Quantile estimation of partially varying coefficient model for panel count data with informative observation times

Panel count data occur in many fields including clinical, demographical and industrial studies and an extensive literature has been established for their regression analysis. However, most of the existing methods apply only to the situations where both covariates and their effects are constant or one of them may be time-dependent. In the first part of this dissertation, we consider a situation where both covariates and their effects may be time-dependent and an estimating equation-based approach is developed for estimating those time-varying effects. In the method, B-spline functions are employed to approximate time-dependent coefficients and the asymptotic properties of the proposed estimators are established. To assess the performance of the proposed approach, an extensive simulation study is conducted and suggests that it works well in practical situations. An application to the China Health and Nutrition Survey (CHNS) study is provided. In practice, there could exist more than one type of event of interest, such as two types of tumor recurrence, leading to multivariate panel count data. The second part of this dissertation considers marginal mean model for multivariate panel count data with time-dependent coefficient and covariate effects, which has limited previous research. Based on the conditional estimating equation method developed for time-dependent covariates, we approximate the coefficients by B-splines, hence allow both coefficients and covariates to be time-dependent. Simulation studies show that the proposed estimation procedures work well for practical situations. The methodology is again applied to the China Health and Nutrition Survey (CHNS) study. When we consider time-varying covariates and coefficients effects, most of the previous study focused on the proportional mean model because the likelihood function under the rate model involves intractable integration. However, the rate model is more realistic and efficient. Hence, in the third part of this dissertation, we propose a semi-parametric MLE method under the rate model for panel count data with time-dependent covariates and time-varying effects. B-spline functions are employed again to approximate time-dependent coefficients and an efficient Expectation-Maximization-type algorithm is developed to overcome the computational difficulty. The resulting estimators are shown to be consistent and asymptotically efficient. Monte Carlo simulation studies demonstrate that the proposed method enjoys desirable finite-sample properties. An application to The Young Women's Project (YWP) is provided.

The Poisson regression model is widely used for count data. This model assumes equidispersion. In practice, equidispersion is seldom reflected in data. However, in real-life data, the variance usually exceeds the mean. This situation is known as overdispersion. Negative binomial distribution and other Poisson mix models are often used to model overdispersion count data. Another extension of the negative binomial distribution in another model for count data is the univariate generalized Waring. In addition, the model developed by Famoye can be used in the analysis of count data. When the count data contains a large number of zeros, it is necessary to use zero-inflated models. In this study, different generalized regression models are emphasized for the analysis of excessive zeros count data. For this purpose, a real data set was analysed with the generalized Poisson model, generalized negative binomial model, generalized negative binomial Famoye, generalized Waring model, and the foregoing zero-inflated models. Log-likelihood, Akaike information criterion, Bayes information criterion, Vuong statistics were used for model comparisons.

One of the most important indicators of dental caries prevalence is the total count of decayed, missing or filled surfaces in a tooth. These count data are often clustered in nature (several count responses clustered within a subject), over-dispersed as well as spatially referenced (a diseased tooth might be positively influencing the decay process of a set of neighbouring teeth). In this article, we develop a multivariate spatial betabinomial (BB) model for these data that accommodates both over-dispersion as well as latent spatial associations. Using a Bayesian paradigm, the re-parameterised marginal mean (as well as variance) under the BB framework are modelled using a regression on subject/tooth-specific co-variables and a conditionally autoregressive prior that models the latent spatial process. The necessity of exploiting spatial associations to model count data arising in dental caries research is demonstrated using a small simulation study. Real data confirms that our spatial BB model provides a superior estimation and model fit as compared to other sub-models that do not consider modelling spatial associations.

The current status and panel count data frequently arise from cancer and tumorigenicity studies when events currently occur. A common and widely used class of two sample tests, for current status and panel count data, is the permutation class. We manipulate the double saddlepoint method to calculate the exact mid-p-values of the underlying permutation distributions of this class of tests. Permutation simulations are replaced by analytical saddlepoint computations which provide extremely accurate mid-p-values that are exact for most practical purposes and almost always more accurate than normal approximations. The method is illustrated using two real tumorigenicity panel count data. To compare the saddlepoint approximation with the normal asymptotic approximation, a simulation study is conducted. The speed and accuracy of the saddlepoint method facilitate the calculation of the confidence interval for the treatment effect. The inversion of the mid-p-values to calculate the confidence interval for the mean rate of development of the recurrent event is discussed.

In analysis of longitudinal data, it is not uncommon that observation times of repeated measurements are subject-specific and correlated with underlying longitudinal outcomes. Taking account of the dependence between observation times and longitudinal outcomes is critical under these situations to assure the validity of statistical inference. In this article, we propose a flexible joint model for longitudinal data analysis in the presence of informative observation times. In particular, the new procedure considers the shared random-effect model and assumes a time-varying coefficient for the latent variable, allowing a flexible way of modeling longitudinal outcomes while adjusting their association with observation times. Estimating equations are developed for parameter estimation. We show that the resulting estimators are consistent and asymptotically normal, with variance-covariance matrix that has a closed form and can be consistently estimated by the usual plug-in method. One additional advantage of the procedure is that it provides a unified framework to test whether the effect of the latent variable is zero, constant, or time-varying. Simulation studies show that the proposed approach is appropriate for practical use. An application to a bladder cancer data is also given to illustrate the methodology.

This paper discusses the formation of one temporal spatial model, the Generalized Space Time Autoregressive (GSTAR) model, if the data of model is count data. The development of the GSTAR model is an update or problem completion of the count data which tends to be stationary and non-normal / Gaussian data, because GSTAR model is assumed normal distributed and stationarity. GSTAR modelling for count data refers to the Time series model for count data, which are the Generalized Autoregressive Moving Average (GARMA) model and modeling Count Time Series which has the Generalized Linear Model (GLM) concept. The model formed is called the Poisson GSTAR model

The purpose of this chapter is to present recent advances in models for genomic-enabled prediction developed for ordinal categorical and count data. For both models we provide details of their corresponding derivation and then apply them to a real data set. The proposed models were derived using a Bayesian framework. Bayesian logistic ordinal regression (BLOR) and Bayesian negative binomial regression (BNBR) make use of the Polya-Gamma distribution to produce an analytic Gibbs, a sampler with similar full conditional distributions of a model with Gaussian response and can be used for complex data sets as those that arise in the context of genomic selection where the sample size usually is smaller than the number of covariates (markers). We illustrate the proposed models using simulation and a real data set. Results indicate that our models for ordinal categorical and count data are a good alternative for analyzing ordinal and count data in the context of genomic-enabled prediction.

Overdispersion is common in models of count data in ecology and evolutionary biology, and can occur due to missing covariates, non-independent (aggregated) data, or an excess frequency of zeroes (zero-inflation). Accounting for overdispersion in such models is vital, as failing to do so can lead to biased parameter estimates, and false conclusions regarding hypotheses of interest. Observation-level random effects (OLRE), where each data point receives a unique level of a random effect that models the extra-Poisson variation present in the data, are commonly employed to cope with overdispersion in count data. However studies investigating the efficacy of observation-level random effects as a means to deal with overdispersion are scarce. Here I use simulations to show that in cases where overdispersion is caused by random extra-Poisson noise, or aggregation in the count data, observation-level random effects yield more accurate parameter estimates compared to when overdispersion is simply ignored. Conversely, OLRE fail to reduce bias in zero-inflated data, and in some cases increase bias at high levels of overdispersion. There was a positive relationship between the magnitude of overdispersion and the degree of bias in parameter estimates. Critically, the simulations reveal that failing to account for overdispersion in mixed models can erroneously inflate measures of explained variance (r2), which may lead to researchers overestimating the predictive power of variables of interest. This work suggests use of observation-level random effects provides a simple and robust means to account for overdispersion in count data, but also that their ability to minimise bias is not uniform across all types of overdispersion and must be applied judiciously.

S ome P oints of D iscussion • What are counts? What are count data? • What is a linear statistical model? • What is the relationship between a probability distribution function (PDF) and a statistical model? • What are the parameters of a statistical model? Where do they come from, and can we ever truly know them? • How does a count model differ from other regression models? • What are the basic count models, and how do they relate with one another? • What is over dispersion, and why is it considered to be the fundamental problem when modeling count data? W hat A re C ounts ? When discussing the modeling of count data, it's important to clarify exactly what is meant by a count, as well as “count data” and “count variable.” The word “count” is typically used as a verb meaning to enumerate units, items, or events. We might count the number of road kills observed on a stretch of highway, how many patients died at a particular hospital within 48 hours of having a myocardial infarction, or how many separate sunspots were observed in March 2013. “Count data,” on the other hand, is a plural noun referring to observations made about events or items that are enumerated. In statistics, count data refer to observations that have only nonnegative integer values ranging from zero to some greater undetermined value. Theoretically, counts can range from zero to infinity, but they are always limited to some lesser distinct value – generally the maximum value of the count data being modeled.

In modern biomedical datasets, it is common for recurrent outcomes data to be collected in an incomplete manner. More specifically, information on recurrent events is routinely recorded as a mixture of recurrent event data, panel count data, and panel binary data; we refer to this structure as general mixed recurrent event data. Although the aforementioned data types are individually well-studied, there does not appear to exist an established approach for regression analysis of the three component combination. Often, ad-hoc measures such as imputation or discarding of data are used to homogenize records prior to the analysis, but such measures lead to obvious concerns regarding robustness, loss of efficiency, and other issues. This work proposes a maximum likelihood regression estimation procedure for the combination of general mixed recurrent event data and establishes the asymptotic properties of the proposed estimators. In addition, we generalize the approach to allow for the existence of terminal events, a common complicating feature in recurrent event analysis. Numerical studies and application to the Childhood Cancer Survivor Study suggest that the proposed procedures work well in practical situations.

Paradigms for substance abuse cue-reactivity research involve pharmacological or stressful stimulation designed to elicit stress and craving responses in cocaine-dependent subjects. It is unclear as to whether stress induced from participation in such studies increases drug-seeking behavior. We propose a 2-state Hidden Markov model to model the number of cocaine abuses per week before and after participation in a stress-and cue-reactivity study. The hypothesized latent state corresponds to 'high' or 'low' use. To account for a preponderance of zeros, we assume a zero-inflated Poisson model for the count data. Transition probabilities depend on the prior week's state, fixed demographic variables, and time-varying covariates. We adopt a Bayesian approach to model fitting, and use the conditional predictive ordinate statistic to demonstrate that the zero-inflated Poisson hidden Markov model outperforms other models for longitudinal count data.

The Poisson regression model is often used as a first model for count data with covariates. Since this model is a GLM with canonical link, regression parameters can be easily fitted using standard software. However the model requires equidispersion, which might not be valid for the data set under consideration. There have been many models proposed in the literature to allow for overdispersion. One such model is the negative binomial regression model. In addition, score tests have been commonly used to detect overdispersion in the data. However these tests do not allow to quantify the effects of overdispersion. In this paper we propose easily interpretable discrepancy measures which allow to quantify the overdispersion effects when comparing a negative binomial regression to Poisson regression. We propose asymptotic $\alpha$-level tests for testing the size of overdispersion effects in terms of the developed discrepancy measures. A graphical display of p-values curves can then be used to allow for an exact quantification of the overdispersion effects. This can lead to a validation of the Poisson regression or a discrimination of the Poisson regression with respect to the negative binomial regression. The proposed asymptotic tests are investigated in small samples using simulation and applied to two examples.

Panel count data often occur in a long-term recurrent event study, where the exact occurrence time of the recurrent events is unknown, but only the occurrence count between any two adjacent observation time points is recorded. Most traditional methods only handle panel count data for a single type of event. In this paper, we propose a Bayesian semiparameteric approach to analyze panel count data for multiple types of events. For each type of recurrent event, the proportional mean model is adopted to model the mean count of the event, where its baseline mean function is approximated by monotone I-splines. The correlation between multiple types of events is modeled by common frailty terms and scale parameters. Unlike many frequentist estimating equation methods, our approach is based on the observed likelihood and makes no assumption on the relationship between the recurrent process and the observation process. Under the Poisson counting process assumption, we develop an efficient Gibbs sampler based on novel data augmentation for the Markov chain Monte Carlo sampling. Simulation studies show good estimation performance of the baseline mean functions and the regression coefficients; meanwhile, the importance of including the scale parameter to flexibly accommodate the correlation between events is also demonstrated. Finally, a skin cancer data example is fully analyzed to illustrate the proposed methods.

A nonparametric test for the equality of counting processes with panel count data

Panel count data typically refer to data arising from studies with recurrent events, in which subjects are observed only at discrete time points rather than under continuous observations. We investigate a general situation where a recurrent event process is eventually truncated by an informative terminal event and we are particularly interested in behaviors of the recurrent event process near the terminal event. We propose a reversed mean model for estimating the mean function of the recurrent event process. We develop a two-stage sieve likelihood-based method to estimate the mean function, which overcomes the computational difficulties arising from a nuisance functional parameter involved in the likelihood. The consistency and the convergence rate of the two-stage estimator are established. Allowing for the convergence rate slower than the standard rate, we develop the general weak convergence theory of M-estimators with a nuisance functional parameter, and then apply it to the proposed estimator for deriving the asymptotic normality. Furthermore, a class of two-sample tests is developed. The proposed methods are evaluated with extensive simulation studies and illustrated with panel count data from the Chinese Longitudinal Healthy Longevity Study.