Analysis of Matched Case–Control Data in Presence of Nonignorable Missing Exposure

Summary.   The present article deals with informative missing (IM) exposure data in matched case–control studies. When the missingness mechanism depends on the unobserved exposure values, modeling the missing data mechanism is inevitable. Therefore, a full likelihood-based approach for handling IM data has been proposed by positing a model for selection probability, and a parametric model for the partially missing exposure variable among the control population along with a disease risk model. We develop an EM algorithm to estimate the model parameters. Three special cases: (a) binary exposure variable, (b) normally distributed exposure variable, and (c) lognormally distributed exposure variable are discussed in detail. The method is illustrated by analyzing a real matched case–control data with missing exposure variable. The performance of the proposed method is evaluated through simulation studies, and the robustness of the proposed method for violation of different types of model assumptions has been considered.     

Conditional likelihood methods for haplotype-based association analysis using matched case-control data

Genetic epidemiologists routinely assess disease susceptibility in relation to haplotypes, that is, combinations of alleles on a single chromosome. We study statistical methods for inferring haplotype-related disease risk using single nucleotide polymorphism (SNP) genotype data from matched case-control studies, where controls are individually matched to cases on some selected factors. Assuming a logistic regression model for haplotype-disease association, we propose two conditional likelihood approaches that address the issue that haplotypes cannot be inferred with certainty from SNP genotype data (phase ambiguity). One approach is based on the likelihood of disease status conditioned on the total number of cases, genotypes, and other covariates within each matching stratum, and the other is based on the joint likelihood of disease status and genotypes conditioned only on the total number of cases and other covariates. The joint-likelihood approach is generally more efficient, particularly for assessing haplotype-environment interactions. Simulation studies demonstrated that the first approach was more robust to model assumptions on the diplotype distribution conditioned on environmental risk variables and matching factors in the control population. We applied the two methods to analyze a matched case-control study of prostate cancer.     

Statistical analysis for haplotype-based matched case-control studies

Using unphased genotype data, we studied statistical inference for association between a disease and a haplotype in matched case-control studies. Statistical inference for haplotype data is complicated due to ambiguity of genotype phases. An estimating equation-based method is developed for estimating odds ratios and testing disease-haplotype association. The method potentially can also be applied to testing haplotype-environment interaction. Simulation studies show that the proposed method has good performance. The performance of the method in the presence of departures from Hardy-Weinberg equilibrium is also studied.     

A semiparametric method for analyzing matched case-control family studies with a continuous outcome and proband sampling

We consider matched case-control familial studies which match a group of patients, called "case probands," with a group of disease-free subjects, called "control probands," using a set of family-level matching variables. Family members of each proband are then recruited into the study. Of interest here is the familial aggregation of the response variable and the effects of subject-specific covariates on the response. We propose an estimating equation approach to jointly estimate the main effects and intrafamilial correlations for matched family studies with a continuous outcome. Only knowledge of the first two joint moments of the response variable is required. The induced estimators for the main effects and intrafamilial correlations are consistent and asymptotically normally distributed. We apply the proposed method to sleep apnea data. A simulation study demonstrates the usefulness of our approach.     

A latent autoregressive model for longitudinal binary data subject to informative missingness

Longitudinal clinical trials often collect long sequences of binary data. Our application is a recent clinical trial in opiate addicts that examined the effect of a new treatment on repeated binary urine tests to assess opiate use over an extended follow-up. The dataset had two sources of missingness: dropout and intermittent missing observations. The primary endpoint of the study was comparing the marginal probability of a positive urine test over follow-up across treatment arms. We present a latent autoregressive model for longitudinal binary data subject to informative missingness. In this model, a Gaussian autoregressive process is shared between the binary response and missing-data processes, thereby inducing informative missingness. Our approach extends the work of others who have developed models that link the various processes through a shared random effect but do not allow for autocorrelation. We discuss parameter estimation using Monte Carlo EM and demonstrate through simulations that incorporating within-subject autocorrelation through a latent autoregressive process can be very important when longitudinal binary data is subject to informative missingness. We illustrate our new methodology using the opiate clinical trial data.     

Bivariate binary data analysis with nonignorably missing outcomes

One of the objectives in the Northern Manhattan Stroke Study is to investigate the impact of stroke subtype on the functional status 2 years after the first ischemic stroke. A challenge in this analysis is that the functional status at 2 years after stroke is not completely observed. In this paper, we propose a method to handle nonignorably missing binary functional status when the baseline value and the covariates are completely observed. The proposed method consists of fitting four separate binary regression models: for the baseline outcome, the outcome 2 years after the stroke, the product of the previous two, and finally, the missingness indicator. We then conduct a sensitivity analysis by varying the assumptions about the third and the fourth binary regression models. Our method belongs to an imputation paradigm and can be an alternative to the weighting method of Rotnitzky and Robins (1997, Statistics in Medicine 16, 81-102). A jackknife variance estimate is proposed for the variance of the resulting estimate. The proposed analysis can be implemented using statistical software such as SAS.     

Point-source modelling using matched case-control data

We describe an extension to matched case-control studies of the parametric modelling framework developed by Diggle (1990) and Diggle and Rowlingson (1994) to investigate raised risk around putative sources of environmental pollution. We use a conditional likelihood approach for the family of risk functions considered in Diggle and Rowlingson (1994). We show that the likelihood surface that results from these models may be highly irregular, and provide a Bayesian analysis in which we investigate the posterior distribution using Markov chain Monte Carlo. An analysis of one-one matched data that were collected to investigate the relationship between respiratory disease and distance to roads in East London is presented.     

Approaches for missing covariate data in logistic regression with MNAR sensitivity analyses

Data with missing covariate values but fully observed binary outcomes are an important subset of the missing data challenge. Common approaches are complete case analysis (CCA) and multiple imputation (MI). While CCA relies on missing completely at random (MCAR), MI usually relies on a missing at random (MAR) assumption to produce unbiased results. For MI involving logistic regression models, it is also important to consider several missing not at random (MNAR) conditions under which CCA is asymptotically unbiased and, as we show, MI is also valid in some cases. We use a data application and simulation study to compare the performance of several machine learning and parametric MI methods under a fully conditional specification framework (MI-FCS). Our simulation includes five scenarios involving MCAR, MAR, and MNAR under predictable and nonpredictable conditions, where “predictable” indicates missingness is not associated with the outcome. We build on previous results in the literature to show MI and CCA can both produce unbiased results under more conditions than some analysts may realize. When both approaches were valid, we found that MI-FCS was at least as good as CCA in terms of estimated bias and coverage, and was superior when missingness involved a categorical covariate. We also demonstrate how MNAR sensitivity analysis can build confidence that unbiased results were obtained, including under MNAR-predictable, when CCA and MI are both valid. Since the missingness mechanism cannot be identified from observed data, investigators should compare results from MI and CCA when both are plausibly valid, followed by MNAR sensitivity analysis.     

A semi-parametric shared parameter model to handle nonmonotone nonignorable missingness

Summary .  Longitudinal studies often generate incomplete response patterns according to a missing not at random mechanism. Shared parameter models provide an appealing framework for the joint modelling of the measurement and missingness processes, especially in the nonmonotone missingness case, and assume a set of random effects to induce the interdependence. Parametric assumptions are typically made for the random effects distribution, violation of which leads to model misspecification with a potential effect on the parameter estimates and standard errors. In this article we avoid any parametric assumption for the random effects distribution and leave it completely unspecified. The estimation of the model is then made using a semi-parametric maximum likelihood method. Our proposal is illustrated on a randomized longitudinal study on patients with rheumatoid arthritis exhibiting nonmonotone missingness.     

Sampling for conditional inference on case-control data

The problem of exact conditional inference for discrete multivariate case-control data has two forms. The first is grouped case-control data, where Monte Carlo computations can be done using the importance sampling method of Booth and Butler (1999, Biometrika86, 321-332), or a proposed alternative sequential importance sampling method. The second form is matched case-control data. For this analysis we propose a new exact sampling method based on the conditional-Poisson distribution for conditional testing with one binary and one integral ordered covariate. This method makes computations on data sets with large numbers of matched sets fast and accurate. We provide detailed derivation of the constraints and conditional distributions for conditional inference on grouped and matched data. The methods are illustrated on several new and old data sets.     

Covariate measurement error adjustment for matched case-control studies

We propose a conditional scores procedure for obtaining bias-corrected estimates of log odds ratios from matched case-control data in which one or more covariates are subject to measurement error. The approach involves conditioning on sufficient statistics for the unobservable true covariates that are treated as fixed unknown parameters. For the case of Gaussian nondifferential measurement error, we derive a set of unbiased score equations that can then be solved to estimate the log odds ratio parameters of interest. The procedure successfully removes the bias in naive estimates, and standard error estimates are obtained by resampling methods. We present an example of the procedure applied to data from a matched case-control study of prostate cancer and serum hormone levels, and we compare its performance to that of regression calibration procedures.     

Haplotype-based association analysis in cohort and nested case-control studies

Genetic epidemiologic studies often collect genotype data at multiple loci within a genomic region of interest from a sample of unrelated individuals. One popular method for analyzing such data is to assess whether haplotypes, i.e., the arrangements of alleles along individual chromosomes, are associated with the disease phenotype or not. For many study subjects, however, the exact haplotype configuration on the pair of homologous chromosomes cannot be derived with certainty from the available locus-specific genotype data (phase ambiguity). In this article, we consider estimating haplotype-specific association parameters in the Cox proportional hazards model, using genotype, environmental exposure, and the disease endpoint data collected from cohort or nested case-control studies. We study alternative Expectation-Maximization algorithms for estimating haplotype frequencies from cohort and nested case-control studies. Based on a hazard function of the disease derived from the observed genotype data, we then propose a semiparametric method for joint estimation of relative-risk parameters and the cumulative baseline hazard function. The method is greatly simplified under a rare disease assumption, for which an asymptotic variance estimator is also proposed. The performance of the proposed estimators is assessed via simulation studies. An application of the proposed method is presented, using data from the Alpha-Tocopherol, Beta-Carotene Cancer Prevention Study.     

Mixed-effect hybrid models for longitudinal data with nonignorable dropout

Summary .  Selection models and pattern-mixture models are often used to deal with nonignorable dropout in longitudinal studies. These two classes of models are based on different factorizations of the joint distribution of the outcome process and the dropout process. We consider a new class of models, called mixed-effect hybrid models (MEHMs), where the joint distribution of the outcome process and dropout process is factorized into the marginal distribution of random effects, the dropout process conditional on random effects, and the outcome process conditional on dropout patterns and random effects. MEHMs combine features of selection models and pattern-mixture models: they directly model the missingness process as in selection models, and enjoy the computational simplicity of pattern-mixture models. The MEHM provides a generalization of shared-parameter models (SPMs) by relaxing the conditional independence assumption between the measurement process and the dropout process given random effects. Because SPMs are nested within MEHMs, likelihood ratio tests can be constructed to evaluate the conditional independence assumption of SPMs. We use data from a pediatric AIDS clinical trial to illustrate the models.     

Semiparametric regression splines in matched case-control studies

We develop semiparametric methods for matched case-control studies using regression splines. Three methods are developed: 1) an approximate cross-validation scheme to estimate the smoothing parameter inherent in regression splines, as well as 2) Monte Carlo expectation maximization (MCEM) and 3) Bayesian methods to fit the regression spline model. We compare the approximate cross-validation approach, MCEM, and Bayesian approaches using simulation, showing that they appear approximately equally efficient; the approximate cross-validation method is computationally the most convenient. An example from equine epidemiology that motivated the work is used to demonstrate our approaches.     

Inverse sampling of controls in a matched case control study

A method of inverse sampling of controls in a matched case-control study is described in which, for each case, controls are sampled until a discordant set is achieved. For a binary exposure, inverse sampling is used to determine the number of controls for each case. When most individuals in a population have the same exposure, standard case-control sampling may result in many case-control sets being concordant with respect to exposure and thus uninformative in the conditional logistic analysis. The method using inverse control sampling is proposed as a solution to this problem in situations when it is practically feasible. In many circumstances, inverse control sampling is found to offer improved statistical efficiency relative to a comparable study with a fixed number of controls per case.