not be plausible in all situations.

CONCLUSIONS

A principal stratification sensitivity analysis provides a meaningful method of accounting for abstinence effects in the evaluation of smoking cessation interventions on withdrawal symptoms. Smoking researchers have previously recommended analyses in subgroups defined by observed abstention status in the evaluation of smoking cessation interventions. We believe that principal stratification analyses should replace such analyses as the preferred means of accounting for post-randomization abstinence effects in the evaluation of smoking cessation programs. Clinical Trials 2010; 7: 286-298. http://ctj.sagepub.com.

Longitudinal Data with Follow-up Truncated by Death: Match the Analysis Method to Research Aims

Diverse analysis approaches have been proposed to distinguish data missing due to death from nonresponse, and to summarize trajectories of longitudinal data truncated by death. We demonstrate how these analysis approaches arise from factorizations of the distribution of longitudinal data and survival information. Models are illustrated using cognitive functioning data for older adults. For unconditional models, deaths do not occur, deaths are independent of the longitudinal response, or the unconditional longitudinal response is averaged over the survival distribution. Unconditional models, such as random effects models fit to unbalanced data, may implicitly impute data beyond the time of death. Fully conditional models stratify the longitudinal response trajectory by time of death. Fully conditional models are effective for describing individual trajectories, in terms of either aging (age, or years from baseline) or dying (years from death). Causal models (principal stratification) as currently applied are fully conditional models, since group differences at one timepoint are described for a cohort that will survive past a later timepoint. Partly conditional models summarize the longitudinal response in the dynamic cohort of survivors. Partly conditional models are serial cross-sectional snapshots of the response, reflecting the average response in survivors at a given timepoint rather than individual trajectories. Joint models of survival and longitudinal response describe the evolving health status of the entire cohort. Researchers using longitudinal data should consider which method of accommodating deaths is consistent with research aims, and use analysis methods accordingly.