Laplace's approximation for relative risk frailty models

Multiple-Membership Survival Analysis and Its Applications in Organizational Behavior and Management Research

Multilevel paradigms have permeated organizational research in recent years, greatly advancing our understanding of organizational behavior and management decisions. Despite the advancements made in multilevel modeling, taking into account complex hierarchical structures in data remains challenging. This is particularly the case for models used for predicting the occurrence and timing of events and decisions—often referred to as survival models. In this study, the authors construct a multilevel survival model that takes into account subjects being nested in multiple environments—known as a multiple-membership structure. Through this article, the authors provide a step-by-step guide to building a multiple-membership survival model, illustrating each step with an application on a real-life, large-scale, archival data set. Easy-to-use R code is provided for each model-building step. The article concludes with an illustration of potential applications of the model to answer alternative research questions in the organizational behavior and management fields.

A Poisson approach to the validation of failure time surrogate endpoints in individual patient data meta-analyses

Surrogate endpoints are often used in clinical trials instead of well-established hard endpoints for practical convenience. The meta-analytic approach relies on two measures of surrogacy: one at the individual level and one at the trial level. In the survival data setting, a two-step model based on copulas is commonly used. We present a new approach which employs a bivariate survival model with an individual random effect shared between the two endpoints and correlated treatment-by-trial interactions. We fit this model using auxiliary mixed Poisson models. We study via simulations the operating characteristics of this mixed Poisson approach as compared to the two-step copula approach. We illustrate the application of the methods on two individual patient data meta-analyses in gastric cancer, in the advanced setting (4069 patients from 20 randomized trials) and in the adjuvant setting (3288 patients from 14 randomized trials).

Log-normal frailty models fitted as Poisson generalized linear mixed models

BACKGROUND AND OBJECTIVES

The equivalence of a survival model with a piecewise constant baseline hazard function and a Poisson regression model has been known since decades. As shown in recent studies, this equivalence carries over to clustered survival data: A frailty model with a log-normal frailty term can be interpreted and estimated as a generalized linear mixed model with a binary response, a Poisson likelihood, and a specific offset. Proceeding this way, statistical theory and software for generalized linear mixed models are readily available for fitting frailty models. This gain in flexibility comes at the small price of (1) having to fix the number of pieces for the baseline hazard in advance and (2) having to "explode" the data set by the number of pieces.

METHODS

In this paper we extend the simulations of former studies by using a more realistic baseline hazard (Gompertz) and by comparing the model under consideration with competing models. Furthermore, the SAS macro %PCFrailty is introduced to apply the Poisson generalized linear mixed approach to frailty models.

RESULTS

The simulations show good results for the shared frailty model. Our new %PCFrailty macro provides proper estimates, especially in case of 4 events per piece.

CONCLUSIONS

The suggested Poisson generalized linear mixed approach for log-normal frailty models based on the %PCFrailty macro provides several advantages in the analysis of clustered survival data with respect to more flexible modelling of fixed and random effects, exact (in the sense of non-approximate) maximum likelihood estimation, and standard errors and different types of confidence intervals for all variance parameters.

Survival analysis of clinical mastitis data using a nested frailty Cox model fit as a mixed-effects Poisson model

Mastitis is a complex disease affecting dairy cows and is considered to be the most costly disease of dairy herds. The hazard of mastitis is a function of many factors, both managerial and environmental, making its control a difficult issue to milk producers. Observational studies of clinical mastitis (CM) often generate datasets with a number of characteristics which influence the analysis of those data: the outcome of interest may be the time to occurrence of a case of mastitis, predictors may change over time (time-dependent predictors), the effects of factors may change over time (time-dependent effects), there are usually multiple hierarchical levels, and datasets may be very large. Analysis of such data often requires expansion of the data into the counting-process format - leading to larger datasets - thus complicating the analysis and requiring excessive computing time. In this study, a nested frailty Cox model with time-dependent predictors and effects was applied to Canadian Bovine Mastitis Research Network data in which 10,831 lactations of 8035 cows from 69 herds were followed through lactation until the first occurrence of CM. The model was fit to the data as a Poisson model with nested normally distributed random effects at the cow and herd levels. Risk factors associated with the hazard of CM during the lactation were identified, such as parity, calving season, herd somatic cell score, pasture access, fore-stripping, and proportion of treated cases of CM in a herd. The analysis showed that most of the predictors had a strong effect early in lactation and also demonstrated substantial variation in the baseline hazard among cows and between herds. A small simulation study for a setting similar to the real data was conducted to evaluate the Poisson maximum likelihood estimation approach with both Gaussian quadrature method and Laplace approximation. Further, the performance of the two methods was compared with the performance of a widely used estimation approach for frailty Cox models based on the penalized partial likelihood. The simulation study showed good performance for the Poisson maximum likelihood approach with Gaussian quadrature and biased variance component estimates for both the Poisson maximum likelihood with Laplace approximation and penalized partial likelihood approaches.

The analysis--hierarchical models: past, present and future

This paper discusses statistical modelling for data with a hierarchical structure, and distinguishes in this context between three different meanings of the term hierarchical model: to account for clustering, to investigate variability and separate predictive equations at different hierarchical levels (multi-level analysis), and in a Bayesian framework to involve multiple layers of data or prior information. Within each of these areas, the paper reviews both past developments and the present state, and offers indications of future directions. In a worked example, previously reported data on piglet lameness are reanalyzed with multi-level methodology for survival analysis, leading to new insights into the data structure and predictor effects. In our view, hierarchical models of all three types discussed have much to offer for data analysis in veterinary epidemiology and other disciplines.

Conditional Akaike information under generalized linear and proportional hazards mixed models

We study model selection for clustered data, when the focus is on cluster specific inference. Such data are often modelled using random effects, and conditional Akaike information was proposed in Vaida & Blanchard (2005) and used to derive an information criterion under linear mixed models. Here we extend the approach to generalized linear and proportional hazards mixed models. Outside the normal linear mixed models, exact calculations are not available and we resort to asymptotic approximations. In the presence of nuisance parameters, a profile conditional Akaike information is proposed. Bootstrap methods are considered for their potential advantage in finite samples. Simulations show that the performance of the bootstrap and the analytic criteria are comparable, with bootstrap demonstrating some advantages for larger cluster sizes. The proposed criteria are applied to two cancer datasets to select models when the cluster-specific inference is of interest.

Interpersonal problems and negative mood as predictors of within-day time to drinking

Using data collected via handheld electronic diaries (EDs), we examined within-day associations between early-day negative moods and stress and subsequent time to drinking. A sample of 97 (n=48 women) adults recruited to participate in a drinking-reduction intervention study used EDs to record mood and interpersonal problems at randomly selected times during each of 3 reporting intervals and drinking as it occurred each day for 21 days. Using multilevel hazard models, we tested associations between early-day stress/negative mood ratings and time to drinking as well as potential moderating effects of drinking to cope (DTC) motives on these associations. Whereas previous analyses of these data showed no associations between early-day negative moods and number of drinks consumed later in the day, here we found significant associations between negative moods and time to drinking. Associations involving negative moods, DTC, and hazard for drinking varied depending on time of day, and some mood effects were moderated by DTC. These findings suggest that time to drinking may be more sensitive to the effects of acute negative mood states than is drinking quantity.