The International Journal of Biostatistics

Causal Inference for Vaccine Effects on Infectiousness

M. Elizabeth Halloran et al. — Fri, 06 Jan 2012 13:51:13 PST

If a vaccine does not protect individuals completely against infection, it could still reduce infectiousness of infected vaccinated individuals to others. Typically, vaccine efficacy for infectiousness is estimated based on contrasts between the transmission risk to susceptible individuals from infected vaccinated individuals compared with that from infected unvaccinated individuals. Such estimates are problematic, however, because they are subject to selection bias and do not have a causal interpretation. Here, we develop causal estimands for vaccine efficacy for infectiousness for four different scenarios of populations of transmission units of size two. These causal estimands incorporate both principal stratification, based on the joint potential infection outcomes under vaccine and control, and interference between individuals within transmission units. In the most general scenario, both individuals can be exposed to infection outside the transmission unit and both can be assigned either vaccine or control. The three other scenarios are special cases of the general scenario where only one individual is exposed outside the transmission unit or can be assigned vaccine. The causal estimands for vaccine efficacy for infectiousness are well defined only within certain principal strata and, in general, are identifiable only with strong unverifiable assumptions. Nonetheless, the observed data do provide some information, and we derive large sample bounds on the causal vaccine efficacy for infectiousness estimands. An example of the type of data observed in a study to estimate vaccine efficacy for infectiousness is analyzed in the causal inference framework we developed.

Meta-Analysis of Observational Studies with Unmeasured Confounders

Lawrence C. McCandless — Fri, 06 Jan 2012 13:51:10 PST

Meta-analysis of observational studies is an exciting new area of innovation in statistical science. Unlike randomized controlled trials, which are the gold standard for proving causation, observational studies are prone to biases including confounding. In this article, we describe a novel Bayesian procedure to control for a confounder that is missing across the sequence of studies in a meta-analysis. We motivate the discussion with the example of a meta-analysis of cohort, case-control and cross-sectional studies examining the relationship between oral contraceptives and endometriosis. An important unmeasured confounder is dysmennoreah, which is an indication for oral contraceptive use. To adjust for unmeasured confounding, we combine random effects models with probabilistic sensitivity analysis techniques. Information about the unmeasured confounder is incorporated into the analysis via prior distributions, and we use MCMC to sample from posterior.

Double-Robust Estimators: Slightly More Bayesian than Meets the Eye?

Paul Gustafson — Fri, 06 Jan 2012 13:51:07 PST

Consider the simple setting of point exposure, outcome and confounding variables, all of which are discrete. As is well known, parametric modeling of outcome given exposure and confounders and also exposure given confounders can yield a double-robust estimator. This has the property of being consistent as long as at least one of the two specified models is correct. Such an estimator can also be cast as arising from a compromise between the parametric outcome model and a nonparametric or saturated outcome model. This brings to mind an alternate compromise based on Bayesian model averaging, and prompts comparisons between the double-robust method and the Bayesian method.

Bounds on the Effect of Vaccine Induced Immune Response on Outcome

Dean Follmann et al. — Fri, 06 Jan 2012 13:51:04 PST

A major goal of vaccine development is the identification of immune responses that are responsible for vaccine efficacy. In theory, modest vaccines could be successfully improved by increasing such immune responses. And for a vaccine with a great benefit in one population, inducing such immune response in a different population could help one conclude the vaccine would have great benefit there. Such identification is tricky because the immune response to vaccination can only be measured in the vaccine group and thus immune responses might only be identifying individuals with a constitutional ability to remain uninfected, rather than being causal. Define the vaccine induced immune response as X(1). The value X(1) is a potential outcome; it is measured directly in vaccinees but unobserved in the placebo group. Our goal is to regress outcome on X(1) separately in the vaccine and placebo groups and to see if the vaccine effect varies with X(1). Put another way, our goal is to see if there is a vaccine by X(1) interaction. Regression of outcome on X(1) is easy to do in the vaccine group, but difficult in the placebo group as X(1) is not observed. In this paper we derive bounds on the regression curve in the placebo group. For a continuous endpoint these bounds can be unhelpful, or can help modestly temper our enthusiasm for a role of X(1) on the vaccine effect. For binary outcomes with 100% placebo infection the bound is very tight but unhelpful as 100% infection precludes identification of any covariate with a differential effect on placebo infection. We apply these methods to experiments of anthrax vaccine in rabbits with survival to challenge as the outcome and demonstrate how to extrapolate the model to humans.

Bias Analysis to Guide New Data Collection

Timothy L. Lash et al. — Fri, 06 Jan 2012 13:51:01 PST

Bias analysis serves multiple objectives in epidemiologic data analysis. The objectives most often emphasized are quantification of uncertainty due to systematic errors and reduction in overconfidence by specifying hypotheses that compete with the causal hypothesis. A third objective is the utility of bias analysis to identify strategies for new data collection that will be productive in evaluating the validity of an association. The authors illustrate the value of this objective using two examples. The first example examines the value of comprehensive CYP2D6 genotyping in a study of tamoxifen resistance. Tamoxifen is metabolized primarily by CYP2D6 to more active forms. More than thirty polymorphisms in the CYP2D6 gene reduce its function. We genotyped the most prevalent CYP2D6 polymorphism and found a null association between genotype and breast cancer recurrence in a Danish population. One possibility is that incomplete genotyping of the multiple functional polymorphisms introduced non-differential misclassification and biased the association toward the null. We used bias analysis to evaluate the plausibility of this explanation and to guide a decision about devoting study resources toward more comprehensive genotyping of other polymorphisms in the CYP2D6 gene. The second example examines the association between vitamin K antagonist (VKA) therapy and the incidence of 24 site-specific cancers, using heart valve replacement as an instrumental variable. Earlier studies suggested a protective association between VKA anticoagulants and the incidence of cancer. We observed a null-centered distribution of associations, which may be due to non-differential misclassification of VKA therapy by the instrument. We used bias analysis to evaluate whether this misclassification was likely to explain the null-centered distribution of associations and to guide decisions about conducting a more expensive validation study. In the first example, the bias analysis showed that new data collection would be required to resolve the uncertainty, whereas the second example showed that new data collection was unlikely to be a productive use of scarce study resources.

Special Issue on Causal Inference in Health Research

Erica E. M. Moodie et al. — Fri, 06 Jan 2012 13:50:57 PST

We provide a brief editorial introduction to a special issue of The International Journal of Biostatistics dedicated to some of the papers presented at a workshop held at the Centre de recherches mathématiques in Montreal, Québec, in May 2011.

Targeted Maximum Likelihood Estimation of Natural Direct Effects

Wenjing Zheng et al. — Fri, 06 Jan 2012 13:44:09 PST

In many causal inference problems, one is interested in the direct causal effect of an exposure on an outcome of interest that is not mediated by certain intermediate variables. Robins and Greenland (1992) and Pearl (2001) formalized the definition of two types of direct effects (natural and controlled) under the counterfactual framework. The efficient scores (under a nonparametric model) for the various natural effect parameters and their general robustness conditions, as well as an estimating equation based estimator using the efficient score, are provided in Tchetgen Tchetgen and Shpitser (2011b). In this article, we apply the targeted maximum likelihood framework of van der Laan and Rubin (2006) and van der Laan and Rose (2011) to construct a semiparametric efficient, multiply robust, substitution estimator for the natural direct effect which satisfies the efficient score equation derived in Tchetgen Tchetgen and Shpitser (2011b). We note that the robustness conditions in Tchetgen Tchetgen and Shpitser (2011b) may be weakened, thereby placing less reliance on the estimation of the mediator density. More precisely, the proposed estimator is asymptotically unbiased if either one of the following holds: i) the conditional mean outcome given exposure, mediator, and confounders, and the mediated mean outcome difference are consistently estimated; (ii) the exposure mechanism given confounders, and the conditional mean outcome are consistently estimated; or (iii) the exposure mechanism and the mediator density, or the exposure mechanism and the conditional distribution of the exposure given confounders and mediator, are consistently estimated. If all three conditions hold, then the effect estimate is asymptotically efficient. Extensions to the natural indirect effect are also discussed.

Designs Combining Instrumental Variables with Case-Control: Estimating Principal Strata Causal Effects

Russell T. Shinohara et al. — Fri, 06 Jan 2012 13:44:04 PST

The instrumental variables framework is commonly used for the estimation of causal effects from cohort samples. However, the combination of instrumental variables with more efficient designs such as case-control sampling requires new methodological consideration. For example, as the use of Mendelian randomization studies is increasing and the cost of genotyping and gene expression data can be high, the analysis of data gathered from more cost-effective sampling designs is of prime interest. We show that the standard instrumental variables analysis does not appropriately estimate the causal effects of interest when the instrumental variables design is combined with the case-control design. We also propose a method that can estimate the causal effects in such combined designs. We illustrate the method with a study in oncology.

Cut-Off Estimation and Medical Decision Making Based on a Continuous Prognostic Factor: The Prediction of Kidney Graft Failure

Yohann Foucher et al. — Fri, 06 Jan 2012 12:41:35 PST

The determination of a cut-off value for a continuous prognostic test is an important problem, which is statistically challenging and practically important for risk assessment. We propose in this paper a method to estimate the optimal cut-off from this type of longitudinal data with censored failure times. The principle is to combine the prognostic error rates of false positives and false negatives with a cost function, which has the advantages to be statistically convenient and to be directly associated with the decision-making. Simulations were performed and the results demonstrate the interest of our approach compared to a reference method. The method is also illustrated by predicting the long-term survival of kidney transplant recipients from the 1-year creatinine clearance.

Dose-Finding Designs: The Role of Convergence Properties

Assaf P. Oron et al. — Thu, 27 Oct 2011 16:32:01 PDT

It is common for novel dose-finding designs to be presented without a study of their convergence properties. In this article we suggest that examination of convergence is a necessary quality check for dose-finding designs. We present a new convergence proof for a nonparametric family of methods called “interval designs,” under certain conditions on the toxicity-frequency function F. We compare these conditions with the convergence conditions for the popular CRM one-parameter Phase I cancer design, via an innovative numerical sensitivity study generating a diverse sample of dose-toxicity scenarios. Only a small fraction of scenarios meet the Shen-O'Quigley convergence conditions for CRM. Conditions for “interval design” convergence are met more often, but still less than half the time. In the discussion, we illustrate how convergence properties and limitations help provide insight about small-sample behavior.

Super Learner Based Conditional Density Estimation with Application to Marginal Structural Models

Iván Díaz Muñoz et al. — Mon, 03 Oct 2011 17:35:03 PDT

In this paper, we present a histogram-like estimator of a conditional density that uses cross-validation to estimate the histogram probabilities, as well as the optimal number and position of the bins. This estimator is an alternative to kernel density estimators when the dimension of the covariate vector is large. We demonstrate its applicability to estimation of Marginal Structural Model (MSM) parameters in which an initial estimator of the exposure mechanism is needed. MSM estimation based on the proposed density estimator results in less biased estimates, when compared to estimates based on a misspecified parametric model.

Ridge Regression for Longitudinal Biomarker Data

Melissa Eliot et al. — Tue, 27 Sep 2011 08:14:41 PDT

Technological advances facilitating the acquisition of large arrays of biomarker data have led to new opportunities to understand and characterize disease progression over time. This creates an analytical challenge, however, due to the large numbers of potentially informative markers, the high degrees of correlation among them, and the time-dependent trajectories of association. We propose a mixed ridge estimator, which integrates ridge regression into the mixed effects modeling framework in order to account for both the correlation induced by repeatedly measuring an outcome on each individual over time, as well as the potentially high degree of correlation among possible predictor variables. An expectation-maximization algorithm is described to account for unknown variance and covariance parameters. Model performance is demonstrated through a simulation study and an application of the mixed ridge approach to data arising from a study of cardiometabolic biomarker responses to evoked inflammation induced by experimental low-dose endotoxemia.

Commentary on "Principal Stratification — a Goal or a Tool?" by Judea Pearl

Peter B. Gilbert et al. — Tue, 20 Sep 2011 12:03:51 PDT

This commentary takes up Pearl's welcome challenge to clearly articulate the scientific value of principal stratification estimands that we and colleagues have investigated, in the area of randomized placebo-controlled preventive vaccine efficacy trials, especially trials of HIV vaccines. After briefly arguing that certain principal stratification estimands for studying vaccine effects on post-infection outcomes are of genuine scientific interest, the bulk of our commentary argues that the “causal effect predictiveness” (CEP) principal stratification estimand for evaluating immune biomarkers as surrogate endpoints is not of ultimate scientific interest, because it evaluates surrogacy restricted to the setting of a particular vaccine efficacy trial, but is nevertheless useful for guiding the selection of primary immune biomarker endpoints in Phase I/II vaccine trials and for facilitating assessment of transportability/bridging surrogacy.

Principal Stratification and Attribution Prohibition: Good Ideas Taken Too Far

Marshall Joffe — Wed, 14 Sep 2011 17:37:43 PDT

Pearl’s article provides a useful springboard for discussing further the benefits and drawbacks of principal stratification and the associated discomfort with attributing effects to post-treatment variables. The basic insights of the approach are important: pay close attention to modification of treatment effects by variables not observable before treatment decisions are made, and be careful in attributing effects to variables when counterfactuals are ill-defined. These insights have often been taken too far in many areas of application of the approach, including instrumental variables, censoring by death, and surrogate outcomes. A novel finding is that the usual principal stratification estimand in the setting of censoring by death is by itself of little practical value in estimating intervention effects.

Antihypertensive Medication Use and Change in Kidney Function in Elderly Adults: A Marginal Structural Model Analysis

Michelle C. Odden et al. — Thu, 08 Sep 2011 02:36:20 PDT

Background: The evidence for the effectiveness of antihypertensive medication use for slowing decline in kidney function in older persons is sparse. We addressed this research question by the application of novel methods in a marginal structural model.

Methods: Change in kidney function was measured by two or more measures of cystatin C in 1,576 hypertensive participants in the Cardiovascular Health Study over 7 years of follow-up (1989-1997 in four U.S. communities). The exposure of interest was antihypertensive medication use. We used a novel estimator in a marginal structural model to account for bias due to confounding and informative censoring.

Results: The mean annual decline in eGFR was 2.41 ± 4.91 mL/min/1.73 m². In unadjusted analysis, antihypertensive medication use was not associated with annual change in kidney function. Traditional multivariable regression did not substantially change these estimates. Based on a marginal structural analysis, persons on antihypertensives had slower declines in kidney function; participants had an estimated 0.88 (0.13, 1.63) ml/min/1.73 m² per year slower decline in eGFR compared with persons on no treatment. In a model that also accounted for bias due to informative censoring, the estimate for the treatment effect was 2.23 (-0.13, 4.59) ml/min/1.73 m² per year slower decline in eGFR.

Conclusion: In summary, estimates from a marginal structural model suggested that antihypertensive therapy was associated with preserved kidney function in hypertensive elderly adults. Confirmatory studies may provide power to determine the strength and validity of the findings.

On Causal Mediation Analysis with a Survival Outcome

Eric J. Tchetgen Tchetgen — Fri, 02 Sep 2011 16:08:32 PDT

Suppose that having established a marginal total effect of a point exposure on a time-to-event outcome, an investigator wishes to decompose this effect into its direct and indirect pathways, also known as natural direct and indirect effects, mediated by a variable known to occur after the exposure and prior to the outcome. This paper proposes a theory of estimation of natural direct and indirect effects in two important semiparametric models for a failure time outcome. The underlying survival model for the marginal total effect and thus for the direct and indirect effects, can either be a marginal structural Cox proportional hazards model, or a marginal structural additive hazards model. The proposed theory delivers new estimators for mediation analysis in each of these models, with appealing robustness properties. Specifically, in order to guarantee ignorability with respect to the exposure and mediator variables, the approach, which is multiply robust, allows the investigator to use several flexible working models to adjust for confounding by a large number of pre-exposure variables. Multiple robustness is appealing because it only requires a subset of working models to be correct for consistency; furthermore, the analyst need not know which subset of working models is in fact correct to report valid inferences. Finally, a novel semiparametric sensitivity analysis technique is developed for each of these models, to assess the impact on inference, of a violation of the assumption of ignorability of the mediator.

Modeling Fetal Weight for Gestational Age: A Comparison of a Flexible Multi-level Spline-based Model with Other Approaches

Luc Villandré et al. — Tue, 23 Aug 2011 09:15:56 PDT

We present a model for longitudinal measures of fetal weight as a function of gestational age. We use a linear mixed model, with a Box-Cox transformation of fetal weight values, and restricted cubic splines, in order to flexibly but parsimoniously model median fetal weight. We systematically compare our model to other proposed approaches. All proposed methods are shown to yield similar median estimates, as evidenced by overlapping pointwise confidence bands, except after 40 completed weeks, where our method seems to produce estimates more consistent with observed data. Sex-based stratification affects the estimates of the random effects variance-covariance structure, without significantly changing sex-specific fitted median values. We illustrate the benefits of including sex-gestational age interaction terms in the model over stratification. The comparison leads to the conclusion that the selection of a model for fetal weight for gestational age can be based on the specific goals and configuration of a given study without affecting the precision or value of median estimates for most gestational ages of interest.

The Relative Performance of Targeted Maximum Likelihood Estimators

Kristin E. Porter et al. — Wed, 17 Aug 2011 15:09:01 PDT

There is an active debate in the literature on censored data about the relative performance of model based maximum likelihood estimators, IPCW-estimators, and a variety of double robust semiparametric efficient estimators. Kang and Schafer (2007) demonstrate the fragility of double robust and IPCW-estimators in a simulation study with positivity violations. They focus on a simple missing data problem with covariates where one desires to estimate the mean of an outcome that is subject to missingness. Responses by Robins, et al. (2007), Tsiatis and Davidian (2007), Tan (2007) and Ridgeway and McCaffrey (2007) further explore the challenges faced by double robust estimators and offer suggestions for improving their stability. In this article, we join the debate by presenting targeted maximum likelihood estimators (TMLEs). We demonstrate that TMLEs that guarantee that the parametric submodel employed by the TMLE procedure respects the global bounds on the continuous outcomes, are especially suitable for dealing with positivity violations because in addition to being double robust and semiparametric efficient, they are substitution estimators. We demonstrate the practical performance of TMLEs relative to other estimators in the simulations designed by Kang and Schafer (2007) and in modified simulations with even greater estimation challenges.

Invited Commentary on Pearl and Principal Stratification

Ross Prentice — Sat, 13 Aug 2011 09:21:56 PDT

Pearl (2011) posed the question of whether confinement of clinical trial analyses involving post-randomization variables to the principal stratum “framework” of Frangakis and Rubin (2002) unduly restricts the scientific questions that can be asked. Frangakis and Rubin illustrated their proposal through examples involving compliance, mediation, and surrogacy. Here the utility of the principal stratum framework, and the potential outcomes formulation from which it derives, are considered for these topics in the specific setting of the Women’s Health Initiative randomized, placebo controlled trials of postmenopausal hormone therapy. It is argued that the essential issues related to study reliability and causal interpretation involve the avoidance of context-specific biases that are typically not closely related to whether or not treatment effects have a representation in terms of potential outcomes contrasts. Also, while the questions posed within principal strata may be of interest, some key questions in the hormone therapy setting would not be addressed if restricted to contrasts within principal strata.

Combining Censored and Uncensored Data in a U-Statistic: Design and Sample Size Implications for Cell Therapy Research

Lemuel A. Moyé et al. — Fri, 22 Jul 2011 14:09:16 PDT

The assumptions that anchor large clinical trials are rooted in smaller, Phase II studies. In addition to specifying the target population, intervention delivery, and patient follow-up duration, physician-scientists who design these Phase II studies must select the appropriate response variables (endpoints). However, endpoint measures can be problematic. If the endpoint assesses the change in a continuous measure over time, then the occurrence of an intervening significant clinical event (SCE), such as death, can preclude the follow-up measurement. Finally, the ideal continuous endpoint measurement may be contraindicated in a fraction of the study patients, a change that requires a less precise substitution in this subset of participants.

A score function that is based on the U-statistic can address these issues of 1) intercurrent SCE’s and 2) response variable ascertainments that use different measurements of different precision. The scoring statistic is easy to apply, clinically relevant, and provides flexibility for the investigators’ prospective design decisions. Sample size and power formulations for this statistic are provided as functions of clinical event rates and effect size estimates that are easy for investigators to identify and discuss. Examples are provided from current cardiovascular cell therapy research.