Stratum Membership Research Articles

How to estimate causal effects associated with family planning? An introduction to Prince BART, a new approach to effect estimation based on principal stratification and Bayesian non-parametric models

There is interest in learning about the causal effect of family planning (FP) on empowerment-related outcomes. Experimental data related to this question are available from trials in which FP programs increase access to FP. However, estimating the effect of interest is challenging because FP and empowerment-related variables are likely to be confounded. We propose a new approach, Prince BART, to estimate the causal effect of FP on other outcomes of interest, among women affected by a FP program. Prince BART combines principal stratification, to account for differences in FP behavior based on exposure to a FP program, with Bayesian Additive Regression Trees (BART), to non-parametrically model stratum membership and relationships between variables. Prince Bart allows for estimation of effect heterogeneity. We use Prince BART to assess the causal effect of modern contraceptive use on employment in six cities in Nigeria, based on quasi-experimental data from a FP program trial during the first half of the 2010s. In the sample, we find a positive overall effect of contraceptive use on employment and substantial differences in effect sizes across subgroups defined by baseline covariates such as marital status, work at baseline, and wealth or education. The modeling approach and results can be used to inform future analyses seeking to understand the causal relationship between family planning use and other outcomes, e.g., related to empowerment.

A BAYESIAN MACHINE LEARNING APPROACH FOR ESTIMATING HETEROGENEOUS SURVIVOR CAUSAL EFFECTS: APPLICATIONS TO A CRITICAL CARE TRIAL.

Assessing heterogeneity in the effects of treatments has become increasingly popular in the field of causal inference and carries important implications for clinical decision-making. While extensive literature exists for studying treatment effect heterogeneity when outcomes are fully observed, there has been limited development in tools for estimating heterogeneous causal effects when patient-centered outcomes are truncated by a terminal event, such as death. Due to mortality occurring during study follow-up, the outcomes of interest are unobservable, undefined, or not fully observed for many participants in which case principal stratification is an appealing framework to draw valid causal conclusions. Motivated by the Acute Respiratory Distress Syndrome Network (ARDSNetwork) ARDS respiratory management (ARMA) trial, we developed a flexible Bayesian machine learning approach to estimate the average causal effect and heterogeneous causal effects among the always-survivors stratum when clinical outcomes are subject to truncation. We adopted Bayesian additive regression trees (BART) to flexibly specify separate mean models for the potential outcomes and latent stratum membership. In the analysis of the ARMA trial, we found that the low tidal volume treatment had an overall benefit for participants sustaining acute lung injuries on the outcome of time to returning home but substantial heterogeneity in treatment effects among the always-survivors, driven most strongly by biologic sex and the alveolar-arterial oxygen gradient at baseline (a physiologic measure of lung function and degree of hypoxemia). These findings illustrate how the proposed methodology could guide the prognostic enrichment of future trials in the field.

Sensitivity analyses for the principal ignorability assumption using multiple imputation.

In the context of clinical trials, there is interest in the treatment effect for subpopulations of patients defined by intercurrent events, namely disease-related events occurring after treatment initiation that affect either the interpretation or the existence of endpoints. With the principal stratum strategy, the ICH E9(R1) guideline introduces a formal framework in drug development for defining treatment effects in such subpopulations. Statistical estimation of the treatment effect can be performed based on the principal ignorability assumption using multiple imputation approaches. Principal ignorability is a conditional independence assumption that cannot be directly verified; therefore, it is crucial to evaluate the robustness of results to deviations from this assumption. As a sensitivity analysis, we propose a joint model that multiply imputes the principal stratum membership and the outcome variable while allowing different levels of violation of the principal ignorability assumption. We illustrate with a simulation study that the joint imputation model-based approaches are superior to naive subpopulation analyses. Motivated by an oncology clinical trial, we implement the sensitivity analysis on a time-to-event outcome to assess the treatment effect in the subpopulation of patients who discontinued due to adverse events using a synthetic dataset. Finally, we explore the potential usage and provide interpretation of such analyses in clinical settings, as well as possible extension of such models in more general cases.

Handling parametric assumptions in principal causal effect estimation using Gaussian mixtures.

Given the latent stratum membership, principal stratification models with continuous outcomes naturally fit in the parametric estimation framework of Gaussian mixtures. However, with models that are not nonparametrically identified, relying on parametric mixture modeling has been mostly discouraged as a way of identifying principal effects. This study revisits this rather deserted use of parametric mixture modeling, which may open up various possibilities in principal stratification modeling. The main problem with using the parametric mixture modeling approach is that it is hard to assess the quality of principal effect estimates given its reliance on parametric conditions. As a way of assessing the estimation quality in this situation, this study proposes that we use parametric mixture modeling in two different ways, with and without the assurance of nonparametric identification. The key identifying assumption employed in this study is the moving exclusion restriction, a flexible version of the standard exclusion restriction assumption. This assumption is used as a temporary vehicle to help assess the quality of principal effect estimates obtained relying on parametric mixture modeling. The study presents promising results, showing the possibility of using parametric mixture modeling as an accessible tool for causal inference.

A Comparison of Methods for Predicting Future Cognitive Status: Mixture Modeling, Latent Class Analysis, and Competitors.

The present work compares various methods for using baseline cognitive performance data to predict eventual cognitive status of longitudinal study participants at the University of Kentucky's Alzheimer's Disease Center. Cox proportional hazards models examined time to cognitive transition as predicted by risk strata derived from normal mixture modeling, latent class analysis, and a 1-SD thresholding approach. An additional comparator involved prediction directly from a numeric value for baseline cognitive performance. A normal mixture model suggested 3 risk strata based on Consortium to Establish a Registry for Alzheimer's Disease (CERAD) T scores: high, intermediate, and low risk. Cox modeling of time to cognitive decline based on posterior probabilities for risk stratum membership yielded an estimated hazard ratio of 4.00 with 95% confidence interval 1.53-10.44 in comparing high risk membership to low risk; for intermediate risk membership versus low risk, the modeling yielded hazard ratio=2.29 and 95% confidence interval=0.98-5.33. Latent class analysis produced 3 groups, which did not have a clear ordering in terms of risk; however, one group exhibited appreciably greater hazard of cognitive decline. All methods for generating predictors of cognitive transition yielded statistically significant likelihood ratio statistics but modest concordance statistics. Posterior probabilities from mixture modeling allow for risk stratification that is data-driven and, in the case of CERAD T scores, modestly predictive of later cognitive decline. Incorporating other covariates may enhance predictions.

Enhancing Power of Marketing Experiments Using Observational Data

Advertising experiments often suffer from noisy responses making precise estimation of the average treatment effect (ATE) and evaluating ROI difficult. We develop a principal stratification model that improves the precision of the ATE by dividing the customers into three strata -- those who buy regardless of ad exposure, those who buy only if exposed to ads and those who do not buy regardless. The method decreases the variance of the ATE by separating out the typically large share of customers who buy and therefore have individual treatment effects that are exactly zero. Applying the procedure to 5 catalog mailing experiments with sample sizes around 140,000 shows a reduction of 36-57% in the variance of the estimate. When we include pre-randomization covariates that predict stratum membership, we find that estimates of customers' past response to similar advertising are a good predictor of stratum membership, even if such estimates are biased because past advertising was targeted. Customers who have not purchased recently are also more likely to be in the never purchase stratum. We provide simple summary statistics that firms can compute from their own experiment data to determine if the procedure is expected to be beneficial before applying it.

Identifying and estimating principal causal effects in a multi-site trial of Early College High Schools

Randomized trials are often conducted with separate randomizations across multiple sites such as schools, voting districts, or hospitals. These sites can differ in important ways, including the site’s implementation quality, local conditions, and the composition of individuals. An important question in practice is whether—and under what assumptions—researchers can leverage this cross-site variation to learn more about the intervention. We address these questions in the principal stratification framework, which describes causal effects for subgroups defined by post-treatment quantities. We show that researchers can estimate certain principal causal effects via the multi-site design if they are willing to impose the strong assumption that the site-specific effects are independent of the site-specific distribution of stratum membership. We motivate this approach with a multi-site trial of the Early College High School Initiative, a unique secondary education program with the goal of increasing high school graduation rates and college enrollment. Our analyses corroborate previous studies suggesting that the initiative had positive effects for students who would have otherwise attended a low-quality high school, although power is limited.

Bayesian Inference for Sequential Treatments Under Latent Sequential Ignorability

We focus on causal inference for longitudinal treatments, where units are assigned to treatments at multiple time points, aiming to assess the effect of different treatment sequences on an outcome observed at a final point. A common assumption in similar studies is sequential ignorability (SI): treatment assignment at each time point is assumed independent of future potential outcomes given past observed outcomes and covariates. SI is questionable when treatment participation depends on individual choices, and treatment assignment may depend on unobservable quantities associated with future outcomes. We rely on principal stratification to formulate a relaxed version of SI: latent sequential ignorability (LSI) assumes that treatment assignment is conditionally independent on future potential outcomes given past treatments, covariates, and principal stratum membership, a latent variable defined by the joint value of observed and missing intermediate outcomes. We evaluate SI and LSI, using theoretical arguments and simulation studies to investigate the performance of the two assumptions when one holds and inference is conducted under both. Simulations show that when SI does not hold, inference performed under SI leads to misleading conclusions. Conversely, LSI generally leads to correct posterior distributions, irrespective of which assumption holds.

Socioeconomic differences in body mass index in Spain: An intersectional multilevel analysis of individual heterogeneity and discriminatory accuracy.

Many studies have demonstrated the existence of simple, unidimensional socioeconomic gradients in body mass index (BMI). However, in the present paper we move beyond such traditional analyses by simultaneously considering multiple demographic and socioeconomic dimensions. Using the Spanish National Health Survey 2011–2012, we apply intersectionality theory and multilevel analysis of individual heterogeneity and discriminatory accuracy (MAIHDA) to analyze 14,190 adults nested within 108 intersectional strata defined by combining categories of gender, age, income, educational achievement and living situation. We develop two multilevel models to obtain information on stratum-specific BMI averages and the degree of clustering of BMI within strata expressed by the intra-class correlation coefficient (ICC). The first model is a simple variance components analysis that provides a detailed mapping of the BMI disparities in the population and measures the accuracy of stratum membership to predict individual BMI. The second model includes the variables used to define the intersectional strata as a way to identify stratum-specific interactions. The first model suggests moderate but meaningful clustering of individual BMI within the intersectional strata (ICC = 12.4%). Compared with the population average (BMI = 26.07 Kg/m2), the stratum of cohabiting 18-35-year-old females with medium income and high education presents the lowest BMI (-3.7 Kg/m2), while cohabiting 36-64-year-old females with low income and low education show the highest BMI (+2.6 Kg/m2). In the second model, the ICC falls to 1.9%, suggesting the existence of only very small stratum specific interaction effects. We confirm the existence of a socioeconomic gradient in BMI. Compared with traditional analyses, the intersectional MAIHDA approach provides a better mapping of socioeconomic and demographic inequalities in BMI. Because of the moderate clustering, public health policies aiming to reduce BMI in Spain should not solely focus on the intersectional strata with the highest BMI, but should also consider whole population polices.

Principal Score Methods: Assumptions, Extensions, and Practical Considerations

Researchers addressing posttreatment complications in randomized trials often turn to principal stratification to define relevant assumptions and quantities of interest. One approach for the subsequent estimation of causal effects in this framework is to use methods based on the “principal score,” the conditional probability of belonging to a certain principal stratum given covariates. These methods typically assume that stratum membership is as good as randomly assigned, given these covariates. We clarify the key assumption in this context, known as principal ignorability, and argue that versions of this assumption are quite strong in practice. We describe these concepts in terms of both one- and two-sided noncompliance and propose a novel approach for researchers to “mix and match” principal ignorability assumptions with alternative assumptions, such as the exclusion restriction. Finally, we apply these ideas to randomized evaluations of a job training program and an early childhood education program. O...

Causal inference in longitudinal comparative effectiveness studies with repeated measures of a continuous intermediate variable

We propose a principal stratification approach to assess causal effects in nonrandomized longitudinal comparative effectiveness studies with a binary endpoint outcome and repeated measures of a continuous intermediate variable. Our method is an extension of the principal stratification approach originally proposed for the longitudinal randomized study "Prevention of Suicide in Primary Care Elderly: Collaborative Trial" to assess the treatment effect on the continuous Hamilton depression score adjusting for the heterogeneity of repeatedly measured binary compliance status. Our motivation for this work comes from a comparison of the effect of two glucose-lowering medications on a clinical cohort of patients with type 2 diabetes. Here, we consider a causal inference problem assessing how well the two medications work relative to one another on two binary endpoint outcomes: cardiovascular disease-related hospitalization and all-cause mortality. Clinically, these glucose-lowering medications can have differential effects on the intermediate outcome, glucose level over time. Ultimately, we want to compare medication effects on the endpoint outcomes among individuals in the same glucose trajectory stratum while accounting for the heterogeneity in baseline covariates (i.e., to obtain 'principal effects' on the endpoint outcomes). The proposed method involves a three-step model estimation procedure. Step 1 identifies principal strata associated with the intermediate variable using hybrid growth mixture modeling analyses. Step 2 obtains the stratum membership using the pseudoclass technique and derives propensity scores for treatment assignment. Step 3 obtains the stratum-specific treatment effect on the endpoint outcome weighted by inverse propensity probabilities derived from Step 2.

Assessing the sensitivity of methods for estimating principal causal effects

The framework of principal stratification provides a way to think about treatment effects conditional on post-randomization variables, such as level of compliance. In particular, the complier average causal effect (CACE) - the effect of the treatment for those individuals who would comply with their treatment assignment under either treatment condition - is often of substantive interest. However, estimation of the CACE is not always straightforward, with a variety of estimation procedures and underlying assumptions, but little advice to help researchers select between methods. In this article, we discuss and examine two methods that rely on very different assumptions to estimate the CACE: a maximum likelihood ('joint') method that assumes the 'exclusion restriction,' (ER) and a propensity score-based method that relies on 'principal ignorability.' We detail the assumptions underlying each approach, and assess each methods' sensitivity to both its own assumptions and those of the other method using both simulated data and a motivating example. We find that the ER-based joint approach appears somewhat less sensitive to its assumptions, and that the performance of both methods is significantly improved when there are strong predictors of compliance. Interestingly, we also find that each method performs particularly well when the assumptions of the other approach are violated. These results highlight the importance of carefully selecting an estimation procedure whose assumptions are likely to be satisfied in practice and of having strong predictors of principal stratum membership.

Modified Sieve Sampling: A Method for Single- and Multi-Stage Probability- Proportional-to-Size Sampling

SUMMARY: Widely used probability-proportional-to-size (PPS) selection methods are not well adapted to circumstances requiring sample augmentation. Limitations include: (1) an inability to augment selections while maintaining PPS properties, (2) a failure to recognize changes in census stratum membership which result from sample augmentation, and (3) imprecise control over line item sample size. This paper presents a new method of PPS selection, a modified version of sieve sampling which overcomes these limitations. Simulations indicate the new method effectively maintains sampling stratum PPS properties in single- and multi-stage samples, appropriately recognizes changes in census stratum membership which result from sample augmentation, and provides precise control over line item sample sizes. In single-stage applications the method provides reliable control of sampling risk over varied tainting levels and error bunching patterns. Tightness and efficiency measures are comparable to randomized systematic sampling and superior to sieve sampling.

On estimation of conditional density models with two-phase sampling

Suppose that the conditional density of a response variable given a vector of explanatory variables is parametrically modelled, and that data are collected by a two-phase sampling design. First, a simple random sample is drawn from the population. The stratum membership in a finite number of strata of the response and explanatory variables is recorded for each unit. Second, a subsample is drawn from the phase-one sample such that the selection probability is determined by the stratum membership. The response and explanatory variables are fully measured at this phase. We synthesize existing results on nonparametric likelihood estimation and present a streamlined approach for the computation and the large sample theory of profile likelihood in four different situations. The amount of information in terms of data and assumptions varies depending on whether the phase-one data are retained, the selection probabilities are known, and/or the stratum probabilities are known. We establish and illustrate numerically the order of efficiency among the maximum likelihood estimators, according to the amount of information utilized, in the four situations.

On the use of propensity scores in principal causal effect estimation.

We examine the practicality of propensity score methods for estimating causal treatment effects conditional on intermediate posttreatment outcomes (principal effects) in the context of randomized experiments. In particular, we focus on the sensitivity of principal causal effect estimates to violation of principal ignorability, which is the primary assumption that underlies the use of propensity score methods to estimate principal effects. Under principal ignorability (PI), principal strata membership is conditionally independent of the potential outcome under control given the pre-treatment covariates; i.e. there are no differences in the potential outcomes under control across principal strata given the observed pretreatment covariates. Under this assumption, principal scores modeling principal strata membership can be estimated based solely on the observed covariates and used to predict strata membership and estimate principal effects. While this assumption underlies the use of propensity scores in this setting, sensitivity to violations of it has not been studied rigorously. In this paper, we explicitly define PI using the outcome model (although we do not actually use this outcome model in estimating principal scores) and systematically examine how deviations from the assumption affect estimates, including how the strength of association between principal stratum membership and covariates modifies the performance. We find that when PI is violated, very strong covariate predictors of stratum membership are needed to yield accurate estimates of principal effects.

An adjustment for patient heterogeneity in the design of two‐stage phase II trials

In single-arm, two-stage phase II clinical trials to evaluate efficacy of cancer treatments using a response endpoint, one typically identifies a single reference response rate to be the null hypothesis benchmark. Patients eligible for the trial are assumed to have this response rate on average under the null hypothesis. When patients arise from subpopulations having different response rates, this single response rate reference may not be appropriate for the particular mix of patients actually enrolled on the trial. As a result, the Type I and Type II error rates conditional on the mix of enrolled patients may differ considerably from the unconditional error rates used to design the trial. We describe a method for designing two-stage Phase II studies that accounts for patient heterogeneity and effectively stabilizes conditional Type I and Type II error over the range of patient mixes that are likely to arise. Use of the design requires good estimates of the expected response rate within each population stratum as well as the stratum membership probabilities, but its properties are similar to and often preferable to the standard two-stage design even in situations where the underlying assumptions do not hold absolutely.

Using latent outcome trajectory classes in causal inference.

In longitudinal studies, outcome trajectories can provide important information about substantively and clinically meaningful underlying subpopulations who may also respond differently to treatments or interventions. Growth mixture analysis is an efficient way of identifying heterogeneous trajectory classes. However, given its exploratory nature, it is unclear how involvement of latent classes should be handled in the analysis when estimating causal treatment effects. In this paper, we propose a 2-step approach, where formulation of trajectory strata and identification of causal effects are separated. In Step 1, we stratify individuals in one of the assignment conditions (reference condition) into trajectory strata on the basis of growth mixture analysis. In Step 2, we estimate treatment effects for different trajectory strata, treating the stratum membership as partly known (known for individuals assigned to the reference condition and missing for the rest). The results can be interpreted as how subpopulations that differ in terms of outcome prognosis under one treatment condition would change their prognosis differently when exposed to another treatment condition. Causal effect estimation in Step 2 is consistent with that in the principal stratification approach (Frangakis and Rubin, 2002) in the sense that clarified identifying assumptions can be employed and therefore systematic sensitivity analyses are possible. Longitudinal development of attention deficit among children from the Johns Hopkins School Intervention Trial (Ialongo et al., 1999) will be presented as an example.

An estimated score method for missing and auxiliary covariates in two stage studies

We consider the following type of two stage studies. First, a full scale study is conducted by observing an outcome Y, imperfect surrogate measure T of the unobserved covariate of interest X, and covariates U for all members of a cohort, where the cohort is a random sample from some infinite, underlying population. Second, a validation study is conducted, in which we ascertain the previously unmeasured covariate X for a subset of the subjects selected at random within each category/stratum of the cohort. Stratum membership can depend on Y, T and U from the full scale study. Several methods of analysis are now available that one can use to estimate the relationship between Y and X in two stage studies, but each method has potential limitations, especially when both Y and X are continuous. We present a method of analysis for two stage study data with continuous and/or discrete variables. The proposed method generalizes the mean score method of Reilly and Pepe (Biometrika 82 (1995) 299) in two ways. First, it uses an estimating equations approach to estimate mean scores. Second, it applies even when the outcome and independent variables are continuous, provided sampling of subjects for the validation study is random within categories. Despite its greater applicability, this new approach retains many of the consistency properties of the mean score method. One can use the method of analysis for effect estimation with many study designs when error in measurement or misclassification of independent variables has occurred. Like the mean score method, this method of analysis uses an estimated score function. We assume some “regularity conditions”, but make no distributional assumptions about the misclassified variable or the measurement error. A limited simulation study was performed to evaluate the proposed method and two examples are presented to illustrate the proposed method.

The Jesus Movement: A Social History of Its First Century

The Jesus Movement: A Social History of Its First Century, by Ekkehard W. Stegemann and Wolfgang Stegemann, trans. 0. C. Dean, Jr. Minneapolis: Fortress Press, 1999. Pp. xx + 532. $45.00. This major work, published in German under the title Urchristliche Sozialgeschichte: Die Anfange im Judentum and die Christusgemeinden in der mediterranen Welt (Stuttgart: Kohlhammer, 1995) deserves a careful reading by all scholars in NT studies and related fields. It represents an achievement of substantial erudition and wide-ranging synthesis. The work divides into four parts: (1) Economy and society in the Mediterranean world of the first century: Ancient Mediterranean societies can be classified, following Lenski, as advanced agrarian societies. Technological level, power structure, population, cities, division of labor, money, writing, and social inequality are all important dimensions of the social picture. The authors examine production, distribution, consumption, and daily living conditions for various social strata. Power is a key variable in agrarian social stratification, and the economy is embedded in politics. The book's seven pyramidal models for stratification support careful discussions of social location. (2) The land of Israel, the social history of Judaism, and the followers of Jesus: The authors trace the development of political economy in Hellenistic-Roman Palestine; include a solid discussion of taxation; follow changes in social structure of HellenisticRoman Palestine; note the problem of debt; examine currents in a situation of crisis; characterize religious institutions and explain why deviance theory is preferable to sect theory; pay attention to various groups in relation to ideology and stratum of origin; and examine the relationship of religion and politics. Discussion then turns to Jesus and his movement. Pre-70 and post-70 C.E. contexts are distinguished. The Stegemanns prefer Weber's notion of charisma over sect or millenarian theoretical approaches to Jesus. They consider the relationship of the Jesus movement to other groups; focus especially on information in Matthew and John about conflict; and discuss the relationship of Christ-followers to synagogue and rabbinic movements. (3) The social history of Christ-confessing communities in the cities of the Roman empire: The urban Christ-confessing assemblies fostered unrestricted social interaction, especially at meals, between Jew and Gentile. Innovation is attributed to charisma. Ancient analogies are found in household, political structures, and voluntary associations, but not in army or ethnos. The authors trace stratum membership of Christ-confessing groups and show a mix of mostly upper non-elite and some lower elite; discuss conflict and persecution due to upper-stratum inclination to criminalize Christianity; consider Roman suspicions of Christians after 70 C.E. as an anti-Roman Jewish group; show political pressure on Diaspora Judaism; and draw on deviance theory as a way to understand social dynamics. (4) The social roles and social situation of women in the Mediterranean world and in early Christianity: With careful nuance in terms of stratum and culture, the authors examine the gendered nature of space and roles for ancient women; assess occupations women could engage in by stratum; and attempt to determine the role and status of women in the earliest Jesus movement. The reputation of such women in the entourage would have been questionable. The Stegemanns show how a certain equality of role and status for women reverted in the later church to conventional norms. The book's carefully constructed framework situates the social history of the groups behind the NT such that the political domain is a primary variable in the discussion. This is the proper emphasis. Attention to social structure is central for the analysis, and the social location of various Jesus and Christ-confessing groups is established in the account of historical developments. …

Semiparametric Methods for Response-Selective and Missing Data Problems in Regression

Summary Suppose that data are generated according to the model f(y|x; θ) g(x), where y is a response and x are covariates. We derive and compare semiparametric likelihood and pseudolikelihood methods for estimating θ for situations in which units generated are not fully observed and in which it is impossible or undesirable to model the covariate distribution. The probability that a unit is fully observed may depend on y, and there may be a subset of covariates which is observed only for a subsample of individuals. Our key assumptions are that the probability that a unit has missing data depends only on which of a finite number of strata that (y, x) belongs to and that the stratum membership is observed for every unit. Applications include case–control studies in epidemiology, field reliability studies and broad classes of missing data and measurement error problems. Our results make fully efficient estimation of θ feasible, and they generalize and provide insight into a variety of methods that have been proposed for specific problems.