Weighted-bootstrap alignment of explanatory variables

Adjustment for covariates is a time-honored tool in statistical analysis and is often implemented by including the covariates that one intends to adjust as additional predictors in a model. This adjustment often does not work well when the underlying model is misspecified. We consider here the situation where we compare a response between two groups. This response may depend on a covariate for which the distribution differs between the two groups one intends to compare. This creates the potential that observed differences are due to differences in covariate levels rather than “genuine” population differences that cannot be explained by covariate differences. We propose a bootstrap-based adjustment method. Bootstrap weights are constructed with the aim of aligning bootstrap–weighted empirical distributions of the covariate between the two groups. Generally, the proposed weighted-bootstrap algorithm can be used to align or match the values of an explanatory variable as closely as desired to those of a given target distribution. We illustrate the proposed bootstrap adjustment method in simulations and in the analysis of data on the fecundity of historical cohorts of French-Canadian women.     

Estimating low pay transition probabilities accounting for endogenous selection mechanisms

Summary.  We propose a model of transitions into and out of low paid employment that accounts for non-ignorable panel dropout, employment retention and base year low pay status ('initial conditions'). The model is fitted to data for men from the British Household Panel Survey. Initial conditions and employment retention are found to be non-ignorable selection processes. Whether panel dropout is found to be ignorable depends on how item non-response on pay is treated. Notwithstanding these results, we also find that models incorporating a simpler approach to accounting for non-ignorable selections provide estimates of covariate effects that differ very little from the estimates from the general model.     

A linear model-based test for the heterogeneity of conditional correlations

Current methods of testing the equality of conditional correlations of bivariate data on a third variable of interest (covariate) are limited due to discretizing of the covariate when it is continuous. In this study, we propose a linear model approach for estimation and hypothesis testing of the Pearson correlation coefficient, where the correlation itself can be modeled as a function of continuous covariates. The restricted maximum likelihood method is applied for parameter estimation, and the corrected likelihood ratio test is performed for hypothesis testing. This approach allows for flexible and robust inference and prediction of the conditional correlations based on the linear model. Simulation studies show that the proposed method is statistically more powerful and more flexible in accommodating complex covariate patterns than the existing methods. In addition, we illustrate the approach by analyzing the correlation between the physical component summary and the mental component summary of the MOS SF-36 form across a fair number of covariates in the national survey data.     

Use of multiple imputation in supersampled nested case-control and case-cohort studies

Nested case-control and case-cohort studies are useful for studying associations between covariates and time-to-event when some covariates are expensive to measure. Full covariate information is collected in the nested case-control or case-cohort sample only, while cheaply measured covariates are often observed for the full cohort. Standard analysis of such case-control samples ignores any full cohort data. Previous work has shown how data for the full cohort can be used efficiently by multiple imputation of the expensive covariate(s), followed by a full-cohort analysis. For large cohorts this is computationally expensive or even infeasible. An alternative is to supplement the case-control samples with additional controls on which cheaply measured covariates are observed. We show how multiple imputation can be used for analysis of such supersampled data. Simulations show that this brings efficiency gains relative to a traditional analysis and that the efficiency loss relative to using the full cohort data is not substantial.     

Age at first marriage in Malawi: a Bayesian multilevel analysis using a discrete time-to-event model

Summary.  The paper presents a hierarchical discrete time survival model for the analysis of the 2000 Malawi Demographic and Health Survey data to assess the determinants of transition to marriage among women in Malawi. The model explicitly accounts for the unobserved heterogeneity by using family and community random effects with cross-level correlation structure. A nonparametric technique is used to model the base-line discrete hazard dynamically. Parameters of the model are computed by using a Markov chain Monte Carlo algorithm. The results show that rising age at marriage is a combination of birth cohort and education effects, depends considerably on the family and to some extent on the community in which a woman resides and the correlation between family and community random effects is negative. These results confirm a downward trend in teenage marriage and that raising women's education levels in sub-Saharan Africa has the beneficial effect of increasing age at marriage, and by implication reducing total fertility rates. The negative correlation between family and community random effects has policy implications in that targeting communities with an intervention to increase age at first marriage may not necessarily yield reduced fertility levels in individual families. A campaign that is geared towards individual families would achieve the desired goals. Overall, the findings point to the need for the Government in Malawi to enact public policies which are geared at vastly improving women's education at higher levels. The variation in marriage rates over families poses problems in delivering the policy, since particular policies must be devised for specific groups of families to accomplish the required social and health objectives.     

Births in remarriages (using data from sample surveys)

Data from sample surveys conducted between 1978 and 1981 are used to examine the fertility of women in second and subsequent marriages in the USSR. The results indicate that women up to age 25 who have been married more than once have higher fertility than women in a first marriage. However, total fertility is higher for women in uninterrupted marriages. The analysis is presented separately for various cohorts and for socioeconomic characteristics such as educational status and rural or urban residence.     

A segmented regression model for event history data: an application to the fertility patterns in Italy

We propose a segmented discrete-time model for the analysis of event history data in demographic research. Through a unified regression framework, the model provides estimates of the effects of explanatory variables and jointly accommodates flexibly non-proportional differences via segmented relationships. The main appeal relies on ready availability of parameters, changepoints, and slopes, which may provide meaningful and intuitive information on the topic. Furthermore, specific linear constraints on the slopes may also be set to investigate particular patterns. We investigate the intervals between cohabitation and first childbirth and from first to second childbirth using individual data for Italian women from the Second National Survey on Fertility. The model provides insights into dramatic decrease of fertility experienced in Italy, in that it detects a ‘common’ tendency in delaying the onset of childbearing for the more recent cohorts and a ‘specific’ postponement strictly depending on the educational level and age at cohabitation.     

Adjusted estimates for time-to-event endpoints

In the analysis of retrospective data or when interpreting results from a single-arm phase II clinical trial relative to historical data, it is often of interest to show plots summarizing time-to-event outcomes comparing treatment groups. If the groups being compared are imbalanced with respect to factors known to influence outcome, these plots can be misleading and seemingly incompatible with results obtained from a regression model that accounts for these imbalances. We consider ways in which covariate information can be used to obtain adjusted curves for time-to-event outcomes. We first review a common model-based method and then suggest another model-based approach that is not as reliant on model assumptions. Finally, an approach that is partially model free is suggested. Each method is applied to an example from hematopoietic cell transplantation.     

A dependent Dirichlet process model for survival data with competing risks

In this paper, we first propose a dependent Dirichlet process (DDP) model using a mixture of Weibull models with each mixture component resembling a Cox model for survival data. We then build a Dirichlet process mixture model for competing risks data without regression covariates. Next we extend this model to a DDP model for competing risks regression data by using a multiplicative covariate effect on subdistribution hazards in the mixture components. Though built on proportional hazards (or subdistribution hazards) models, the proposed nonparametric Bayesian regression models do not require the assumption of constant hazard (or subdistribution hazard) ratio. An external time-dependent covariate is also considered in the survival model. After describing the model, we discuss how both cause-specific and subdistribution hazard ratios can be estimated from the same nonparametric Bayesian model for competing risks regression. For use with the regression models proposed, we introduce an omnibus prior that is suitable when little external information is available about covariate effects. Finally we compare the models’ performance with existing methods through simulations. We also illustrate the proposed competing risks regression model with data from a breast cancer study. An R package “DPWeibull” implementing all of the proposed methods is available at CRAN.

     

Joint Modeling of Event Time and Nonignorable Missing Longitudinal Data

Survival studies usually collect on each participant, both duration until some terminal event and repeated measures of a time-dependent covariate. Such a covariate is referred to as an internal time-dependent covariate. Usually, some subjects drop out of the study before occurence of the terminal event of interest. One may then wish to evaluate the relationship between time to dropout and the internal covariate. The Cox model is a standard framework for that purpose. Here, we address this problem in situations where the value of the covariate at dropout is unobserved. We suggest a joint model which combines a first-order Markov model for the longitudinaly measured covariate with a time-dependent Cox model for the dropout process. We consider maximum likelihood estimation in this model and show how estimation can be carried out via the EM-algorithm. We state that the suggested joint model may have applications in the context of longitudinal data with nonignorable dropout. Indeed, it can be viewed as generalizing Diggle and Kenward's model (1994) to situations where dropout may occur at any point in time and may be censored. Hence we apply both models and compare their results on a data set concerning longitudinal measurements among patients in a cancer clinical trial.     

Estimation and tests for a longitudinal regression model based on the Markov chain

In this paper, the dependence of transition probabilities on covariates and a test procedure for covariate dependent Markov models are examined. The nonparametric test for the role of waiting time proposed by Jones and Crowley [M. Jones, J. Crowley, Nonparametric tests of the Markov model for survival data Biometrika 79 (3) (1992) 513–522] has been extended here to transitions and reverse transitions. The limitation of the Jones and Crowley method is that it does not take account of other covariates that might have association with the probabilities of transition. A simple test procedure is proposed that can be employed for testing: (i) the significance of association between covariates and transition probabilities, and (ii) the impact of waiting time on the transition probabilities. The procedure is illustrated using panel data on hospitalization of the elderly population in the USA from the Health and Retirement Survey (HRS).     

A Smooth Test in Proportional Hazard Survival Models Using Local Partial Likelihood Fitting

Proportional hazard models for survival data, even though popular and numerically handy, suffer from the restrictive assumption that covariate effects are constant over survival time. A number of tests have been proposed to check this assumption. This paper contributes to this area by employing local estimates allowing to fit hazard models in which covariate effects are smoothly varying with time. A formal test is derived to check for proportional hazards against smooth hazards as alternative. The test proves to possess omnibus power in that it is powerful against arbitrary but smooth alternatives. Comparative simulations and two data examples accompany the presentation. Extensions are provided to multiple covariate settings, where the focus of interest is to decide which of the covariate effects vary with time.     

Bayesian adjustment for unidirectional misclassification in ordinal covariates

In this paper, we study the identification of Bayesian regression models, when an ordinal covariate is subject to unidirectional misclassification. Xia and Gustafson [Bayesian regression models adjusting for unidirectional covariate misclassification. Can J Stat. 2016;44(2):198–218] obtained model identifiability for non-binary regression models, when there is a binary covariate subject to unidirectional misclassification. In the current paper, we establish the moment identifiability of regression models for misclassified ordinal covariates with more than two categories, based on forms of observable moments. Computational studies are conducted that confirm the theoretical results. We apply the method to two datasets, one from the Medical Expenditure Panel Survey (MEPS), and the other from Translational Research Investigating Underlying Disparities in Acute Myocardial infarction Patients Health Status (TRIUMPH).     

Goodness-of-fit tests for additive mean residual life model under right censoring

The mean residual life (MRL) measures the remaining life expectancy and is useful in actuarial studies, biological experiments and clinical trials. To assess the covariate effect, an additive MRL regression model has been proposed in the literature. In this paper, we focus on the topic of model checking. Specifically, we develop two goodness-of-fit tests to test the additive MRL model assumption. We explore the large sample properties of the test statistics and show that both of them are based on asymptotic Gaussian processes so that resampling approaches can be applied to find the rejection regions. Simulation studies indicate that our methods work reasonably well for sample sizes ranging from 50 to 200. Two empirical data sets are analyzed to illustrate the approaches.     

Issues in Fitting Inverse Gaussian First Hitting Time Regression Models for Lifetime Data

Inverse Gaussian first hitting time regression models sometimes provide an attractive representation of lifetime data. Various authors comment that dependence of both parameters on the same covariate may imply multicollinearity. The frequent appearance of conflicting signs for the two coefficients of the same covariate may be related to this. We carry out simulation studies to examine the reality of this possible multicollinearity. Although there is some dependence between estimates, multicollinearity does not seem to be a major problem. Fitting this model to data generated by a Weibull regression suggests that conflicting signs of estimates may be due to model misspecification.     

Nonparametric Modeling Auxiliary Covariates in Random Coefficient Models

Random coefficient model (RCM) is a powerful statistical tool in analyzing correlated data collected from studies with different clusters or from longitudinal studies. In practice, there is a need for statistical methods that allow biomedical researchers to adjust for the measured and unmeasured covariates that might affect the regression model. This article studies two nonparametric methods dealing with auxiliary covariate data in linear random coefficient models. We demonstrate how to estimate the coefficients of the models and how to predict the random effects when the covariates are missing or mismeasured. We employ empirical estimator and kernel smoother to handle a discrete and continuous auxiliary, respectively. Simulation results show that the proposed methods perform better than an alternative method that only uses data in the validation data set and ignores the random effects in the random coefficient model.     

An additive–multiplicative mean model for panel count data with dependent observation and dropout processes

This paper discusses regression analysis of panel count data with dependent observation and dropout processes. For the problem, a general mean model is presented that can allow both additive and multiplicative effects of covariates on the underlying point process. In addition, the proportional rates model and the accelerated failure time model are employed to describe possible covariate effects on the observation process and the dropout or follow‐up process, respectively. For estimation of regression parameters, some estimating equation‐based procedures are developed and the asymptotic properties of the proposed estimators are established. In addition, a resampling approach is proposed for estimating a covariance matrix of the proposed estimator and a model checking procedure is also provided. Results from an extensive simulation study indicate that the proposed methodology works well for practical situations, and it is applied to a motivating set of real data.     

Issues for stratified randomization based on a factor derived from a continuous baseline variable

Stratified randomization based on the baseline value of the primary analysis variable is common in clinical trial design. We illustrate from a theoretical viewpoint the advantage of such a stratified randomization to achieve balance of the baseline covariate. We also conclude that the estimator for the treatment effect is consistent when including both the continuous baseline covariate and the stratification factor derived from the baseline covariate. In addition, the analysis of covariance model including both the continuous covariate and the stratification factor is asymptotically no less efficient than including either only the continuous baseline value or only the stratification factor. We recommend that the continuous baseline covariate should generally be included in the analysis model. The corresponding stratification factor may also be included in the analysis model if one is not confident that the relationship between the baseline covariate and the response variable is linear. In spite of the above recommendation, one should always carefully examine relevant historical data to pre-specify the most appropriate analysis model for a perspective study.     

Marginalized transition random effect models for multivariate longitudinal binary data

Generalized linear models with random effects and/or serial dependence are commonly used to analyze longitudinal data. However, the computation and interpretation of marginal covariate effects can be difficult. This led Heagerty (1999, 2002) to propose models for longitudinal binary data in which a logistic regression is first used to explain the average marginal response. The model is then completed by introducing a conditional regression that allows for the longitudinal, within‐subject, dependence, either via random effects or regressing on previous responses. In this paper, the authors extend the work of Heagerty to handle multivariate longitudinal binary response data using a triple of regression models that directly model the marginal mean response while taking into account dependence across time and across responses. Markov Chain Monte Carlo methods are used for inference. Data from the Iowa Youth and Families Project are used to illustrate the methods.     

ESTIMATING THE NUMBER OF VISITS TO THE DOCTOR

The frequency of doctor consultations has direct consequences for health care budgets, yet little statistical analysis of the determinants of doctor visits has been reported. We consider the distribution of the number of visits to the doctor and, in particular, we model its dependence on a number of demographic factors. Examination of the Australian 1995 National Health Survey data reveals that generalized linear Poisson or negative binomial models are inadequate for modelling the mean as a function of covariates, because of excessive zero counts, and a mean‐variance relationship that varies enormously over covariate values. A negative binomial model is used, with parameter values estimated in subgroups according to the discrete combinations of the covariate values. Smoothing splines are then used to smooth and interpolate the parameter values. In effect the mean and the shape parameters are each modelled as (different) functions of gender, age and geographical factors. The estimated regressions for the mean have simple and intuitive interpretations. However, the dependence of the (negative binomial) shape parameter on the covariates is more difficult to interpret and is subject to influence by extreme observations. We illustrate the use of the model by estimating the distribution of the number of doctor consultations in the Statistical Local Area of Ryde, based on population numbers from the 1996 census.