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).     

The effect of conditional heteroskedasticity on common statistical procedures for means and variances

Summary: Commonly used standard statistical procedures for means and variances (such as the t–test for means or the F–test for variances and related confidence procedures) require observations from independent and identically normally distributed variables. These procedures are often routinely applied to financial data, such as asset or currency returns, which do not share these properties. Instead, they are nonnormal and show conditional heteroskedasticity, hence they are dependent. We investigate the effect of conditional heteroskedasticity (as modelled by GARCH(1,1)) on the level of these tests and the coverage probability of the related confidence procedures. It can be seen that conditional heteroskedasticity has no effect on procedures for means (at least in large samples). There is, however, a strong effect of conditional heteroskedasticity on procedures for variances. These procedures should therefore not be used if conditional heteroskedasticity is prevalent in the data.*We are grateful to the referees for their useful and constructive comments.     

An alternative method for evaluating stationarity in transition models

Transition models are an important framework that can be used to model longitudinal categorical data. A relevant issue in applying these models is the condition of stationarity, or homogeneity of transition probabilities over time. We propose two tests to assess stationarity in transition models: Wald and likelihood-ratio tests, which do not make use of transition probabilities, using only the estimated parameters of the models in contrast to the classical test available in the literature. In this paper, we present two motivating studies, with ordinal longitudinal data, to which proportional odds transition models are fitted and the two proposed tests are applied as well as the classical test. Additionally, their performances are assessed through simulation studies. The results show that the proposed tests have good performance, being better for control of type-I error and they present equivalent power functions asymptotically. Also, the correlations between the Wald, likelihood-ratio and the classical test statistics are positive and large, an indicator of general concordance. Additionally, both of the proposed tests are more flexible and can be applied in studies with qualitative and quantitative covariates.     

Large sample properties and confidence bands for component-wise varying-coefficient regression with longitudinal dependent variable

Longitudinal studies with repeatedly measured dependent variable (out-come) and time-invariant covariates are common in biomedical and epidemi-ological studies. A useful statistical tool to evaluate the effects of covariates on the outcome variable over time is the varying-coefficient regression, which considers a linear relationship between the covariates and the outcome at a specific time point but assumes the linear coefficients to be smooth curves over time. In order to provide adequate smoothing for each coefficient curve, Wu and Chiang ( 1999 ) proposed a class of component-wise kernel estimators and determined the large sample convergence rates and some of the constant terms of the mean squared errors of their estimators. In this paper we calcu¬late the explicit large sample mean squared errors, including the convergence rates and ail the constant terms, and the asymptotic distributions of the kernel estimators of Wu and Chiang ( 1999 ). These asymptotic distributions are used to construct point-wise confidence intervals and Bonferroni-type confidence bands for the coefficient curves. Through a Monte Carlo simulation, wre show that our confidence regions have adequate coverage probabilities. Applying our procedures to a NIH fetal growth study, we show that our procedures are useful to determine the effects of maternal height, cigarette smoking and al¬cohol consumption on the growth of fetal abdominal circumference over time during pregnancy.     

Regression Analysis for Multistate Models Based on a Pseudo-value Approach, with Applications to Bone Marrow Transplantation Studies

Abstract.  Typically, regression analysis for multistate models has been based on regression models for the transition intensities. These models lead to highly nonlinear and very complex models for the effects of covariates on state occupation probabilities. We present a technique that models the state occupation or transition probabilities in a multistate model directly. The method is based on the pseudo-values from a jackknife statistic constructed from non-parametric estimators for the probability in question. These pseudo-values are used as outcome variables in a generalized estimating equation to obtain estimates of model parameters. We examine this approach and its properties in detail for two special multistate model probabilities, the cumulative incidence function in competing risks and the current leukaemia-free survival used in bone marrow transplants. The latter is the probability a patient is alive and in either a first or second post-transplant remission. The techniques are illustrated on a dataset of leukaemia patients given a marrow transplant. We also discuss extensions of the model that are of current research interest.     

Semiparametric lack-of-fit tests in an additive hazard regression model

In the semiparametric additive hazard regression model of McKeague and Sasieni (Biometrika 81: 501–514), the hazard contributions of some covariates are allowed to change over time, without parametric restrictions (Aalen model), while the contributions of other covariates are assumed to be constant. In this paper, we develop tests that help to decide which of the covariate contributions indeed change over time. The remaining covariates may be modelled with constant hazard coefficients, thus reducing the number of curves that have to be estimated nonparametrically. Several bootstrap tests are proposed. The behavior of the tests is investigated in a simulation study. In a practical example, the tests consistently identify covariates with constant and with changing hazard contributions.     

Model checks for nonparametric regression with missing data: a comparative study

ABSTRACT

This paper analyses the behaviour of the goodness-of-fit tests for regression models. To this end, it uses statistics based on an estimation of the integrated regression function with missing observations either in the response variable or in some of the covariates. It proposes several versions of one empirical process, constructed from a previous estimation, that uses only the complete observations or replaces the missing observations with imputed values. In the case of missing covariates, a link model is used to fill the missing observations with other complete covariates. In all the situations, Bootstrap methodology is used to calibrate the distribution of the test statistics. A broad simulation study compares the different procedures based on empirical regression methodology, with smoothed tests previously studied in the literature. The comparison reflects the effect of the correlation between the covariates in the tests based on the imputed sample for missing covariates. In addition, the paper proposes a computational binning strategy to evaluate the tests based on an empirical process for large data sets. Finally, two applications to real data illustrate the performance of the tests.     

Testing for State Dependence with Time-Variant Transition Probabilities

We derive a simple result that allows us to test for the presence of state dependence in a dynamic Logit model with time-variant transition probabilities and an arbitrary distribution of the unobserved heterogeneity. Monte Carlo evidence suggests that this test has desirable properties even when there are some violations of the model's assumptions. We also consider alternative tests that will have desirable properties only when the transition probabilities do not depend on time and provide evidence that there is an “acceptable” range in which ignoring time-dependence does not matter too much. We conclude with an application to the Barker Hypothesis.     

Inverse probability weighted estimators for single-index models with missing covariates

Abstract

In this article, we consider the inverse probability weighted estimators for a single-index model with missing covariates when the selection probabilities are known or unknown. It is shown that the estimator for the index parameter by using estimated selection probabilities has a smaller asymptotic variance than that with true selection probabilities, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for the index parameter in single index model. However, this difference disappears for the estimators of the link function. Some numerical examples and a real data application are also conducted to illustrate the performances of the estimators.     

Rule generation for categorical time series with Markov assumptions

Several procedures of sequential pattern analysis are designed to detect frequently occurring patterns in a single categorical time series (episode mining). Based on these frequent patterns, rules are generated and evaluated, for example, in terms of their confidence. The confidence value is commonly interpreted as an estimate of a conditional probability, so some kind of stochastic model has to be assumed. The model is identified as a variable length Markov model. With this assumption, the usual confidences are maximum likelihood estimates of the transition probabilities of the Markov model. We discuss possibilities of how to efficiently fit an appropriate model to the data. Based on this model, rules are formulated. It is demonstrated that this new approach generates noticeably less and more reliable rules.     

Testing for State Dependence with Time-Variant Transition Probabilities

We derive a simple result that allows us to test for the presence of state dependence in a dynamic Logit model with time-variant transition probabilities and an arbitrary distribution of the unobserved heterogeneity. Monte Carlo evidence suggests that this test has desirable properties even when there are some violations of the model's assumptions. We also consider alternative tests that will have desirable properties only when the transition probabilities do not depend on time and provide evidence that there is an “acceptable” range in which ignoring time-dependence does not matter too much. We conclude with an application to the Barker Hypothesis.     

Application of a non-homogeneous Markov chain with seasonal transition probabilities to ozone data

In this work, we assume that the sequence recording whether or not an ozone exceedance of an environmental threshold has occurred in a given day is ruled by a non-homogeneous Markov chain of order one. In order to account for the possible presence of cycles in the empirical transition probabilities, a parametric form incorporating seasonal components is considered. Results show that even though some covariates (namely, relative humidity and temperature) are not included explicitly in the model, their influence is captured in the behavior of the transition probabilities. Parameters are estimated using the Bayesian point of view via Markov chain Monte Carlo algorithms. The model is applied to ozone data obtained from the monitoring network of Mexico City, Mexico. An analysis of how the methodology could be used as an aid in the decision-making is also given.     

Logistic regression and other discrete data models for serially correlated observations

Summary We consider the analysis of discrete serially correlated data in the presence of time dependent covariates. If the interest is to relate the covariates to the marginal distribution of the data, Markov chains are an obvious tool to consider, but their use is complicated by the fact that they are expressed in terms of transitional rather than marginal probabilities. We show how to parametrize the transition matrix in a suitable way so that interpretation is as desired. The focus is on binary and Poisson data, but the methodology can be adopted also with other discrete data distributions.     

Asymptotic theory for the Cox semi-Markov illness-death model

Irreversible illness-death models are used to model disease processes and in cancer studies to model disease recovery. In most applications, a Markov model is assumed for the multistate model. When there are covariates, a Cox (1972, J Roy Stat Soc Ser B 34:187–220) model is used to model the effect of covariates on each transition intensity. Andersen et al. (2000, Stat Med 19:587–599) proposed a Cox semi-Markov model for this problem. In this paper, we study the large sample theory for that model and provide the asymptotic variances of various probabilities of interest. A Monte Carlo study is conducted to investigate the robustness and efficiency of Markov/Semi-Markov estimators. A real data example from the PROVA (1991, Hepatology 14:1016–1024) trial is used to illustrate the theory.     

Detecting change in a hazard regression model with right-censoring

The hazard function plays an important role in survival analysis and reliability, since it quantifies the instantaneous failure rate of an individual at a given time point t, given that this individual has not failed before t. In some applications, abrupt changes in the hazard function are observed, and it is of interest to detect the location of such a change. In this paper, we consider testing of existence of a change in the parameters of an exponential regression model, based on a sample of right-censored survival times and the corresponding covariates. Likelihood ratio type tests are proposed and non-asymptotic bounds for the type II error probability are obtained. When the tests lead to acceptance of a change, estimators for the location of the change are proposed. Non-asymptotic upper bounds of the underestimation and overestimation probabilities are obtained. A short simulation study illustrates these results.     

Global and local tests to assess stationarity of Markov transition models

We present global and local likelihood-based tests to evaluate stationarity in transition models. Three motivational studies are considered. A simulation study was carried out to assess the performance of the proposed tests. The results showed that they present good performance with the control of the type-I error, especially for ordinal responses, and control of the type-II error, especially for the nominal case. Asymptotically they are close to the classical test performance. They can be executed in a single framework without the need to estimate the transition probabilities, incorporating both categorical and continuous covariates, and used to identify sources of non-stationarity.     

The Coverage Probability of Confidence Intervals in One‐Way Analysis of Covariance after Two F Tests

Volume 3 of Analysis of Messy Data by Milliken & Johnson (2002) provides detailed recommendations about sequential model development for the analysis of covariance. In his review of this volume, Koehler (2002) asks whether users should be concerned about the effect of this sequential model development on the coverage probabilities of confidence intervals for comparing treatments. We present a general methodology for the examination of these coverage probabilities in the context of the two‐stage model selection procedure that uses two F tests and is proposed in Chapter 2 of Milliken & Johnson (2002). We apply this methodology to an illustrative example from this volume and show that these coverage probabilities are typically very far below nominal. Our conclusion is that users should be very concerned about the coverage probabilities of confidence intervals for comparing treatments constructed after this two‐stage model selection procedure.     

Robust confidence regions for multinomial probabilities

Generally, confidence regions for the probabilities of a multinomial population are constructed based on the Pearson χ² statistic. Morales et al. (Bootstrap confidence regions in multinomial sampling. Appl Math Comput. 2004;155:295–315) considered the bootstrap and asymptotic confidence regions based on a broader family of test statistics known as power-divergence test statistics. In this study, we extend their work and propose penalized power-divergence test statistics-based confidence regions. We only consider small sample sizes where asymptotic properties fail and alternative methods are needed. Both bootstrap and asymptotic confidence regions are constructed. We consider the percentile and the bias corrected and accelerated bootstrap confidence regions. The latter confidence region has not been studied previously for the power-divergence statistics much less for the penalized ones. Designed simulation studies are carried out to calculate average coverage probabilities. Mean absolute deviation between actual and nominal coverage probabilities is used to compare the proposed confidence regions.     

Semi-parametric extended Poisson process models for count data

A general framework for the analysis of count data (with covariates) is proposed using formulations for the transition rates of a state-dependent birth process. The form for the transition rates incorporates covariates proportionally, with the residual distribution determined from a smooth non-parametric state-dependent form. Computation of the resulting probabilities is discussed, leading to model estimation using a penalized likelihood function. Two data sets are used as illustrative examples, one representing underdispersed Poisson-like data and the other overdispersed binomial-like data.     

Exact tests for matched pairs:studies with covariates

In prospective or retrospective studies with matched pairs one often wishes to control for covariates other than those used in the matching process.Large sample procedures assuming a logistic model are available for this problem.The present paper presents some exact permutation tests which are uniformly most powerful unbiased within a large class of tests.