Numeric score‐based conditional and overall change‐in‐status indices for ordered categorical data

Planned interventions and/or natural conditions often effect change on an ordinal categorical outcome (e.g., symptom severity). In such scenarios, it is sometimes desirable to assign a priori scores to observed changes in status, typically giving higher weight to changes of greater magnitude. We define change indices for such data based upon a multinomial model for each row of a c × c table, where the rows represent the baseline status categories. We distinguish an index designed to assess conditional changes within each baseline category from two others designed to capture overall change. One of these overall indices measures expected change across a target population. The other is scaled to capture the proportion of total possible change in the direction indicated by the data, so that it ranges from ?1 (when all subjects finish in the least favorable category) to +1 (when all finish in the most favorable category). The conditional assessment of change can be informative regardless of how subjects are sampled into the baseline categories. In contrast, the overall indices become relevant when subjects are randomly sampled at baseline from the target population of interest, or when the investigator is able to make certain assumptions about the baseline status distribution in that population. We use a Dirichlet‐multinomial model to obtain Bayesian credible intervals for the conditional change index that exhibit favorable small‐sample frequentist properties. Simulation studies illustrate the methods, and we apply them to examples involving changes in ordinal responses for studies of sleep deprivation and activities of daily living. Copyright © 2015 John Wiley & Sons, Ltd.     

Global goodness‐of‐fit tests for group testing regression models

In a variety of biomedical applications, particularly those involving screening for infectious diseases, testing individuals (e.g. blood/urine samples, etc.) in pools has become a standard method of data collection. This experimental design, known as group testing (or pooled testing), can provide a large reduction in testing costs and can offer nearly the same precision as individual testing. To account for covariate information on individual subjects, regression models for group testing data have been proposed recently. However, there are currently no tools available to check the adequacy of these models. In this paper, we present various global goodness‐of‐fit tests for regression models with group testing data. We use simulation to examine the small‐sample size and power properties of the tests for different pool composition strategies. We illustrate our methods using two infectious disease data sets, one from an HIV study in Kenya and one from the Infertility Prevention Project. Copyright © 2009 John Wiley & Sons, Ltd.     

Extension of the Peters–Belson method to estimate health disparities among multiple groups using logistic regression with survey data

Determining the extent of a disparity, if any, between groups of people, for example, race or gender, is of interest in many fields, including public health for medical treatment and prevention of disease. An observed difference in the mean outcome between an advantaged group (AG) and disadvantaged group (DG) can be due to differences in the distribution of relevant covariates. The Peters–Belson (PB) method fits a regression model with covariates to the AG to predict, for each DG member, their outcome measure as if they had been from the AG. The difference between the mean predicted and the mean observed outcomes of DG members is the (unexplained) disparity of interest. We focus on applying the PB method to estimate the disparity based on binary/multinomial/proportional odds logistic regression models using data collected from complex surveys with more than one DG. Estimators of the unexplained disparity, an analytic variance–covariance estimator that is based on the Taylor linearization variance–covariance estimation method, as well as a Wald test for testing a joint null hypothesis of zero for unexplained disparities between two or more minority groups and a majority group, are provided. Simulation studies with data selected from simple random sampling and cluster sampling, as well as the analyses of disparity in body mass index in the National Health and Nutrition Examination Survey 1999–2004, are conducted. Empirical results indicate that the Taylor linearization variance–covariance estimation is accurate and that the proposed Wald test maintains the nominal level. Copyright © 2014 John Wiley & Sons, Ltd.     

A goodness‐of‐fit test for the proportional odds regression model

We examine goodness‐of‐fit tests for the proportional odds logistic regression model—the most commonly used regression model for an ordinal response variable. We derive a test statistic based on the Hosmer–Lemeshow test for binary logistic regression. Using a simulation study, we investigate the distribution and power properties of this test and compare these with those of three other goodness‐of‐fit tests. The new test has lower power than the existing tests; however, it was able to detect a greater number of the different types of lack of fit considered in this study. Moreover, the test allows for the results to be summarized in a contingency table of observed and estimated frequencies, which is a useful supplementary tool to assess model fit. We illustrate the ability of the tests to detect lack of fit using a study of aftercare decisions for psychiatrically hospitalized adolescents. The test proposed in this paper is similar to a recently developed goodness‐of‐fit test for multinomial logistic regression. A unified approach for testing goodness of fit is now available for binary, multinomial, and ordinal logistic regression models. Copyright © 2012 John Wiley & Sons, Ltd.     

Goodness-of-fit tests for ordinal response regression models

It is well documented that the commonly used Pearson chi-square and deviance statistics are not adequate for assessing goodness-of-fit in logistic regression models when continuous covariates are modelled. In recent years, several methods have been proposed which address this shortcoming in the binary logistic regression setting or assess model fit differently. However, these techniques have typically not been extended to the ordinal response setting and few techniques exist to assess model fit in that case. We present the modified Pearson chi-square and deviance tests that are appropriate for assessing goodness-of-fit in ordinal response models when both categorical and continuous covariates are present. The methods have good power to detect omitted interaction terms and reasonable power to detect failure of the proportional odds assumption or modelling the wrong functional form of a continuous covariate. These tests also provide immediate information as to where a model may not fit well. In addition, the methods are simple to understand and implement, and are non-specific. That is, they do not require prespecification of a type of lack-of-fit to detect.     

A general score‐independent test for order‐restricted inference

In the analysis of ordered categorical data, the categories are often assigned a set of subjectively chosen order‐restricted scores. To overcome the arbitrariness involved in the assignment of the scores, several score‐independent tests have been proposed. However, these methods are limited to 2 × K contingency tables, where K is the number of ordered categories. We present an efficiency robust score‐independent test that is applicable to more general situations. The test is embedded into a flexible framework for conditional inference and provides a natural generalization of many familiar tests involving ordered categorical data, such as the generalized Cochran‐Mantel‐Haenszel test for singly or doubly ordered contingency tables, the Page test for randomized block designs and the Tarone‐Ware trend test for survival data. The proposed method is illustrated by several numerical examples.     

A score‐type test for heterogeneity in zero‐inflated models in a stratified population

We propose a score‐type statistic to evaluate heterogeneity in zero‐inflated models for count data in a stratified population, where heterogeneity is defined as instances in which the zero counts are generated from two sources. Evaluating heterogeneity in this class of models has attracted considerable attention in the literature, but existing testing procedures have primarily relied on the constancy assumption under the alternative hypothesis. In this paper, we extend the literature by describing a score‐type test to evaluate homogeneity against general alternatives that do not neglect the stratification information under the alternative hypothesis. The limiting null distribution of the proposed test statistic is a mixture of chi‐squared distributions that can be well approximated by a simple parametric bootstrap procedure. Our numerical simulation studies show that the proposed test can greatly improve efficiency over tests of heterogeneity that ignore the stratification information. An empirical application to dental caries data in early childhood further shows the importance and practical utility of the methodology in using the stratification profile to detect heterogeneity in the population. Copyright © 2014 John Wiley & Sons, Ltd.     

Model validation and influence diagnostics for regression models with missing covariates

Missing covariate values are prevalent in regression applications. While an array of methods have been developed for estimating parameters in regression models with missing covariate data for a variety of response types, minimal focus has been given to validation of the response model and influence diagnostics. Previous research has mainly focused on estimating residuals for observations with missing covariates using expected values, after which specialized techniques are needed to conduct proper inference. We suggest a multiple imputation strategy that allows for the use of standard methods for residual analyses on the imputed data sets or a stacked data set. We demonstrate the suggested multiple imputation method by analyzing the Sleep in Mammals data in the context of a linear regression model and the New York Social Indicators Status data with a logistic regression model.     

Goodness‐of‐fit test for proportional subdistribution hazards model

This paper concerns using modified weighted Schoenfeld residuals to test the proportionality of subdistribution hazards for the Fine–Gray model, similar to the tests proposed by Grambsch and Therneau for independently censored data. We develop a score test for the time‐varying coefficients based on the modified Schoenfeld residuals derived assuming a certain form of non‐proportionality. The methods perform well in simulations and a real data analysis of breast cancer data, where the treatment effect exhibits non‐proportional hazards. Copyright © 2013 John Wiley & Sons, Ltd.     

Multinomial goodness-of-fit tests for logistic regression models.

We examine the properties of several tests for goodness-of-fit for multinomial logistic regression. One test is based on a strategy of sorting the observations according to the complement of the estimated probability for the reference outcome category and then grouping the subjects into g equal-sized groups. A g x c contingency table, where c is the number of values of the outcome variable, is constructed. The test statistic, denoted as Cg, is obtained by calculating the Pearson chi2 statistic where the estimated expected frequencies are the sum of the model-based estimated logistic probabilities. Simulations compare the properties of Cg with those of the ungrouped Pearson chi2 test (X2) and its normalized test (z). The null distribution of Cg is well approximated by the chi2 distribution with (g-2) x (c-1) degrees of freedom. The sampling distribution of X2 is compared with a chi2 distribution with n x (c-1) degrees of freedom but shows erratic behavior. With a few exceptions, the sampling distribution of z adheres reasonably well to the standard normal distribution. Power simulations show that Cg has low power for a sample of 100 observations, but satisfactory power for a sample of 400. The tests are illustrated using data from a study of cytological criteria for the diagnosis of breast tumors.     

Semi‐varying coefficient multinomial logistic regression for disease progression risk prediction

This paper proposes a risk prediction model using semi‐varying coefficient multinomial logistic regression. We use a penalized local likelihood method to do the model selection and estimate both functional and constant coefficients in the selected model. The model can be used to improve predictive modelling when non‐linear interactions between predictors are present. We conduct a simulation study to assess our method's performance, and the results show that the model selection procedure works well with small average numbers of wrong‐selection or missing‐selection. We illustrate the use of our method by applying it to classify the patients with early rheumatoid arthritis at baseline into different risk groups in future disease progression. We use a leave‐one‐out cross‐validation method to assess its correct prediction rate and propose a recalibration framework to evaluate how reliable are the predicted risks. Copyright © 2016 John Wiley & Sons, Ltd.     

Kernel density‐based likelihood ratio tests for linear regression models

In this article, we develop a so‐called profile likelihood ratio test (PLRT) based on the estimated error density for the multiple linear regression model. Unlike the existing likelihood ratio test (LRT), our proposed PLRT does not require any specification on the error distribution. The asymptotic properties are developed and the Wilks phenomenon is studied. Simulation studies are conducted to examine the performance of the PLRT. It is observed that our proposed PLRT generally outperforms the existing LRT, empirical likelihood ratio test and the weighted profile likelihood ratio test in sense that (i) its type I error rates are closer to the prespecified nominal level; (ii) it generally has higher powers; (iii) it performs satisfactorily when moments of the error do not exist (eg, Cauchy distribution); and (iv) it has higher probability of correctly selecting the correct model in the multiple testing problem. A mammalian eye gene expression dataset and a concrete compressive strength dataset are analyzed to illustrate our methodologies.     

Dynamic probability control limits for risk‐adjusted CUSUM charts based on multiresponses

For a patient who has survived a surgery, there could be several levels of recovery. Thus, it is reasonable to consider more than two outcomes when monitoring surgical outcome quality. The risk‐adjusted cumulative sum (CUSUM) chart based on multiresponses has been developed for monitoring a surgical process with three or more outcomes. However, there is a significant effect of varying risk distributions on the in‐control performance of the chart when constant control limits are applied. To overcome this disadvantage, we apply the dynamic probability control limits to the risk‐adjusted CUSUM charts for multiresponses. The simulation results demonstrate that the in‐control performance of the charts with dynamic probability control limits can be controlled for different patient populations because these limits are determined for each specific sequence of patients. Thus, the use of dynamic probability control limits for risk‐adjusted CUSUM charts based on multiresponses allows each chart to be designed for the corresponding patient sequence of a surgeon or a hospital and therefore does not require estimating or monitoring the patients' risk distribution. Copyright © 2017 John Wiley & Sons, Ltd.     

Utility‐based designs for randomized comparative trials with categorical outcomes

A general utility‐based testing methodology for design and conduct of randomized comparative clinical trials with categorical outcomes is presented. Numerical utilities of all elementary events are elicited to quantify their desirabilities. These numerical values are used to map the categorical outcome probability vector of each treatment to a mean utility, which is used as a one‐dimensional criterion for constructing comparative tests. Bayesian tests are presented, including fixed sample and group sequential procedures, assuming Dirichlet‐multinomial models for the priors and likelihoods. Guidelines are provided for establishing priors, eliciting utilities, and specifying hypotheses. Efficient posterior computation is discussed, and algorithms are provided for jointly calibrating test cutoffs and sample size to control overall type I error and achieve specified power. Asymptotic approximations for the power curve are used to initialize the algorithms. The methodology is applied to re‐design a completed trial that compared two chemotherapy regimens for chronic lymphocytic leukemia, in which an ordinal efficacy outcome was dichotomized, and toxicity was ignored to construct the trial's design. The Bayesian tests also are illustrated by several types of categorical outcomes arising in common clinical settings. Freely available computer software for implementation is provided. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Logistic‐AFT location‐scale mixture regression models with nonsusceptibility for left‐truncated and general interval‐censored data

In conventional survival analysis there is an underlying assumption that all study subjects are susceptible to the event. In general, this assumption does not adequately hold when investigating the time to an event other than death. Owing to genetic and/or environmental etiology, study subjects may not be susceptible to the disease. Analyzing nonsusceptibility has become an important topic in biomedical, epidemiological, and sociological research, with recent statistical studies proposing several mixture models for right‐censored data in regression analysis. In longitudinal studies, we often encounter left, interval, and right‐censored data because of incomplete observations of the time endpoint, as well as possibly left‐truncated data arising from the dissimilar entry ages of recruited healthy subjects. To analyze these kinds of incomplete data while accounting for nonsusceptibility and possible crossing hazards in the framework of mixture regression models, we utilize a logistic regression model to specify the probability of susceptibility, and a generalized gamma distribution, or a log‐logistic distribution, in the accelerated failure time location‐scale regression model to formulate the time to the event. Relative times of the conditional event time distribution for susceptible subjects are extended in the accelerated failure time location‐scale submodel. We also construct graphical goodness‐of‐fit procedures on the basis of the Turnbull–Frydman estimator and newly proposed residuals. Simulation studies were conducted to demonstrate the validity of the proposed estimation procedure. The mixture regression models are illustrated with alcohol abuse data from the Taiwan Aboriginal Study Project and hypertriglyceridemia data from the Cardiovascular Disease Risk Factor Two‐township Study in Taiwan. Copyright © 2013 John Wiley & Sons, Ltd.     

Calibration weighted estimation of semiparametric transformation models for two‐phase sampling

Two‐phase designs are commonly used to subsample subjects from a cohort in order to study covariates that are too expensive to ascertain for everyone in the cohort. This is particularly true for the study of immune response biomarkers in vaccine immunology, where new, elaborate assays are constantly being developed to improve our understanding of the human immune responses to vaccines and how the immune response may protect humans from virus infection. It has long being recognized that if there exist variables that are correlated with expensive variables and can be measured for every subject in the cohort, they can be leveraged to improve the estimation efficiency for the effects of the expensive variables. In this research article, we developed an improved inverse probability weighted estimation approach for semiparametric transformation models with a two‐phase study design. Semiparametric transformation models are a class of models that include the Cox PH and proportional odds models. They provide an attractive way to model the effects of immune response biomarkers as human immune responses generally wane over time. Our approach is based on weights calibration, which has its origin in survey statistics and was used by Breslow et al. 1 , 2 to improve inverse probability weighted estimation of the Cox regression model. We develop asymptotic theory for our estimator and examine its performance through simulation studies. We illustrate the proposed method with application to two HIV‐1 vaccine efficacy trials. Copyright © 2015 John Wiley & Sons, Ltd.     

Antedependence models for nonstationary categorical longitudinal data with ignorable missingness: likelihood‐based inference

Time index‐ordered random variables are said to be antedependent (AD) of order (p₁,p₂, … ,p_n) if the kth variable, conditioned on the p_k immediately preceding variables, is independent of all further preceding variables. Inferential methods associated with AD models are well developed for continuous (primarily normal) longitudinal data, but not for categorical longitudinal data. In this article, we develop likelihood‐based inferential procedures for unstructured AD models for categorical longitudinal data. Specifically, we derive maximum likelihood estimators (MLEs) of model parameters; penalized likelihood criteria and likelihood ratio tests for determining the order of antedependence; and likelihood ratio tests for homogeneity across groups, time invariance of transition probabilities, and strict stationarity. We give closed‐form expressions for MLEs and test statistics, which allow for the possibility of empty cells and monotone missing data, for all cases save strict stationarity. For data with an arbitrary missingness pattern, we derive an efficient restricted expectation–maximization algorithm for obtaining MLEs. We evaluate the performance of the tests by simulation. We apply the methods to longitudinal studies of toenail infection severity (measured on a binary scale) and Alzheimer's disease severity (measured on an ordinal scale). The analysis of the toenail infection severity data reveals interesting nonstationary behavior of the transition probabilities and indicates that an unstructured first‐order AD model is superior to stationary and other structured first‐order AD models that have previously been fit to these data. The analysis of the Alzheimer's severity data indicates that the antedependence is second order with time‐invariant transition probabilities, suggesting the use of a second‐order autoregressive cumulative logit model. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximate score‐based testing with application to multivariate trait association analysis

For genome‐wide association studies and DNA sequencing studies, several powerful score‐based tests, such as kernel machine regression and sum of powered score tests, have been proposed in the last few years. However, extensions of these score‐based tests to more complex models, such as mixed‐effects models for analysis of multiple and correlated traits, have been hindered by the unavailability of the score vector, due to either no output from statistical software or no closed‐form solution at all. We propose a simple and general method to asymptotically approximate the score vector based on an asymptotically normal and consistent estimate of a parameter vector to be tested and its (consistent) covariance matrix. The proposed method is applicable to both maximum‐likelihood estimation and estimating function‐based approaches. We use the derived approximate score vector to extend several score‐based tests to mixed‐effects models. We demonstrate the feasibility and possible power gains of these tests in association analysis of multiple and correlated quantitative or binary traits with both real and simulated data. The proposed method is easy to implement with a wide applicability.