Using local correlation in kernel-based smoothers for dependent data

We consider the general problem of smoothing correlated data to estimate the nonparametric mean function when a random, but bounded, number of measurements is available for each independent subject. We propose a simple extension to the local polynomial regression smoother that retains the asymptotic properties of the working independence estimator, while typically reducing both the conditional bias and variance for practical sample sizes, as demonstrated by exact calculations for some particular models. We illustrate our method by smoothing longitudinal functional decline data for 100 patients with Huntington's disease. The class of local polynomial kernel-based estimating equations previously considered in the literature is shown to use the global correlation structure in an apparently detrimental way, which explains why some previous attempts to incorporate correlation were found to be asymptotically inferior to the working independence estimator.     

Applications of additive semivarying coefficient models: monthly suicide data from Hong Kong

A semivarying coefficient model for the monthly numbers of suicides in Hong Kong is developed and a new estimation procedure for estimating the parametric component is proposed. The estimators are examined in a small simulation study and fitted to monthly suicide data to estimate a nonparametric long-term trend and parametric seasonal and socioeconomic effects. Fitting the model detected interpretable structure in the data that is consistent with that driving public health policy. While exploratory, the analysis motivates the collection of more detailed data and the development of more sophisticated models to help determine target groups and strategies to reduce the suicide rate in Hong Kong.     

Generalized varying coefficient models for longitudinal data

We propose a generalization of the varying coefficient modelfor longitudinal data to cases where not only current but alsorecent past values of the predictor process affect current response.More precisely, the targeted regression coefficient functionsof the proposed model have sliding window supports around currenttime t. A variant of a recently proposed two-step estimationmethod for varying coefficient models is proposed for estimationin the context of these generalized varying coefficient models,and is found to lead to improvements, especially for the caseof additive measurement errors in both response and predictors.The proposed methodology for estimation and inference is alsoapplicable for the case of additive measurement error in thecommon versions of varying coefficient models that relate onlycurrent observations of predictor and response processes toeach other. Asymptotic distributions of the proposed estimatorsare derived, and the model is applied to the problem of predictingprotein concentrations in a longitudinal study. Simulation studiesdemonstrate the efficacy of the proposed estimation procedure.     

Generalized linear mixed models with varying coefficients for longitudinal data

The routinely assumed parametric functional form in the linear predictor of a generalized linear mixed model for longitudinal data may be too restrictive to represent true underlying covariate effects. We relax this assumption by representing these covariate effects by smooth but otherwise arbitrary functions of time, with random effects used to model the correlation induced by among-subject and within-subject variation. Due to the usually intractable integration involved in evaluating the quasi-likelihood function, the double penalized quasi-likelihood (DPQL) approach of Lin and Zhang (1999, Journal of the Royal Statistical Society, Series B61, 381-400) is used to estimate the varying coefficients and the variance components simultaneously by representing a nonparametric function by a linear combination of fixed effects and random effects. A scaled chi-squared test based on the mixed model representation of the proposed model is developed to test whether an underlying varying coefficient is a polynomial of certain degree. We evaluate the performance of the procedures through simulation studies and illustrate their application with Indonesian children infectious disease data.     

Modeling longitudinal data with ordinal response by varying coefficients

This paper presents a smooth regression model for ordinal data with longitudinal dependence structure. A marginal model with cumulative logit link is applied to cope with the ordinal scale and the main and covariate effects in the model are allowed to vary with time. Local fitting is pursued and asymptotic properties of the estimates are discussed. In a second step, the longitudinal dependence of the observations is considered. Cumulative log odds ratios are fitted locally, which allows investigation of how the longitudinal dependence of the ordinal observations changes with time.     

Negative binomial additive models

The generalized additive model is extended to handle negative binomial responses. The extension is complicated by the fact that the negative binomial distribution has two parameters and is not in the exponential family. The methodology is applied to data involving DNA adduct counts and smoking variables among ex-smokers with lung cancer. A more detailed investigation is made of the parametric relationship between the number of adducts and years since quitting while retaining a smooth relationship between adducts and the other covariates.     

Mixtures of varying coefficient models for longitudinal data with discrete or continuous nonignorable dropout

The analysis of longitudinal repeated measures data is frequently complicated by missing data due to informative dropout. We describe a mixture model for joint distribution for longitudinal repeated measures, where the dropout distribution may be continuous and the dependence between response and dropout is semiparametric. Specifically, we assume that responses follow a varying coefficient random effects model conditional on dropout time, where the regression coefficients depend on dropout time through unspecified nonparametric functions that are estimated using step functions when dropout time is discrete (e.g., for panel data) and using smoothing splines when dropout time is continuous. Inference under the proposed semiparametric model is hence more robust than the parametric conditional linear model. The unconditional distribution of the repeated measures is a mixture over the dropout distribution. We show that estimation in the semiparametric varying coefficient mixture model can proceed by fitting a parametric mixed effects model and can be carried out on standard software platforms such as SAS. The model is used to analyze data from a recent AIDS clinical trial and its performance is evaluated using simulations.     

Respiratory health and air pollution: additive mixed model analyses

We conduct a reanalysis of data from the Utah Valley respiratory health/air pollution study of Pope and co-workers (Pope et al., 1991) using additive mixed models. A relatively recent statistical development (e.g. Wang, 1998; Verbyla et al., 1999; Lin and Zhang, 1999), the methods allow for smooth functional relationships, subject-specific effects and time series error structure. All three of these are apparent in the Utah Valley data.     

Median regression models for longitudinal data with dropouts

Summary .  Recently, median regression models have received increasing attention. When continuous responses follow a distribution that is quite different from a normal distribution, usual mean regression models may fail to produce efficient estimators whereas median regression models may perform satisfactorily. In this article, we discuss using median regression models to deal with longitudinal data with dropouts. Weighted estimating equations are proposed to estimate the median regression parameters for incomplete longitudinal data, where the weights are determined by modeling the dropout process. Consistency and the asymptotic distribution of the resultant estimators are established. The proposed method is used to analyze a longitudinal data set arising from a controlled trial of HIV disease ( Volberding et al., 1990 , The New England Journal of Medicine 322, 941–949). Simulation studies are conducted to assess the performance of the proposed method under various situations. An extension to estimation of the association parameters is outlined.     

Prediction using partly conditional time-varying coefficients regression models

Data collected longitudinally in time provide the opportunity to develop predictive models of future observations given current data for an individual. Such models may be of particular value in defining individuals at high risk and thereby in suggesting subgroups for targeting of prevention intervention research efforts. In this paper, we propose a method for estimating predictive functions. The method uses an extension of the marginal regression analysis methods of Liang and Zeger (1986, Biometrika 73, 13-22) and is implemented using simple estimating equations. A key feature of the models is that regression coefficients are modelled as smooth functions of the times both at and for prediction. Data from a study of obesity in childhood and early adulthood is used to demonstrate the methodology. Criteria for defining individuals to be at high risk can be defined on the basis of estimated predictive functions. We suggest methods for evaluating the diagnostic accuracy (sensitivity and specificity) of such rules using cross-validation. The method holds promise as a robust and technically easy way of evaluating information about future prognosis that may be gleaned from a patient's current and past clinical status.     

On models for binomial data with random numbers of trials

A binomial outcome is a count s of the number of successes out of the total number of independent trials n=s+f, where f is a count of the failures. The n are random variables not fixed by design in many studies. Joint modeling of (s, f) can provide additional insight into the science and into the probability pi of success that cannot be directly incorporated by the logistic regression model. Observations where n= 0 are excluded from the binomial analysis yet may be important to understanding how pi is influenced by covariates. Correlation between s and f may exist and be of direct interest. We propose Bayesian multivariate Poisson models for the bivariate response (s, f), correlated through random effects. We extend our models to the analysis of longitudinal and multivariate longitudinal binomial outcomes. Our methodology was motivated by two disparate examples, one from teratology and one from an HIV tertiary intervention study.     

Clustered mixed nonhomogeneous Poisson process spline models for the analysis of recurrent event panel data

Summary .   A flexible semiparametric model for analyzing longitudinal panel count data arising from mixtures is presented. Panel count data refers here to count data on recurrent events collected as the number of events that have occurred within specific follow-up periods. The model assumes that the counts for each subject are generated by mixtures of nonhomogeneous Poisson processes with smooth intensity functions modeled with penalized splines. Time-dependent covariate effects are also incorporated into the process intensity using splines. Discrete mixtures of these nonhomogeneous Poisson process spline models extract functional information from underlying clusters representing hidden subpopulations. The motivating application is an experiment to test the effectiveness of pheromones in disrupting the mating pattern of the cherry bark tortrix moth. Mature moths arise from hidden, but distinct, subpopulations and monitoring the subpopulation responses was of interest. Within-cluster random effects are used to account for correlation structures and heterogeneity common to this type of data. An estimating equation approach to inference requiring only low moment assumptions is developed and the finite sample properties of the proposed estimating functions are investigated empirically by simulation.     

Linear mixed models with flexible distributions of random effects for longitudinal data

Normality of random effects is a routine assumption for the linear mixed model, but it may be unrealistic, obscuring important features of among-individual variation. We relax this assumption by approximating the random effects density by the seminonparameteric (SNP) representation of Gallant and Nychka (1987, Econometrics 55, 363-390), which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior, controlled by a user-chosen tuning parameter. An advantage is that the marginal likelihood may be expressed in closed form, so inference may be carried out using standard optimization techniques. We demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality, and we illustrate the approach via simulation and using longitudinal data from the Framingham study.     

Cox proportional hazards models with left truncation and time‐varying coefficient: Application of age at event as outcome in cohort studies

When analyzing time‐to‐event cohort data, two different ways of choosing a time scale have been discussed in the literature: time‐on‐study or age at onset of disease. One advantage of choosing the latter is interpretability of the hazard ratio as a function of age. To handle the analysis of age at onset in a principled manner, we present an analysis of the Cox Proportional Hazards model with time‐varying coefficient for left‐truncated and right‐censored data. In the analysis of Northern Manhattan Study (NOMAS) with age at onset of stroke as outcome, we demonstrate that well‐established risk factors may be important only around a certain age span and less established risk factors can have a strong effect in a certain age span.