Analysis of matched case-control data with multiple ordered disease states: possible choices and comparisons

In an individually matched case-control study, effects of potential risk factors are ascertained through conditional logistic regression (CLR). Extension of CLR to situations with multiple disease or reference categories has been made through polychotomous CLR and is shown to be more efficient than carrying out separate CLRs for each subgroup. In this paper, we consider matched case-control studies where there is one control group, but there are multiple disease states with a natural ordering among themselves. This scenario can be observed when the cases can be further classified in terms of the seriousness or progression of the disease, for example, according to different stages of cancer. We explore several popular models for ordered categorical data in this context. We first adopt a cumulative logit or equivalently, a proportional-odds model to account for the ordinal nature of the data. The important distinction of this model from a stratified dichotomous and polychotomous logistic regression model is that the stratum-specific nuisance parameters cannot be eliminated in this model via the conditional-likelihood approach. We discuss a Mantel-Haenszel approach for analysing such data. We point out possible difficulties with standard likelihood-based approaches with the cumulative logit model when applied to case-control data. We then consider an alternative conditional adjacent-category logit model. We illustrate the methods by analysing data from a matched case-control study on low birthweight in newborns where infants are classified according to low and very low birthweight and a child with normal birthweight serves as a control. A simulation study compares the different ordinal methods with methods ignoring sub-classification of the ordered disease states.     

Assessment of stratum-covariate interactions in Cox's proportional hazards regression model

We assess stratum (e.g. treatment) interactions with covariates and with the baseline hazard function in the proportional hazards (PH) regression model for lifetime data. We consider models incorporating stratum interactions both with and without stratification of the risk sets in the likelihood function, and describe likelihood ratio statistics for tests of the presence of these interactions. We also present step-down methods for building reduced models which include stratum-specific parameters corresponding to covariates which interact with treatment. We apply PH models with such interactions to a clinical trial of DES in the treatment of prostate cancer to determine optimal treatment conditional on each patient's covariates.     

Likelihood approaches for proportional likelihood ratio model with right‐censored data

Regression methods for survival data with right censoring have been extensively studied under semiparametric transformation models such as the Cox regression model and the proportional odds model. However, their practical application could be limited because of possible violation of model assumption or lack of ready interpretation for the regression coefficients in some cases. As an alternative, in this paper, the proportional likelihood ratio model introduced by Luo and Tsai is extended to flexibly model the relationship between survival outcome and covariates. This model has a natural connection with many important semiparametric models such as generalized linear model and density ratio model and is closely related to biased sampling problems. Compared with the semiparametric transformation model, the proportional likelihood ratio model is appealing and practical in many ways because of its model flexibility and quite direct clinical interpretation. We present two likelihood approaches for the estimation and inference on the target regression parameters under independent and dependent censoring assumptions. Based on a conditional likelihood approach using uncensored failure times, a numerically simple estimation procedure is developed by maximizing a pairwise pseudo‐likelihood. We also develop a full likelihood approach, and the most efficient maximum likelihood estimator is obtained by a profile likelihood. Simulation studies are conducted to assess the finite‐sample properties of the proposed estimators and compare the efficiency of the two likelihood approaches. An application to survival data for bone marrow transplantation patients of acute leukemia is provided to illustrate the proposed method and other approaches for handling non‐proportionality. The relative merits of these methods are discussed in concluding remarks. Copyright © 2014 John Wiley & Sons, Ltd.     

Graphical diagnostics to check model misspecification for the proportional odds regression model

The cumulative logit or the proportional odds regression model is commonly used to study covariate effects on ordinal responses. This paper provides some graphical and numerical methods for checking the adequacy of the proportional odds regression model. The methods focus on evaluating functional misspecification for specific covariate effects, but misspecification of the link function can also be dealt with under the same framework. For the logistic regression model with binary responses, Arbogast and Lin (Statist. Med. 2005; 24:229-247) developed similar graphical and numerical methods for assessing the adequacy of the model using the cumulative sums of residuals. The paper generalizes their methods to ordinal responses and illustrates them using an example from the VA Normative Aging Study. Simulation studies comparing the performance of the different diagnostic methods indicate that some of the graphical methods are more powerful in detecting model misspecification than the Hosmer-Lemeshow-type goodness-of-fit statistics for the class of models studied.     

POPULATION-AVERAGED AND CLUSTER-SPECIFIC MODELS FOR CLUSTERED ORDINAL RESPONSE DATA

We compare population-averaged and cluster-specific models for clustered ordinal data. We consider generalized estimating equations and constrained equations maximum likelihood estimation of population-averaged cumulative logit regression models, and mixed effects estimation of cluster-specific cumulative logit regression models. A previously reported relationship between population-averaged and cluster-specific parameters for the binary logistic link appears to hold for analogous parameters under the cumulative logit link. We address these issues in the context of data from two cross-over clinical trials.     

Conditional pseudolikelihood methods for clustered ordinal,multinomial, or count outcomes with complex survey data

In order to adjust individual‐level covariate effects for confounding due to unmeasured neighborhood characteristics, we have recently developed conditional pseudolikelihood methods to estimate the parameters of a proportional odds model for clustered ordinal outcomes with complex survey data. The methods require sampling design joint probabilities for each within‐neighborhood pair. In the present article, we develop a similar methodology for a baseline category logit model for clustered multinomial outcomes and for a loglinear model for clustered count outcomes. All of the estimators and asymptotic sampling distributions we present can be conveniently computed using standard logistic regression software for complex survey data, such as sas proc surveylogistic . We demonstrate validity of the methods theoretically and also empirically by using simulations. We apply the new method for clustered multinomial outcomes to data from the 2008 Florida Behavioral Risk Factor Surveillance System survey in order to investigate disparities in frequency of dental cleaning both unadjusted and adjusted for confounding by neighborhood. Copyright © 2012 John Wiley & Sons, Ltd.     

Modelling ordinal responses from co-twin control studies

The co-twin control design has been widely used in studying the effects of environmental factors on the development of diseases. For binary outcomes that arise from co-twin control studies, the conditional likelihood method is commonly used. This approach, however, does not readily extend to ordinal response data because the standard conditional likelihood does not exist for cumulative logit or proportional odds models. In this paper, we investigate the applicability of the random-effects and GEE approaches in analysing ordinal response data from co-twin control studies. Using both approaches, we re-analyse data from a co-twin control study of the impact of military services during the Vietnam era on post-traumatic stress disorders (PTSD). The ordinal models have considerably increased power in detecting the effects of exposure when compared to the analyses using a dichotomized response. We discuss the interpretation of the estimates from GEE and random-effects models in the context of the twin data. © 1998 John Wiley & Sons, Ltd.     

Assessment of skewed exposure in case-control studies with pooling

Pooling-based strategies that combine samples from multiple participants for laboratory assays have been proposed for epidemiologic investigations of biomarkers to address issues including cost, efficiency, detection, and when minimal sample volume is available. A modification of the standard logistic regression model has been previously described to allow use with pooled data; however, this model makes assumptions regarding exposure distribution and logit-linearity of risk (i.e., constant odds ratio) that can be violated in practice. We were motivated by a nested case-control study of miscarriage and inflammatory factors with highly skewed distributions to develop a more flexible model for analysis of pooled data. Using characteristics of the gamma distribution and the relation between models of binary outcome conditional on exposure and of exposure conditional on outcome, we use a modified logistic regression to accommodate nonlinearity because of unequal shape parameters in gamma distributed exposure for cases and controls. Using simulations, we compare our approach with existing methods for logistic regression for pooled data considering: (1) constant and dose-dependent effects; (2) gamma and log-normal distributed exposure; (3) effect size; and (4) the proportions of biospecimens pooled. We show that our approach allows estimation of odds ratios that vary with exposure level, yet has minimal loss of efficiency compared with existing approaches when exposure effects are dose-invariant. Our model performed similarly to a maximum likelihood estimation approach in terms of bias and efficiency, and provides an easily implemented approach for estimation with pooled biomarker data when effects may not be constant across exposure. Copyright ? 2012 John Wiley & Sons, Ltd.     

Relative risk,risk difference and rate difference models for sparse stratified data: A pseudo likelihood approach

We consider a relative risk and a risk difference model for binomial data, and a rate difference model for Poisson )person year( data. It is assumed that the data are stratified in a large number of small strata. If each stratum has its own parameter in the model, then, due to the large number of parameters, straightforward maximum likelihood leads to inconsistent estimates of the relevant parameters. By contrast to the logistic model, conditioning on the number of events per stratum does not help in eliminating the stratum nuisance parameters. We propose a pseudo likelihood method to overcome these consistency problems. The resulting pseudo maximum likelihood estimates can easily be computed with standard statistical software. Our approach gives a more general framework for the Mantel–Haenszel type estimators proposed in the literature. In the special case of a series of 2 × 2 tables, for the risk and rate difference models, our approach yields exactly these ad hoc Mantel–Haenszel estimators, while for the relative risk model it gives a close approximation of the Mantel–Haenszel relative risk estimator. For the regression models corresponding to the association measures relative risk, risk difference and rate difference, our method provides analogues of conditional logistic regression, which were not previously available.     

A multistate Markov chain model for longitudinal, categorical quality-of-life data subject to non-ignorable missingness

Quality-of-life (QOL) is an important outcome in clinical research, particularly in cancer clinical trials. Typically, data are collected longitudinally from patients during treatment and subsequent follow-up. Missing data are a common problem, and missingness may arise in a non-ignorable fashion. In particular, the probability that a patient misses an assessment may depend on the patient's QOL at the time of the scheduled assessment. We propose a Markov chain model for the analysis of categorical outcomes derived from QOL measures. Our model assumes that transitions between QOL states depend on covariates through generalized logit models or proportional odds models. To account for non-ignorable missingness, we incorporate logistic regression models for the conditional probabilities of observing measurements, given their actual values. The model can accommodate time-dependent covariates. Estimation is by maximum likelihood, summing over all possible values of the missing measurements. We describe options for selecting parsimonious models, and we study the finite-sample properties of the estimators by simulation. We apply the techniques to data from a breast cancer clinical trial in which QOL assessments were made longitudinally, and in which missing data frequently arose.     

Bayesian inference for the stereotype regression model: Application to a case–control study of prostate cancer

The stereotype regression model for categorical outcomes, proposed by Anderson (J. Roy. Statist. Soc. B. 1984; 46 :1–30) is nested between the baseline‐category logits and adjacent category logits model with proportional odds structure. The stereotype model is more parsimonious than the ordinary baseline‐category (or multinomial logistic) model due to a product representation of the log‐odds‐ratios in terms of a common parameter corresponding to each predictor and category‐specific scores. The model could be used for both ordered and unordered outcomes. For ordered outcomes, the stereotype model allows more flexibility than the popular proportional odds model in capturing highly subjective ordinal scaling which does not result from categorization of a single latent variable, but are inherently multi‐dimensional in nature. As pointed out by Greenland (Statist. Med. 1994; 13 :1665–1677), an additional advantage of the stereotype model is that it provides unbiased and valid inference under outcome‐stratified sampling as in case–control studies. In addition, for matched case–control studies, the stereotype model is amenable to classical conditional likelihood principle, whereas there is no reduction due to sufficiency under the proportional odds model. In spite of these attractive features, the model has been applied less, as there are issues with maximum likelihood estimation and likelihood‐based testing approaches due to non‐linearity and lack of identifiability of the parameters. We present comprehensive Bayesian inference and model comparison procedure for this class of models as an alternative to the classical frequentist approach. We illustrate our methodology by analyzing data from The Flint Men's Health Study, a case–control study of prostate cancer in African‐American men aged 40–79 years. We use clinical staging of prostate cancer in terms of Tumors, Nodes and Metastasis as the categorical response of interest. Copyright © 2009 John Wiley & Sons, Ltd.     

Mediation analysis for common binary outcomes

Mediation analysis provides an attractive causal inference framework to decompose the total effect of an exposure on an outcome into natural direct effects and natural indirect effects acting through a mediator. For binary outcomes, mediation analysis methods have been developed using logistic regression when the binary outcome is rare. These methods will not hold in practice when a disease is common. In this paper, we develop mediation analysis methods that relax the rare disease assumption when using logistic regression. We calculate the natural direct and indirect effects for common diseases by exploiting the relationship between logit and probit models. Specifically, we derive closed-form expressions for the natural direct and indirect effects on the odds ratio scale. Mediation models for both continuous and binary mediators are considered. We demonstrate through simulation that the proposed method performs well for common binary outcomes. We apply the proposed methods to analyze the Normative Aging Study to identify DNA methylation sites that are mediators of smoking behavior on the outcome of obstructed airway function.     

Efficient semiparametric mean-association estimation for longitudinal binary responses

Semiparametric methods for longitudinal data with association within subjects have recently received considerable attention. However, existing methods for semiparametric longitudinal binary regression modeling (i) mainly concern mean structures with association parameters treated as nuisance; (ii) generally require a correct specification of the covariance structure for misspecified covariance structure may lead to inefficient mean parameter estimates; and (iii) usually run into computation and estimation problems when the time points are irregularly and possibly subject specific. In this article, we propose a semiparametric logistic regression model, which simultaneously takes into account both the mean and response-association structures (via conditional log-odds ratio) for multivariate longitudinal binary outcomes. Our main interest lies in efficient estimation of both the marginal and association parameters. The estimators of the parameters are obtained via the profile kernel approach. We evaluate the proposed methodology through simulation studies and apply it to a real dataset. Both theoretical and empirical results demonstrate that the proposed method yields highly efficient estimators and performs satisfactorily.     

Multivariate analysis for matched case-control studies

A multivariate method based on the linear logistic model is presented for the analysis of case-control studies with pairwise matching. This technique enables one to investigate the effect of several variables simultaneously in the analysis while allowing for the matched design. The odds ratio is used as the basic measure of risk. One is able to control for variables which are not matching variables while investigating the odds ratio for a particular factor, and to estimate the change in the odds ratio as the level of one or more interval variables changes. The computing methods used for obtaining maximum conditional likelihood estimates of the parameters of interest are modifications of standard programs for logit regression.     

Doubly robust conditional logistic regression

Epidemiologic research often aims to estimate the association between a binary exposure and a binary outcome, while adjusting for a set of covariates (eg, confounders). When data are clustered, as in, for instance, matched case-control studies and co-twin-control studies, it is common to use conditional logistic regression. In this model, all cluster-constant covariates are absorbed into a cluster-specific intercept, whereas cluster-varying covariates are adjusted for by explicitly adding these as explanatory variables to the model. In this paper, we propose a doubly robust estimator of the exposure-outcome odds ratio in conditional logistic regression models. This estimator protects against bias in the odds ratio estimator due to misspecification of the part of the model that contains the cluster-varying covariates. The doubly robust estimator uses two conditional logistic regression models for the odds ratio, one prospective and one retrospective, and is consistent for the exposure-outcome odds ratio if at least one of these models is correctly specified, not necessarily both. We demonstrate the properties of the proposed method by simulations and by re-analyzing a publicly available dataset from a matched case-control study on induced abortion and infertility.     

Directly modelling matched case-control data

Matching in case-control studies is a situation in which one wishes to make inferences about a parameter of interest in the presence of nuisance parameters. The usual approach is to apply a conditional likelihood. A bivariate latent class log-linear model for binomial responses is shown to yield a standard likelihood identical to the usual conditional one. This extension of the Rasch model for binary responses gives consistent estimates and a suitable likelihood function for cases matched with any fixed number of controls.     

Simulation from a known Cox MSM using standard parametric models for the g‐formula

It is routinely argued that, unlike standard regression‐based estimates, inverse probability weighted (IPW) estimates of the parameters of a correctly specified Cox marginal structural model (MSM) may remain unbiased in the presence of a time‐varying confounder affected by prior treatment. Previously proposed methods for simulating from a known Cox MSM lack knowledge of the law of the observed outcome conditional on the measured past. Although unbiased IPW estimation does not require this knowledge, standard regression‐based estimates rely on correct specification of this law. Thus, in typical high‐dimensional settings, such simulation methods cannot isolate bias due to complex time‐varying confounding as it may be conflated with bias due to misspecification of the outcome regression model. In this paper, we describe an approach to Cox MSM data generation that allows for a comparison of the bias of IPW estimates versus that of standard regression‐based estimates in the complete absence of model misspecification. This approach involves simulating data from a standard parametrization of the likelihood and solving for the underlying Cox MSM. We prove that solutions exist and computations are tractable under many data‐generating mechanisms. We show analytically and confirm in simulations that, in the absence of model misspecification, the bias of standard regression‐based estimates for the parameters of a Cox MSM is indeed a function of the coefficients in observed data models quantifying the presence of a time‐varying confounder affected by prior treatment. We discuss limitations of this approach including that implied by the ‘g‐null paradox’. Copyright © 2013 John Wiley & Sons, Ltd.     

Proportional‐odds models for repeated composite and long ordinal outcome scales

In many medical studies, researchers widely use composite or long ordinal scores, that is, scores that have a large number of categories and a natural ordering often resulting from the sum of a number of short ordinal scores, to assess function or quality of life. Typically, we analyse these using unjustified assumptions of normality for the outcome measure, which are unlikely to be even approximately true. Scores of this type are better analysed using methods reserved for more conventional (short) ordinal scores, such as the proportional‐odds model. We can avoid the need for a large number of cut‐point parameters that define the divisions between the score categories for long ordinal scores in the proportional‐odds model by the inclusion of orthogonal polynomial contrasts. We introduce the repeated measures proportional‐odds logistic regression model and describe for long ordinal outcomes modifications to the generalized estimating equation methodology used for parameter estimation. We introduce data from a trial assessing two surgical interventions, briefly describe and re‐analyse these using the new model and compare inferences from the new analysis with previously published results for the primary outcome measure (hip function at 12 months postoperatively). We use a simulation study to illustrate how this model also has more general application for conventional short ordinal scores, to select amongst competing models of varying complexity for the cut‐point parameters. Copyright © 2013 John Wiley & Sons, Ltd.     

Computationally simple analysis of matched,outcome‐based studies of ordinal disease states

Outcome‐based sampling is an efficient study design for rare conditions, such as glioblastoma. It is often used in conjunction with matching, for increased efficiency and to potentially avoid bias due to confounding. A study was conducted at the Massachusetts General Hospital that involved retrospective sampling of glioblastoma patients with respect to multiple‐ordered disease states, as defined by three categories of overall survival time. To analyze such studies, we posit an adjacent categories logit model and exploit its allowance for prospective analysis of a retrospectively sampled study and its advantageous removal of set and level specific nuisance parameters through conditioning on sufficient statistics. This framework allows for any sampling design and is not limited to one level of disease within each set, such as in previous publications. We describe how this ordinal conditional model can be fit using standard conditional logistic regression procedures. We consider an alternative pseudo‐likelihood approach that potentially offers robustness under partial model misspecification at the expense of slight loss of efficiency under correct model specification for small sample sizes. We apply our methods to the Massachusetts General Hospital glioblastoma study. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian transformation cure frailty models with multivariate failure time data

We propose a class of transformation cure frailty models to accommodate a survival fraction in multivariate failure time data. Established through a general power transformation, this family of cure frailty models includes the proportional hazards and the proportional odds modeling structures as two special cases. Within the Bayesian paradigm, we obtain the joint posterior distribution and the corresponding full conditional distributions of the model parameters for the implementation of Gibbs sampling. Model selection is based on the conditional predictive ordinate statistic and deviance information criterion. As an illustration, we apply the proposed method to a real data set from dentistry.