Obtaining marginal estimates from conditional categorical repeated measurements models with missing data

The most commonly used models for categorical repeated measurement data are log-linear models. Not only are they easy to fit with standard software but they include such useful models as Markov chains and graphical models. However, these are conditional models and one often also requires the marginal probabilities of responses, for example, at each time point in a longitudinal study. Here a simple method of matrix manipulation is used to derive the maximum likelihood estimates of the marginal probabilities from any such conditional categorical repeated measures model. The technique is applied to the classical Muscatine data set, taking into account the dependence of missingness on previous observed values, as well as serial dependence and a random effect.     

Extended mixture factor analysis model with covariates for mixed binary and continuous responses

Finite mixture factor analysis provides a parsimonious model to explore latent group structures of high-dimensional data. In this modeling framework, we can explore latent structures for continuous responses. However, dichotomous items are often used to define latent domains in practice. This paper proposes an extended finite mixture factor analysis model with covariates to model mixed continuous and binary responses. We use a Monte Carlo expectation-maximization (MCEM) algorithm to estimate the model. In the E step, closed-form solutions are not available for the conditional expectation of complete data log likelihood, so it is approximated by sample means, which are in turn generated by the Gibbs sampler from the joint conditional distribution of latent variables. To monitor the convergence of the MCEM algorithm, we use bridge sampling to calculate the log likelihood ratio of two successive iterations. We adopt a diagnostic plot of the log likelihood ratio against iterations for monitoring the convergence of the MCEM algorithm. We compare different models based on BIC, in which we approximate the observed data log likelihood by using a Monte Carlo method. We investigate the computational properties of the MCEM algorithm by simulation studies. We use a real data example to illustrate the practical usefulness of the model. Finally, we discuss limitations and possible extensions.     

Distribution-free fitting of logit models with random effects for repeated categorical responses

This article discusses random effects models for within-subject comparisons of repeated responses on the same categorical scale. The models account for the correlation that normally occurs between repeated responses. The standard way of fitting such models maximizes the marginal likelihood after integrating with respect to a distribution for the random effect. An alternative non-parametric approach does not assume a distributional form for the random effects. Recent literature shows that for certain simple logit models, this approach yields essentially the same model parameter estimates as conditional maximum likelihood. Moreover, these estimates also result from fitting corresponding quasi-symmetric log-linear models. For simple data sets in which primary interest relates to subject-specific comparisons of the repeated responses, one can easily obtain the estimates with standard software for log-linear models. Examples include data from crossover designs and from comparisons of treatment and control groups regarding the change between baseline and follow-up observations.     

Semi‐parametric estimation of random effects in a logistic regression model using conditional inference

This paper describes a new approach to the estimation in a logistic regression model with two crossed random effects where special interest is in estimating the variance of one of the effects while not making distributional assumptions about the other effect. A composite likelihood is studied. For each term in the composite likelihood, a conditional likelihood is used that eliminates the influence of the random effects, which results in a composite conditional likelihood consisting of only one‐dimensional integrals that may be solved numerically. Good properties of the resulting estimator are described in a small simulation study. Copyright © 2015 John Wiley & Sons, Ltd.     

Computing back-calculation estimates of aids epidemic

The method of back-calculation estimates the number of HIV infections from AIDS incidence data and projects future AIDS incidence. We explore a conditional likelihood approach for computing estimates of the number of HIV infections and the parameters in the epidemic density. This method is asymptotically equivalent to the usual likelihood method. The asymptotic normal distribution of the estimates facilitates the computation of confidence intervals. We compute standard deviations for the estimates of HIV incidence and project AIDS incidence from the underlying multinomial distributions. We illustrate the methods with applications to AIDS data in the United States.     

Conditional mixed models adjusting for non-ignorable drop-out with administrative censoring in longitudinal studies

In this paper, a class of conditional mixed models is proposed to adjust for non-ignorable drop-out, while also accommodating unequal follow-up due to staggered entry and administrative censoring in longitudinal studies. Conditional linear and quadratic models which model subject-specific slopes as linear or quadratic functions of the time-to-drop-out, as well as pattern mixture models are both special cases of this approach. We illustrate these models and compare them with the usual maximum likelihood approach assuming ignorable drop-out using data from a multi-centre randomized clinical trial of renal disease. Simulations under various scenarios where the drop-out mechanism is ignorable and non-ignorable are employed to evaluate the performance of these models.     

On the estimation of the binomial probability in multistage clinical trials

Due to the optional sampling effect in a sequential design, the maximum likelihood estimator (MLE) following sequential tests is generally biased. In a typical two-stage design employed in a phase II clinical trial in cancer drug screening, a fixed number of patients are enrolled initially. The trial may be terminated for lack of clinical efficacy of treatment if the observed number of treatment responses after the first stage is too small. Otherwise, an additional fixed number of patients are enrolled to accumulate additional information on efficacy as well as on safety. There have been numerous suggestions for design of such two-stage studies. Here we establish that under the two-stage design the sufficient statistic, i.e. stopping stage and the number of treatment responses, for the parameter of the binomial distribution is also complete. Then, based on the Rao-Blackwell theorem, we derive the uniformly minimum variance unbiased estimator (UMVUE) as the conditional expectation of an unbiased estimator, which in this case is simply the maximum likelihood estimator based only on the first stage data, given the complete sufficient statistic. Our results generalize to a multistage design. We will illustrate features of the UMVUE based on two-stage phase II clinical trial design examples and present results of numerical studies on the properties of the UMVUE in comparison to the usual MLE.     

HGLM versus conditional estimators for the analysis of clustered binary data

Clustered binary data arise frequently in medical research such as cross-over clinical trials and twin studies. For the analysis of such data either a random-effects model or a conditional likelihood approach can be used. In this paper, we compare numerically the random-effects model estimator and the conditional likelihood estimator and discuss their relative merits for the analysis of binary data.     

类别归因比例的Bayes估计与极大似然估计比较

目的：比较类别归因比例的Ｂａｙｅｓ图模型估计与极大似然估计方法。方法：应用Ｇｉｂｂｓ抽样和迭代蒙特卡罗方法得到参数后验分布,得到类别归因比例的模型法估计,应用婴儿低出生体重资料分析孕妇吸烟状况的类别归因比例。结果：类别归因比例的Ｂａｙｅｓ估计与极大似然估计一致性较好。结论：Ｂａｙｅｓ方法避免了复杂伯解析和高维积分运算,对于复杂模型估计,比极大似然估计有更实际的可行性。     

Modeling hazard functions in families

A genetic frailty model is presented for censored age of onset data in nuclear families where individuals carrying a genetic susceptibility gene have an increased risk of becoming affected. We use maximum likelihood via the EM algorithm to estimate the genetic relative risk and the allele frequency under a dominant susceptibility type and a proportional hazards model. When sampling is from a disease registry, likelihood corrections are necessary for reducing bias in the parameter estimates. In these biased samples, the full conditional likelihood is approximated by a likelihood conditional on the proband's age of onset. For unbiased samples, simulations show the distributions of the estimates are similar under both a semiparametric and the correctly specified parametric likelihoods. For biased samples, simulations under the approximate conditional likelihood show the median estimates of the allele frequency and genetic relative risk tend to under- and overestimate, respectively, the true values; however, the approximation is better for rarer allele frequencies (0.0033 vs. 0.01). In practice, large samples or more complex ascertainment corrections are recommended. Using the approximate conditional likelihood on familial breast cancer onset data collected as part of a case-control study at the Fred Hutchinson Cancer Research Center in Seattle, Washington, we estimate an allele frequency of 0.0009 (approximate 95% CI 0.0006–0.002) and a genetic relative risk of 104 (approximate 95% CI 55–181). Genet. Epidemiol. 15:147–171,1998. © 1998 Wiley-Liss, Inc.     

Simple methods for assessing haplotype-environment interactions in case-only and case-control studies

For investigating haplotype-environment interactions in case-control studies, one can implement statistical methods based either on a retrospective likelihood (modeling the probability of haplotype and environment conditional on disease status) or a prospective likelihood (modeling the probability of disease status conditional on haplotype and environment). Retrospective approaches are generally more powerful than prospective approaches, but require an explicit model of the joint distribution of haplotype and environmental factors in the sample with the latter being particularly unattractive to specify. To resolve this issue, we propose a number of simple retrospective procedures for haplotype-environment interaction analysis that do not require explicit modeling of environmental covariates in the sample. We first consider a cases-only procedure, followed by a simple likelihood for case-control data that is proportional to the full-retrospective likelihood. Finally, we consider a retrospective procedure for inference on haplotype-environment interaction effects in matched or finely-stratified case-control studies. Our methods are based on the assumptions that haplotypes and environmental covariates are independent in the target population and that disease is rare. We illustrate our approaches using case-control data from the Finland-United States Investigation of Non-Insulin Dependent Diabetes Mellitus (FUSION) genetic study and simulated data.     

Parametric bootstrap and penalized quasi-likelihood inference in conditional autoregressive models

This paper discusses a variety of conditional autoregressive (CAR) models for mapping disease rates, beyond the usual first-order intrinsic CAR model. We illustrate the utility and scope of such models for handling different types of data structures. To encourage their routine use for map production at statistical and health agencies, a simple algorithm for fitting such models is presented. This is derived from penalized quasi-likelihood (PQL) inference which uses an analogue of best-linear unbiased estimation for the regional risk ratios and restricted maximum likelihood for the variance components. We offer the practitioner here the use of the parametric bootstrap for inference. It is more reliable than standard maximum likelihood asymptotics for inference purposes since relevant hypotheses for the mapping of rates lie on the boundary of the parameter space. We illustrate the parametric bootstrap test of the practically relevant and important simplifying hypothesis that there is no spatial autocorrelation. Although the parametric bootstrap requires computational effort, it is straightforward to implement and offers a wealth of information relating to the estimators and their properties. The proposed methodology is illustrated by analysing infant mortality in the province of British Columbia in Canada.     

Proportional hazards model with random effects

We propose a general proportional hazards model with random effects for handling clustered survival data. This generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk, in a way similar to the modelling of random effects in linear, generalized linear and non-linear mixed models. The distribution of the random effects is generally assumed to be multivariate normal, but other (preferably symmetrical) distributions are also possible. Maximum likelihood estimates of the regression parameters, the variance components and the baseline hazard function are obtained via the EM algorithm. The E-step of the algorithm involves computation of the conditional expectations of functions of the random effects, for which we use Markov chain Monte Carlo (MCMC) methods. Approximate variances of the estimates are computed by Louis' formula, and posterior expectations and variances of the individual random effects can be obtained as a by-product of the estimation. The inference procedure is exemplified on two data sets.     

Fitting stratified proportional odds models by amalgamating conditional likelihoods

Classical methods for fitting a varying intercept logistic regression model to stratified data are based on the conditional likelihood principle to eliminate the stratum-specific nuisance parameters. When the outcome variable has multiple ordered categories, a natural choice for the outcome model is a stratified proportional odds or cumulative logit model. However, classical conditioning techniques do not apply to the general K-category cumulative logit model (K>2) with varying stratum-specific intercepts as there is no reduction due to sufficiency; the nuisance parameters remain in the conditional likelihood. We propose a methodology to fit stratified proportional odds model by amalgamating conditional likelihoods obtained from all possible binary collapsings of the ordinal scale. The method allows for categorical and continuous covariates in a general regression framework. We provide a robust sandwich estimate of the variance of the proposed estimator. For binary exposures, we show equivalence of our approach to the estimators already proposed in the literature. The proposed recipe can be implemented very easily in standard software. We illustrate the methods via three real data examples related to biomedical research. Simulation results comparing the proposed method with a random effects model on the stratification parameters are also furnished.     

A conditional likelihood approach for regression analysis using biomarkers measured with batch‐specific error

Measurement error is common in epidemiological and biomedical studies. When biomarkers are measured in batches or groups, measurement error is potentially correlated within each batch or group. In regression analysis, most existing methods are not applicable in the presence of batch‐specific measurement error in predictors. We propose a robust conditional likelihood approach to account for batch‐specific error in predictors when batch effect is additive and the predominant source of error, which requires no assumptions on the distribution of measurement error. Although a regression model with batch as a categorical covariable yields the same parameter estimates as the proposed conditional likelihood approach for linear regression, this result does not hold in general for all generalized linear models, in particular, logistic regression. Our simulation studies show that the conditional likelihood approach achieves better finite sample performance than the regression calibration approach or a naive approach without adjustment for measurement error. In the case of logistic regression, our proposed approach is shown to also outperform the regression approach with batch as a categorical covariate. In addition, we also examine a ‘hybrid’ approach combining the conditional likelihood method and the regression calibration method, which is shown in simulations to achieve good performance in the presence of both batch‐specific and measurement‐specific errors. We illustrate our method by using data from a colorectal adenoma study. Copyright © 2012 John Wiley & Sons, Ltd.     

An improved CML estimation procedure for the Rasch model with item response data

Ordinal response data are commonly observed in health and medical investigations that include several items. The primary goal in the modelling of item response data is to find a unique measurement of the person's abilities and of the item difficulties that satisfies the properties of the fundamental measurement. One such analytic method in item response theory is the Rasch measurement, which is a way to convert ordinal observations into linear measures. Current estimation strategies assume the independence of the Rasch model parameters. In this paper, based on the conditional maximum likelihood, we implemented a simultaneous estimation method that can compare the Rasch parameters more efficiently. We also obtained the asymptotic properties of these estimators and developed the conditional likelihood ratio test for the goodness-of-fit of the model. Simulation studies were used to demonstrate the improved performance of our estimators as compared to that of currently used conditional method known as the CON procedure. We conclude that our estimation method outperforms CON in both model fit and the precision of the Rasch estimators.     

A transitional model of barrier methods compliance with unbalanced loss to follow-up

Differential loss to follow-up can substantially influence the evaluation of treatment effects on a dichotomous outcome of interest in longitudinal trials. The use of transitional models incorporating loss to follow-up as an additional category of response and the nature of the correlated responses can provide a comprehensive view of a trial with unbalanced loss to follow-up. Under the Markov assumption, transitional models estimate the probability of changing from one outcome to another outcome between follow-up visits. Patterns of the response variable can be described by the estimated transition probabilities. The effects of intervention and covariates on the outcome of interest can also be estimated using a conditional likelihood function or a multinomial logit regression. Data from a randomized barrier method study designed to compare the proportion of participants using barrier methods consistently in two counselling groups are used to illustrate the proposed model.     

Meta‐regression with partial information on summary trial or patient characteristics

We present a model for meta‐regression in the presence of missing information on some of the study level covariates, obtaining inferences using Bayesian methods. In practice, when confronted with missing covariate data in a meta‐regression, it is common to carry out a complete case or available case analysis. We propose to use the full observed data, modelling the joint density as a factorization of a meta‐regression model and a conditional factorization of the density for the covariates. With the inclusion of several covariates, inter‐relations between these covariates are modelled. Under this joint likelihood‐based approach, it is shown that the lesser assumption of the covariates being Missing At Random is imposed, instead of the more usual Missing Completely At Random (MCAR) assumption. The model is easily programmable in WinBUGS, and we examine, through the analysis of two real data sets, sensitivity and robustness of results to the MCAR assumption. Copyright © 2010 John Wiley & Sons, Ltd.     

Likelihood-based concordance tests for analysis of ascertained twin pair multinomial data

The classic twin model design has a wide application in human genetics. Under the assumption that nongenetic effects are shared to the same degree by monozygotic (MZ) and dizygotic (DZ) twin pairs, a test of the equality of casewise concordances between MZ and DZ twins provides a clue to the influence of genetic and environmental factors on a disease. The casewise concordance is the conditional probability that given that one member of a twin pair is affected, the other is also affected. When disease prevalence is low or cost-effectiveness is considered, collection of twin pairs by ascertainment for performing casewise concordance analysis is required. In this article, by defining an overall casewise concordance parameter, several likelihood-based tests, such as likelihood ratio test LR, score test Score, the usual Wald test Wald and an alternative Wald test WaldA are investigated for a test of the equality of concordances between ascertained MZ and DZ twin pairs under multinomial models. Simulation studies were conducted for data with small sample sizes. The results show that the type I error rates and power of LR and Score are stable only when the overall casewise concordances are not extremely small or large. The Wald has higher power performance in most cases but would slightly inflate type I error rates; the WaldA is the most robust and recommended approach.     

Multinomial regression models based on continuation ratios

This paper concerns continuation ratio models for multinomial responses. These are conditional probabilities used in logit models to define the dependence of the multinomial proportions on explanatory variables and unknown parameters. A distinctive feature of these models is that if one models the various continuation ratios separately, then resulting estimates and test statistics are asymptotically independent. This allows the partitioning of likelihood ratio statistics and the search for effects in specific categories of an ordinal response variable. Models that use the same parameters for different continuation ratios are suitable for estimating more global differences. The fitting of these models to actual data is illustrated, including an example from a pharmaceutical study. The results show that different models are suitable for modelling complementary sorts of differences between multinomial response distributions.