Robustness of the latent variable model for correlated binary data

The marginal regression model offers a useful alternative to conditional approaches to analyzing binary data (Liang, Zeger, and Qaqish, 1992, Journal of the Royal Statistical Society, Series B 54, 3-40). Instead of modelling the binary data directly as do Liang and Zeger (1986, Biometrika 73, 13-22), the parametric marginal regression model developed by Qu et al. (1992, Biometrics 48, 1095-1102) assumes that there is an underlying multivariate normal vector that gives rise to the observed correlated binary outcomes. Although this parametric approach provides a flexible way to model different within-cluster correlation structures and does not restrict the parameter space, it is of interest to know how robust the parameter estimates are with respect to choices of the latent distribution. We first extend the latent modelling to include multivariate t-distributed latent vectors and assess the robustness in this class of distributions. Then we show through a simulation that the parameter estimates are robust with respect to the latent distribution even if latent distribution is skewed. In addtion to this empirical evidence for robustness, we show through the iterative algorithm that the robustness of the regression coefficents with respect to misspecifications of covariance structure in Liang and Zeger's model in fact indicates robustness with respect to underlying distributional assumptions of the latent vector in the latent variable model.     

A Generalized Logistic Model for Quantal Response Bioassay

In bioassay, where different levels of the stimulus may represent different doses of a drug, the binary response is the death or survival of an individual receiving a specified dose. In such applications, it is common to model the probability of a positive response P at the stimulus level x by P = F(x′β), where F is a cumulative distribution function and β is a vector of unknown parameters which characterize the response function. The two most popular models used for modelling binary response bioassay involve the probit model [BLISS (1935), FINNEY (1978)], and the logistic model [BERKSON (1944), BROWN (1982)]. However, these models have some limitations. The use of the probit model involves the inverse of the standard normal distribution function, making it rather intractable. The logistic model has a simple form and a closed expression for the inverse distribution function, however, neither the logistic nor the probit can provide a good fit to response functions which are not symmetric or are symmetric but have a steeper or gentler incline in the central probability region. In this paper we introduce a more realistic model for the analysis of quantal response bioassay. The proposed model, which we refer to it as the generalized logistic model, is a family of response curves indexed by shape parameters m₁ and m₂. This family is rich enough to include the probit and logistic models as well as many others as special cases or limiting distributions. In particular, we consider the generalized logistic three parameter model where we assume that m₁ = m, m is a positive real number, and m₂ = 1. We apply this model to various sets of data, comparing the fit results to those obtained previously by other dose-response curves such as the logistic and probit, and showing that the fit can be improved by using the generalized logistic.     

A stochastic gompertz model with hereditary effect

Summary In this paper we have studied a stochastic version of the Gompertz model for population growth of a single species after incorporating the aspect of heredity. Various statistical characteristics-the mean-value function, covariance-kernel, etc.-are evaluated for a delta-correlated process and their asymptotic values obtained. The effect of the hereditary kernel on the various statistics is discussed and it is found that it is to shift the distribution towards the origin.     

Flexible generalized t-link models for binary response data

A critical issue in modelling binary response data is the choiceof the links. We introduce a new link based on the generalizedt-distribution. There are two parameters in the generalizedt-link: one parameter purely controls the heaviness of the tailsof the link and the second parameter controls the scale of thelink. Two major advantages are offered by the generalized t-links.First, a symmetric generalized t-link with an unknown shapeparameter is much more identifiable than a Student t-link withunknown degrees of freedom and a known scale parameter. Secondly,skewed generalized t-links with both unknown shape and scaleparameters provide much more flexible and improved skewed linkregression models than the existing skewed links. Various theoreticalproperties and attractive features of the proposed links areexamined and explored in detail. An efficient Markov chain MonteCarlo algorithm is developed for sampling from the posteriordistribution. The deviance information criterion measure isused for guiding the choice of links. The proposed methodologyis motivated and illustrated by prostate cancer data.     

Variable selection for nonparametric additive Cox model with interval-censored data

The standard Cox model is perhaps the most commonly used model for regression analysis of failure time data but it has some limitations such as the assumption on linear covariate effects. To relax this, the nonparametric additive Cox model, which allows for nonlinear covariate effects, is often employed, and this paper will discuss variable selection and structure estimation for this general model. For the problem, we propose a penalized sieve maximum likelihood approach with the use of Bernstein polynomials approximation and group penalization. To implement the proposed method, an efficient group coordinate descent algorithm is developed and can be easily carried out for both low- and high-dimensional scenarios. Furthermore, a simulation study is performed to assess the performance of the presented approach and suggests that it works well in practice. The proposed method is applied to an Alzheimer's disease study for identifying important and relevant genetic factors.     

Dynamic logistic regression and dynamic model averaging for binary classification

We propose an online binary classification procedure for cases when there is uncertainty about the model to use and parameters within a model change over time. We account for model uncertainty through dynamic model averaging, a dynamic extension of Bayesian model averaging in which posterior model probabilities may also change with time. We apply a state-space model to the parameters of each model and we allow the data-generating model to change over time according to a Markov chain. Calibrating a "forgetting" factor accommodates different levels of change in the data-generating mechanism. We propose an algorithm that adjusts the level of forgetting in an online fashion using the posterior predictive distribution, and so accommodates various levels of change at different times. We apply our method to data from children with appendicitis who receive either a traditional (open) appendectomy or a laparoscopic procedure. Factors associated with which children receive a particular type of procedure changed substantially over the 7 years of data collection, a feature that is not captured using standard regression modeling. Because our procedure can be implemented completely online, future data collection for similar studies would require storing sensitive patient information only temporarily, reducing the risk of a breach of confidentiality.     

Use of summary measures to adjust for informative missingness in repeated measures data with random effects

We discuss how to apply the conditional informative missing model of Wu and Bailey (1989, Biometrics 45, 939-955) to the setting where the probability of missing a visit depends on the random effects of the primary response in a time-dependent fashion. This includes the case where the probability of missing a visit depends on the true value of the primary response. Summary measures for missingness that are weighted sums of the indicators of missed visits are derived for these situations. These summary measures are then incorporated as covariates in a random effects model for the primary response. This approach is illustrated by analyzing data collected from a trial of heroin addicts where missed visits are informative about drug test results. Simulations of realistic experiments indicate that these time-dependent summary measures also work well under a variety of informative censoring models. These summary measures can achieve large reductions in estimation bias and mean squared errors relative to those obtained by using other summary measures.     

A hybrid model for nonignorable dropout in longitudinal binary responses

This article presents a likelihood-based method for handling nonignorable dropout in longitudinal studies with binary responses. The methodology developed is appropriate when the target of inference is the marginal distribution of the response at each occasion and its dependence on covariates. A hybrid model is formulated, which is designed to retain advantageous features of the selection and pattern-mixture model approaches. This formulation accommodates a variety of assumed forms of nonignorable dropout, while maintaining transparency of the constraints required for identifying the overall model. Once appropriate identifying constraints have been imposed, likelihood-based estimation is conducted via the EM algorithm. The article concludes by applying the approach to data from a randomized clinical trial comparing two doses of a contraceptive.     

A two variable delay model for the circadian rhythm of Neurospora crassa

A two variable model with delay in both the variables, is proposed for the circadian oscillations of protein concentrations in the fungal species Neurospora crassa. The dynamical variables chosen are the concentrations of FRQ and WC-1 proteins. Our model is a two variable simplification of the detailed model of Smolen et al. (J. Neurosci. 21 (2001) 6644) modeling circadian oscillations with interlocking positive and negative feedback loops, containing 23 variables. In our model, as in the case of Smolen's model, a sustained limit cycle oscillation takes place in both FRQ and WC-1 protein in continuous darkness, and WC-1 is anti-phase to FRQ protein, as observed in experiments. The model accounts for various characteristic features of circadian rhythms such as entrainment to light dark cycles, phase response curves and robustness to parameter variation and molecular fluctuations. Simulations are carried out to study the effect of periodic forcing of circadian oscillations by light-dark cycles. The periodic forcing resulted in a rich bifurcation diagram that includes quasiperiodicity and chaotic oscillations, depending on the magnitude of the periodic changes in the light controlled parameter. When positive feedback is eliminated, our model reduces to the generic one dimensional delay model of Lema et al. (J. Theor. Biol. 204 (2000) 565), delay model of the circadian pace maker with FRQ protein as the dynamical variable which represses its own production. This one-dimensional model also exhibits all characteristic features of circadian oscillations and gives rise to circadian oscillations which are reasonably robust to parameter variations and molecular noise.     

Offspring-parent regression for a binary trait

Summary Offspring-parent regression is often used to estimate the heritability of a quantitative trait. It is shown that for a purely binary trait, the regression of offspring on one parent is always linear, while that on both parents or mid-parent is generally nonlinear. However, the regressions are linear on a logistic scale.     

A bivariate pseudolikelihood for incomplete longitudinal binary data with nonignorable nonmonotone missingness

For analyzing longitudinal binary data with nonignorable and nonmonotone missing responses, a full likelihood method is complicated algebraically, and often requires intensive computation, especially when there are many follow-up times. As an alternative, a pseudolikelihood approach has been proposed in the literature under minimal parametric assumptions. This formulation only requires specification of the marginal distributions of the responses and missing data mechanism, and uses an independence working assumption. However, this estimator can be inefficient for estimating both time-varying and time-stationary effects under moderate to strong within-subject associations among repeated responses. In this article, we propose an alternative estimator, based on a bivariate pseudolikelihood, and demonstrate in simulations that the proposed method can be much more efficient than the previous pseudolikelihood obtained under the assumption of independence. We illustrate the method using longitudinal data on CD4 counts from two clinical trials of HIV-infected patients.     

A longitudinal measurement error model with a semicontinuous covariate

Covariate measurement error in regression is typically assumed to act in an additive or multiplicative manner on the true covariate value. However, such an assumption does not hold for the measurement error of sleep-disordered breathing (SDB) in the Wisconsin Sleep Cohort Study (WSCS). The true covariate is the severity of SDB, and the observed surrogate is the number of breathing pauses per unit time of sleep, which has a nonnegative semicontinuous distribution with a point mass at zero. We propose a latent variable measurement error model for the error structure in this situation and implement it in a linear mixed model. The estimation procedure is similar to regression calibration but involves a distributional assumption for the latent variable. Modeling and model-fitting strategies are explored and illustrated through an example from the WSCS.