Instrumental variable estimation of heteroskedasticity adaptive error component models

The linear panel data estimator proposed by Hausman and Taylor relaxes the hypothesis of exogenous regressors that is assumed by generalized least squares methods but, unlike the Fixed Effects estimator, it can handle endogenous time invariant explanatory variables in the regression equation. One of the assumptions underlying the estimator is the homoskedasticity of the error components. This can be restrictive in applications, and therefore in this paper the assumption is relaxed and more efficient adaptive versions of the estimator are presented.     

Choice of parametric models in survival analysis: applications to monotherapy for epilepsy and cerebral palsy

Summary. In the analysis of medical survival data, semiparametric proportional hazards models are widely used. When the proportional hazards assumption is not tenable, these models will not be suitable. Other models for covariate effects can be useful. In particular, we consider accelerated life models, in which the effect of covariates is to scale the quantiles of the base-line distribution. Solomon and Hutton have suggested that there is some robustness to misspecification of survival regression models. They showed that the relative importance of covariates is preserved under misspecification with assumptions of small coefficients and orthogonal transformation of covariates. We elucidate these results by applications to data from five trials which compare two common anti-epileptic drugs (carbamazepine versus sodium valporate monotherapy for epilepsy) and to survival of a cohort of people with cerebral palsy. Results on the robustness against model misspecification depend on the assumptions of small coefficients and on the underlying distribution of the data. These results hold in cerebral palsy but do not hold in epilepsy data which have early high hazard rates. The orthogonality of coefficients is not important. However, the choice of model is important for an estimation of the magnitude of effects, particularly if the base-line shape parameter indicates high initial hazard rates.     

Moment-based estimation of nonlinear regression models with boundary outcomes and endogeneity,with applications to nonnegative and fractional responses

In this article, we suggest simple moment-based estimators to deal with unobserved heterogeneity in a special class of nonlinear regression models that includes as main particular cases exponential models for nonnegative responses and logit and complementary loglog models for fractional responses. The proposed estimators: (i) treat observed and omitted covariates in a similar manner; (ii) can deal with boundary outcomes; (iii) accommodate endogenous explanatory variables without requiring knowledge on the reduced form model, although such information may be easily incorporated in the estimation process; (iv) do not require distributional assumptions on the unobservables, a conditional mean assumption being enough for consistent estimation of the structural parameters; and (v) under the additional assumption that the dependence between observables and unobservables is restricted to the conditional mean, produce consistent estimators of partial effects conditional only on observables.     

Estimation of Some Nonlinear Panel Data Models With Both Time-Varying and Time-Invariant Explanatory Variables

The so-called “fixed effects” approach to the estimation of panel data models suffers from the limitation that it is not possible to estimate the coefficients on explanatory variables that are time-invariant. This is in contrast to a “random effects” approach, which achieves this by making much stronger assumptions on the relationship between the explanatory variables and the individual-specific effect. In a linear model, it is possible to obtain the best of both worlds by making random effects-type assumptions on the time-invariant explanatory variables while maintaining the flexibility of a fixed effects approach when it comes to the time-varying covariates. This article attempts to do the same for some popular nonlinear models.     

Outliers in multilevel data

This paper offers the data analyst a range of practical procedures for dealing with outliers in multilevel data. It first develops several techniques for data exploration for outliers and outlier analysis and then applies these to the detailed analysis of outliers in two large scale multilevel data sets from educational contexts. The techniques include the use of deviance reduction, measures based on residuals, leverage values, hierarchical cluster analysis and a measure called DFITS. Outlier analysis is more complex in a multilevel data set than in, say, a univariate sample or a set of regression data, where the concept of an outlying value is straightforward. In the multilevel situation one has to consider, for example, at what level or levels a particular response is outlying, and in respect of which explanatory variables; furthermore, the treatment of a particular response at one level may affect its status or the status of other units at other levels in the model.     

A generalization of the logistic linear'model

Consider the logistic linear model, with some explanatory variables overlooked. Those explanatory variables may be quantitative or qualitative. In either case, the resulting true response variable is not a binomial or a beta-binomial but a sum of binomials. Hence, standard computer packages for logistic regression can be inappropriate even if an overdispersion factor is incorporated. Therefore, a discrete exponential family assumption is considered to broaden the class of sampling models. Likelihood and Bayesian analyses are discussed. Bayesian computation techniques such as Laplacian approximations and Markov chain simulations are used to compute posterior densities and moments. Approximate conditional distributions are derived and are shown to be accurate. The Markov chain simulations are performed effectively to calculate posterior moments by using the approximate conditional distributions. The methodology is applied to Keeler's hardness of winter wheat data for checking binomial assumptions and to Matsumura's Accounting exams data for detailed likelihood and Bayesian analyses.     

Robust regression: an inferential method for determining which independent variables are most important

Consider the usual linear regression model consisting of two or more explanatory variables. There are many methods aimed at indicating the relative importance of the explanatory variables. But in general these methods do not address a fundamental issue: when all of the explanatory variables are included in the model, how strong is the empirical evidence that the first explanatory variable is more or less important than the second explanatory variable? How strong is the empirical evidence that the first two explanatory variables are more important than the third explanatory variable? The paper suggests a robust method for dealing with these issues. The proposed technique is based on a particular version of explanatory power used in conjunction with a modification of the basic percentile method.     

A Bayesian hierarchical model for categorical longitudinal data from a social survey of immigrants

Summary.  The paper investigates a Bayesian hierarchical model for the analysis of categorical longitudinal data from a large social survey of immigrants to Australia. Data for each subject are observed on three separate occasions, or waves, of the survey. One of the features of the data set is that observations for some variables are missing for at least one wave. A model for the employment status of immigrants is developed by introducing, at the first stage of a hierarchical model, a multinomial model for the response and then subsequent terms are introduced to explain wave and subject effects. To estimate the model, we use the Gibbs sampler, which allows missing data for both the response and the explanatory variables to be imputed at each iteration of the algorithm, given some appropriate prior distributions. After accounting for significant covariate effects in the model, results show that the relative probability of remaining unemployed diminished with time following arrival in Australia.     

A Class of Parametric Dynamic Survival Models

A class of parametric dynamic survival models are explored in which only limited parametric assumptions are made, whilst avoiding the assumption of proportional hazards. Both the log-baseline hazard and covariate effects are modelled by piecewise constant and correlated processes. The method of estimation is to use Markov chain Monte Carlo simulations Gibbs sampling with a Metropolis–Hastings step. In addition to standard right censored data sets, extensions to accommodate interval censoring and random effects are included. The model is applied to two well known and illustrative data sets, and the dynamic variability of covariate effects investigated.     

Marginal Likelihood Estimation for Proportional Odds Models with Right Censored Data

One majoraspect in medical research is to relate the survival times ofpatients with the relevant covariates or explanatory variables.The proportional hazards model has been used extensively in thepast decades with the assumption that the covariate effects actmultiplicatively on the hazard function, independent of time.If the patients become more homogeneous over time, say the treatmenteffects decrease with time or fade out eventually, then a proportionalodds model may be more appropriate. In the proportional oddsmodel, the odds ratio between patients can be expressed as afunction of their corresponding covariate vectors, in which,the hazard ratio between individuals converges to unity in thelong run. In this paper, we consider the estimation of the regressionparameter for a semiparametric proportional odds model at whichthe baseline odds function is an arbitrary, non-decreasing functionbut is left unspecified. Instead of using the exact survivaltimes, only the rank order information among patients is used.A Monte Carlo method is used to approximate the marginal likelihoodfunction of the rank invariant transformation of the survivaltimes which preserves the information about the regression parameter.The method can be applied to other transformation models withcensored data such as the proportional hazards model, the generalizedprobit model or others. The proposed method is applied to theVeteran's Administration lung cancer trial data.     

Multi-state Models in Epidemiology

I first discuss the main assumptions which can be made for multi-state models: the time-homogeneity and semi-Markov assumptions, the problem of choice of the time scale, the assumption of homogeneity of the population and also assumptions about the way the observations are incomplete, leading to truncation and censoring. The influence of covariates and different durations and time-dependent variables are synthesized using explanatory processes, and a general additive model for transition intensities presented. Different inference approaches, including penalized likelihood, are considered. Finally three examples of application in epidemiology are presented and some references to other works are given.     

Consistency of Procrustes Estimators

Lele has shown that the Procrustes estimator of form is inconsistent and raised the question about the consistency of the Procrustes estimator of shape. In this paper the consistency of estimators of form and shape is studied under various assumptions. In particular, it is shown that the Procrustes estimator of shape is consistent under the assumption of an isotropic error distribution and that consistency breaks down if the assumption of isotropy is relaxed. The relevance of these results for practical shape analysis is discussed. As a by-product, some new results are derived for the offset uniform distribution from directional data.     

Score test of homogeneity for survival data

If follow-up is made for subjects which are grouped into units, such as familial or spatial units then it may be interesting to test whether the groups are homogeneous (or independent for given explanatory variables). The effect of the groups is modelled as random and we consider a frailty proportional hazards model which allows to adjust for explanatory variables. We derive the score test of homogeneity from the marginal partial likelihood and it turns out to be the sum of a pairwise correlation term of martingale residuals and an overdispersion term. In the particular case where the sizes of the groups are equal to one, this statistic can be used for testing overdispersion. The asymptotic variance of this statistic is derived using counting process arguments. An extension to the case of several strata is given. The resulting test is computationally simple; its use is illustrated using both simulated and real data. In addition a decomposition of the score statistic is proposed as a sum of a pairwise correlation term and an overdispersion term. The pairwise correlation term can be used for constructing a statistic more robust to departure from the proportional hazard model, and the overdispesion term for constructing a test of fit of the proportional hazard model.     

A nonparametric test to compare survival distributions with covariate adjustment

Summary.  The analysis of covariance is a technique that is used to improve the power of a k -sample test by adjusting for concomitant variables. If the end point is the time of survival, and some observations are right censored, the score statistic from the Cox proportional hazards model is the method that is most commonly used to test the equality of conditional hazard functions. In many situations, however, the proportional hazards model assumptions are not satisfied. Specifically, the relative risk function is not time invariant or represented as a log-linear function of the covariates. We propose an asymptotically valid k -sample test statistic to compare conditional hazard functions which does not require the assumption of proportional hazards, a parametric specification of the relative risk function or randomization of group assignment. Simulation results indicate that the performance of this statistic is satisfactory. The methodology is demonstrated on a data set in prostate cancer.     

Generalized log-gamma regression models with cure fraction

In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.     

BAYESIAN PREDICTION FOR SPATIAL GENERALISED LINEAR MIXED MODELS WITH CLOSED SKEW NORMAL LATENT VARIABLES

Spatial generalised linear mixed models are used commonly for modelling non‐Gaussian discrete spatial responses. In these models, the spatial correlation structure of data is modelled by spatial latent variables. Most users are satisfied with using a normal distribution for these variables, but in many applications it is unclear whether or not the normal assumption holds. This assumption is relaxed in the present work, using a closed skew normal distribution for the spatial latent variables, which is more flexible and includes normal and skew normal distributions. The parameter estimates and spatial predictions are calculated using the Markov Chain Monte Carlo method. Finally, the performance of the proposed model is analysed via two simulation studies, followed by a case study in which practical aspects are dealt with. The proposed model appears to give a smaller cross‐validation mean square error of the spatial prediction than the normal prior in modelling the temperature data set.     

Model diagnostics for the proportional hazards model with length-biased data

Length-biased data are frequently encountered in prevalent cohort studies. Many statistical methods have been developed to estimate the covariate effects on the survival outcomes arising from such data while properly adjusting for length-biased sampling. Among them, regression methods based on the proportional hazards model have been widely adopted. However, little work has focused on checking the proportional hazards model assumptions with length-biased data, which is essential to ensure the validity of inference. In this article, we propose a statistical tool for testing the assumed functional form of covariates and the proportional hazards assumption graphically and analytically under the setting of length-biased sampling, through a general class of multiparameter stochastic processes. The finite sample performance is examined through simulation studies, and the proposed methods are illustrated with the data from a cohort study of dementia in Canada.

     

Semi-parametric survival analysis via Dirichlet process mixtures of the First Hitting Time model

Time-to-event data often violate the proportional hazards assumption inherent in the popular Cox regression model. Such violations are especially common in the sphere of biological and medical data where latent heterogeneity due to unmeasured covariates or time varying effects are common. A variety of parametric survival models have been proposed in the literature which make more appropriate assumptions on the hazard function, at least for certain applications. One such model is derived from the First Hitting Time (FHT) paradigm which assumes that a subject’s event time is determined by a latent stochastic process reaching a threshold value. Several random effects specifications of the FHT model have also been proposed which allow for better modeling of data with unmeasured covariates. While often appropriate, these methods often display limited flexibility due to their inability to model a wide range of heterogeneities. To address this issue, we propose a Bayesian model which loosens assumptions on the mixing distribution inherent in the random effects FHT models currently in use. We demonstrate via simulation study that the proposed model greatly improves both survival and parameter estimation in the presence of latent heterogeneity. We also apply the proposed methodology to data from a toxicology/carcinogenicity study which exhibits nonproportional hazards and contrast the results with both the Cox model and two popular FHT models.

     

On the estimation of a binary response model in a selected population

A generalization of the Probit model is presented, with the extended skew-normal cumulative distribution as a link function, which can be used for modelling a binary response variable in the presence of selectivity bias. The estimate of the parameters via ML is addressed, and inference on the parameters expressing the degree of selection is discussed. The assumption underlying the model is that the selection mechanism influences the unmeasured factors and does not affect the explanatory variables. When this assumption is violated, but other conditional independencies hold, then the model proposed here is derived. In particular, the instrumental variable formula still applies and the model results at the second stage of the estimating procedure.     

Partial orders with respect to continuous covariates and tests for the proportional hazards model

Several omnibus tests of the proportional hazards assumption have been proposed in the literature. In the two-sample case, tests have also been developed against ordered alternatives like monotone hazard ratio and monotone ratio of cumulative hazards. Here we propose a natural extension of these partial orders to the case of continuous and potentially time varying covariates, and develop tests for the proportional hazards assumption against such ordered alternatives. The work is motivated by applications in biomedicine and economics where covariate effects often decay over lifetime. The proposed tests do not make restrictive assumptions on the underlying regression model, and are applicable in the presence of time varying covariates, multiple covariates and frailty. Small sample performance and an application to real data highlight the use of the framework and methodology to identify and model the nature of departures from proportionality.