Akaike's information criterion in generalized estimating equations

A predictive model relating daily fluctuations in summer temperatures and mortality rates

Background

In the context of climate change, an efficient alert system to prevent the risk associated with summer heat is necessary. The authors' objective was to describe the temperature-mortality relationship in France over a 29-year period and to define and validate a combination of temperature factors enabling optimum prediction of the daily fluctuations in summer mortality.

Methods

The study addressed the daily mortality rates of subjects aged over 55 years, in France as a whole, from 1975 to 2003. The daily minimum and maximum temperatures consisted in the average values recorded by 97 meteorological stations. For each day, a cumulative variable for the maximum temperature over the preceding 10 days was defined.

The mortality rate was modelled using a Poisson regression with over-dispersion and a first-order autoregressive structure and with control for long-term and within-summer seasonal trends. The lag effects of temperature were accounted for by including the preceding 5 days. A "backward" method was used to select the most significant climatic variables. The predictive performance of the model was assessed by comparing the observed and predicted daily mortality rates on a validation period (summer 2003), which was distinct from the calibration period (1975–2002) used to estimate the model.

Results

The temperature indicators explained 76% of the total over-dispersion. The greater part of the daily fluctuations in mortality was explained by the interaction between minimum and maximum temperatures, for a day t and the day preceding it. The prediction of mortality during extreme events was greatly improved by including the cumulative variables for maximum temperature, in interaction with the maximum temperatures. The correlation between the observed and estimated mortality ratios was 0.88 in the final model.

Conclusion

Although France is a large country with geographic heterogeneity in both mortality and temperatures, a strong correlation between the daily fluctuations in mortality and the temperatures in summer on a national scale was observed. The model provided a satisfactory quantitative prediction of the daily mortality both for the days with usual temperatures and for the days during intense heat episodes. The results may contribute to enhancing the alert system for intense heat waves.

Fuzzy Clusterwise Growth Curve Models via Generalized Estimating Equations: An Application to the Antisocial Behavior of Children

The growth curve model has been a useful tool for the analysis of repeated measures data. However, it is designed for an aggregate-sample analysis based on the assumption that the entire sample of respondents are from a single homogenous population. Thus, this method may not be suitable when heterogeneous subgroups exist in the population with qualitatively distinct patterns of trajectories. In this paper, the growth curve model is generalized to a fuzzy clustering framework, which explicitly accounts for such group-level heterogeneity in trajectories of change over time. Moreover, the proposed method estimates parameters based on generalized estimating equations thereby relaxing the assumption of correct specification of the population covariance structure among repeated responses. The performance of the proposed method in recovering parameters and the number of clusters is investigated based on two Monte Carlo analyses involving synthetic data. In addition, the empirical usefulness of the proposed method is illustrated by an application concerning the antisocial behavior of a sample of children.

Longitudinal variable selection by cross-validation in the case of many covariates

Longitudinal models are commonly used for studying data collected on individuals repeatedly through time. While there are now a variety of such models available (marginal models, mixed effects models, etc.), far fewer options exist for the closely related issue of variable selection. In addition, longitudinal data typically derive from medical or other large-scale studies where often large numbers of potential explanatory variables and hence even larger numbers of candidate models must be considered. Cross-validation is a popular method for variable selection based on the predictive ability of the model. Here, we propose a cross-validation Markov chain Monte Carlo procedure as a general variable selection tool which avoids the need to visit all candidate models. Inclusion of a 'one-standard error' rule provides users with a collection of good models as is often desired. We demonstrate the effectiveness of our procedure both in a simulation setting and in a real application.

Model selection for incomplete and design-based samples

The Akaike information criterion, AIC, is one of the most frequently used methods to select one or a few good, optimal regression models from a set of candidate models. In case the sample is incomplete, the naive use of this criterion on the so-called complete cases can lead to the selection of poor or inappropriate models. A similar problem occurs when a sample based on a design with unequal selection probabilities, is treated as a simple random sample. In this paper, we consider a modification of AIC, based on reweighing the sample in analogy with the weighted Horvitz-Thompson estimates. It is shown that this weighted AIC-criterion provides better model choices for both incomplete and design-based samples. The use of the weighted AIC-criterion is illustrated on data from the Belgian Health Interview Survey, which motivated this research. Simulations show its performance in a variety of settings.

Modeling adolescent drug-use patterns in cluster-unit trials with multiple sources of correlation using robust latent class regressions

PURPOSE

The purpose of the study is to examine variation in adolescent drug-use patterns by using latent class regression analysis and evaluate the properties of an estimating-equations approach under different cluster-unit trial designs.

METHODS

A set of second-order estimating equations for latent class models under the cluster-unit trial design are proposed. This approach models the correlation within subclusters (drug-use behaviors), but ignores the correlation within clusters (communities). A robust covariance estimator is proposed that accounts for within-cluster correlation. Performance of this approach is addressed through a Monte Carlo simulation study, and practical implications are illustrated by using data from the National Evaluation of the Enforcing Underage Drinking Laws Randomized Community Trial.

RESULTS

The example shows that the proposed method provides useful information about the heterogeneous nature of drug use by identifying two subtypes of adolescent problem drinkers. A Monte Carlo simulation study supports the proposed estimation method by suggesting that the latent class model parameters were unbiased for 30 or more clusters. Consistent with other studies of generalized estimating equation (GEE) estimators, the robust covariance estimator tended to underestimate the true variance of regression parameters, but the degree of inflation in the test size was relatively small for 70 clusters and only slightly inflated for 30 clusters.

CONCLUSIONS

The proposed model for studying adolescent drug use provides an alternative to standard diagnostic criteria, focusing on the nature of the drug-use profile, rather than relying on univariate symptom counts. The second-order GEE-type estimation procedure provided a computationally feasible approach that performed well for a moderate number of clusters and was consistent with prior studies of GEE under the generalized linear model framework.

Proximate and landscape factors influence grassland bird distributions

Ecologists increasingly recognize that birds can respond to features well beyond their normal areas of activity, but little is known about the relative importance of landscapes and proximate factors or about the scales of landscapes that influence bird distributions. We examined the influences of tree cover at both proximate and landscape scales on grassland birds, a group of birds of high conservation concern, in the Sheyenne National Grassland in North Dakota, USA. The Grassland contains a diverse array of grassland and woodland habitats. We surveyed breeding birds on 2015 100 m long transect segments during 2002 and 2003. We modeled the occurrence of 19 species in relation to habitat features (percentages of grassland, woodland, shrubland, and wetland) within each 100-m segment and to tree cover within 200-1600 m of the segment. We used information-theoretic statistical methods to compare models and variables. At the proximate scales, tree cover was the most important variable, having negative influences on 13 species and positive influences on two species. In a comparison of multiple scales, models with only proximate variables were adequate for some species, but models combining proximate with landscape information were best for 17 of 19 species. Landscape-only models were rarely competitive. Combined models at the largest scales (800-1600 m) were best for 12 of 19 species. Seven species had best models including 1600-m landscapes plus proximate factors in at least one year. These were Wilson's Phalarope (Phalaropus tricolor), Sedge Wren (Cistothorus platensis), Field Sparrow (Spizella pusilla), Grasshopper Sparrow (Ammodramus savannarum), Bobolink (Dolychonix oryzivorus), Red-winged Blackbird (Agelaius phoeniceus), and Brown-headed Cowbird (Molothrus ater). These seven are small-bodied species; thus larger-bodied species do not necessarily respond most to the largest landscapes. Our findings suggest that birds respond to habitat features at a variety of scales. Models with only landscape-scale tree cover were rarely competitive, indicating that broad-scale modeling alone, such as that based solely on remotely sensed data, is likely to be inadequate in explaining species distributions.

Long-term avifaunal impoverishment in an isolated tropical woodlot

Long-term (> 50 years) extinction patterns and processes in isolated tropical forest patches are poorly understood. Considering that forest fragments are rapidly becoming the common feature of most tropical landscapes, data on the long-term conservation value of such fragments are urgently needed. We report on avifaunal turnover in a tropical woodlot (Bogor Botanical Gardens; 86 ha; 54% native and 46% introduced plants; mean 83,649 visitors/month) that has been surveyed several times before and after its isolation in 1936. By 2004 the original avifaunal richness of this woodlot declined by 59% (97 to 40 species) and its forest-dependent avifauna declined by 60% (30 to 12 species). Large-bodied birds were particularly prone to extinction before 1987, but following this time none of the species traits we studied could be considered predictive of extinction proneness. All seven forest-dependent bird species that attempted to colonize this woodlot by 1987 perished thereafter. Our results show that area reduction, isolation, intense human use, and perverse management (e.g., understory removal) of this patch have probably negatively affected the long-term sustainability of its forest avifauna.

Enrollment of racial/ethnic minorities in NIAID-funded networks of HIV vaccine trials in the United States, 1988 to 2002

OBJECTIVE

The purpose of this study was to analyze enrollment of racial/ethnic minorities in Phase I and Phase II HIV vaccine trials in the U.S. conducted by National Institute of Allergy and Infectious Diseases (NIAID)-funded networks from 1988 to 2002.

METHODS

A centralized database was searched for all NIAID-funded networks of HIV vaccine trial enrollment data in the U.S. from 1988 through 2002. The authors reviewed data from Phase I or Phase II preventive HIV vaccine trials that included HIV-1 uninfected participants at low to moderate or high risk for HIV infection based on self-reported risk behaviors. Of 66 identified trials, 55 (52 Phase I, 3 Phase II) met selection criteria and were used for analyses. Investigators extracted data on participant demographics using statistical software.

RESULTS

A total of 3,731 volunteers enrolled in U.S. NIAID-funded network HIV vaccine trials from 1988 to 2002. Racial/ethnic minority participants represented 17% of the overall enrollment. By pooling data across all NIAID-funded networks from 1988 to 2002, the proportion of racial/ethnic minority participants was significantly greater (Fisher's exact test p-value < 0.001) in Phase II trials (278/1,061 or 26%) than in Phase I trials (347/2,670 or 13%). By generalized estimating equations, the proportion of minorities in Phase I trials increased over time (p = 0.017), indicating a significant increase in racial/ethnic minority participants from 1988 to 2002.

CONCLUSIONS

There has been a gradual increase in racial/ethnic minority participation in NIAID-funded network HIV vaccine trials in the U.S. since 1988. In the light of recent efficacy trial results, it is essential to continue to increase the enrollment of diverse populations in HIV vaccine research.

Variable Selection for Marginal Longitudinal Generalized Linear Models

Variable selection is an essential part of any statistical analysis and yet has been somewhat neglected in the context of longitudinal data analysis. In this article, we propose a generalized version of Mallows's C(p) (GC(p)) suitable for use with both parametric and nonparametric models. GC(p) provides an estimate of a measure of model's adequacy for prediction. We examine its performance with popular marginal longitudinal models (fitted using GEE) and contrast results with what is typically done in practice: variable selection based on Wald-type or score-type tests. An application to real data further demonstrates the merits of our approach while at the same time emphasizing some important robust features inherent to GC(p).

Effects of Variance-Function Misspecification in Analysis of Longitudinal Data

The approach of generalized estimating equations (GEE) is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within-subject correlations. The variance is often assumed to be a known function of the mean. This article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses. Our numerical studies indicate that (1) correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified; (2) misspecification of the variance function impacts much more on estimators for within-cluster covariates than for cluster-level covariates; and (3) if the variance function is misspecified, correct choice of the correlation structure may not necessarily improve estimation efficiency. We illustrate impacts of different variance functions using a real data set from cow growth.

Spontaneous revegetation of mined peatlands in eastern Canada

Many North American peatlands previously mined for horticultural peat have been abandoned recently, allowing natural recolonization to occur. The two dominant methods for peat extraction, hand block-cutting and vacuum-mining, have created distinctly different abandoned surfaces, leading to different recolonization patterns. Both types of exploitation can be found throughout eastern Canada where we conducted a vast survey of 26 abandoned mined peatlands in the provinces of Québec and New Brunswick. The aim of this study is to describe the revegetation patterns and to assess the impact of local and regional variables as well as the time since abandonment on Sphagnum re-colonization. We inventoried the vegetation structure in all trenches (2571) and baulks (2595) of abandoned block-cut areas as well as in all vacuum fields (395) of the mechanically mined areas. We also conducted detailed species relevés in 242 of these peat fields. In comparison to vacuum-mined peatlands, block-cut peatlands regenerated remarkably well. Approximately 80% of all baulks and trenches in block-cut peatlands had 50% or higher cover of ericaceous shrubs compared with only 16% found on vacuum fields. Herb cover in the three types of abandoned fields was similar to that in natural peatlands. However, Sphagnum percent cover was below 2% in baulks and vacuum fields and was 30% on average in the trenches, which is clearly below cover estimates in natural peatlands. Sphagnum cover and richness were both higher in trenches with thin residual peat deposit, and Sphagnum richness increased with latitude. Our surveys revealed that abandoned mined peatlands have a high diversity of peatland vascular plants species and a low diversity of non-peatland species.Key words: cutover peatlands, regeneration, milled peatlands, block-cut peatlands, vacuum-mined peatlands, colonization patterns.

Around-the-clock: parent work schedules and children's well-being in a 24-h economy

Family life in developed economies has undergone a fundamental change--shifting from single-breadwinner households (typical of the post war decades) to families where both parents are employed. Equally dramatic has been the emergence of around-the-clock economies, altering the way work is organised, especially working time. Many more children now live in households where one or both parents work non-standard hours (evenings, nights or on weekends). Are there any implications for children's well-being when parents work non-standard schedules? There has been virtually no investigation of how children are faring in these around-the-clock households, despite evidence that non-standard work times affect family functioning and are stressful for parents. Using data from a representative sample of 4433 dual-earner Canadian families and their 2--11-year-old children (N children=6361), we compared families where both parents worked standard hours, with families where one or both worked non-standard times (evenings, nights or weekends). In nearly three-quarters of the families one or both parents regularly worked non-standard times. We found associations between children's well-being and parent work schedules, with higher odds ratios for child difficulties when parents worked non-standard times. These associations persisted after adjusting for several confounding factors including socio-economic status, parent part-time or full-time work, and childcare use, and were evident whether mothers, fathers or both parents worked non-standard times. The findings raise questions about the implications for children of the 24-h economy.

Comparison of methods for analyzing longitudinal binary outcomes: cognitive status as an example

Longitudinal data generate correlated observations. Ignoring correlation can lead to incorrect estimation of standard errors, resulting in incorrect inferences of parameters. In the example used here, standard logistic regression, a population-averaged (PA) model fit using generalized estimating equations (GEE), and random-intercept models are used to model binary outcomes at baseline, three and six years later. The outcomes indicate cognitive impairment versus no cognitive impairment in a sample of community dwelling elders. The models include both time-invariant (age, gender) and time-varying (time, interactions with time) covariates. The absolute estimates from random-intercept models are larger than those of both standard logistic and GEE models. Compared to the model fit using GEE that accounts for time dependency, standard logistic regression models overestimate standard errors of time-varying covariates (such as time, and time by problems with activities of daily living), and underestimate the standard errors of time-invariant covariates (such as age and gender). The standard errors from the random-intercept model are larger than those from logistic regression and GEE models. The choice of models, GEE or random-intercept, depends on the research question and the nature of the covariates. Population-averaged methods are appropriate when between-subjects effects are of interest, and random-effects are useful when subject-specific effects are important.

Mapping quantitative trait Loci using generalized estimating equations

A number of statistical methods are now available to map quantitative trait loci (QTL) relative to markers. However, no existing methodology can simultaneously map QTL for multiple nonnormal traits. In this article we rectify this deficiency by developing a QTL-mapping approach based on generalized estimating equations (GEE). Simulation experiments are used to illustrate the application of the GEE-based approach.

Model selection in estimating equations

Model selection is a necessary step in many practical regression analyses. But for methods based on estimating equations, such as the quasi-likelihood and generalized estimating equation (GEE) approaches, there seem to be few well-studied model selection techniques. In this article, we propose a new model selection criterion that minimizes the expected predictive bias (EPB) of estimating equations. A bootstrap smoothed cross-validation (BCV) estimate of EPB is presented and its performance is assessed via simulation for overdispersed generalized linear models. For illustration, the method is applied to a real data set taken from a study of the development of ewe embryos.