Working covariance model selection for generalized estimating equations

We investigate methods for data-based selection of working covariance models in the analysis of correlated data with generalized estimating equations. We study two selection criteria: Gaussian pseudolikelihood and a geodesic distance based on discrepancy between model-sensitive and model-robust regression parameter covariance estimators. The Gaussian pseudolikelihood is found in simulation to be reasonably sensitive for several response distributions and noncanonical mean-variance relations for longitudinal data. Application is also made to a clinical dataset. Assessment of adequacy of both correlation and variance models for longitudinal data should be routine in applications, and we describe open-source software supporting this practice.

Discussion of “Generalized estimating equations: Notes on the choice of the working correlation matrix”

Objective To discuss generalized estimating equations as an extension of generalized linear models by commenting on the paper of Ziegler and Vens "Generalized Estimating Equations. Notes on the Choice of the Working Correlation Matrix". Methods Inviting an international group of experts to comment on this paper. Results Several perspectives have been taken by the discussants. Econometricians have established parallels to the generalized method of moments (GMM). Statisticians discussed model assumptions and the aspect of missing data Applied statisticians; commented on practical aspects in data analysis. Conclusions In general, careful modeling correlation is encouraged when considering estimation efficiency and other implications, and a comparison of choosing instruments in GMM and generalized estimating equations, (GEE) would be worthwhile. Some theoretical drawbacks of GEE need to be further addressed and require careful analysis of data This particularly applies to the situation when data are missing at random.

Working-correlation-structure identification in generalized estimating equations

Selecting an appropriate working correlation structure is pertinent to clustered data analysis using generalized estimating equations (GEE) because an inappropriate choice will lead to inefficient parameter estimation. We investigate the well-known criterion of QIC for selecting a working correlation Structure. and have found that performance of the QIC is deteriorated by a term that is theoretically independent of the correlation structures but has to be estimated with an error. This leads LIS to propose a correlation information criterion (CIC) that substantially improves the QIC performance. Extensive simulation studies indicate that the CIC has remarkable improvement in selecting the correct correlation structures. We also illustrate our findings using a data set from the Madras Longitudinal Schizophrenia Study.

Iterative estimating equations: Linear convergence and asymptotic properties

We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size increases to infinity. Furthermore, we show that the limiting estimator is consistent and asymptotically efficient, as expected. The method applies to semiparametric regression models with unspecified covariances among the observations. In the special case of linear models, the procedure reduces to iterative reweighted least squares. Finite sample performance of the procedure is studied by simulations, and compared with other methods. A numerical example from a medical study is considered to illustrate the application of the method.

Unbiased estimating equations from working correlation models for irregularly timed repeated measures

The method of generalized estimating equation-, (GEEs) has been criticized recently for a failure to protect against misspecification of working correlation models, which in some cases leads to loss of efficiency or infeasibility of solutions. However, the feasibility and efficiency of GEE methods can be enhanced considerably by using flexible families of working correlation models. We propose two ways of constructing unbiased estimating equations from general correlation models for irregularly timed repeated measures to supplement and enhance GEE. The supplementary estimating equations are obtained by differentiation of the Cholesky decomposition of the working correlation, or as score equations for decoupled Gaussian pseudolikelihood. The estimating equations are solved with computational effort equivalent to that required for a first-order GEE. Full details and analytic expressions are developed for a generalized Markovian model that was evaluated through simulation. Large-sample ".sandwich" standard errors for working correlation parameter estimates are derived and shown to have good performance. The proposed estimating functions are further illustrated in an analysis of repeated measures of pulmonary function in children.

Analysing commercial catch and effort data from a penaeid trawl fishery: A comparison of linear models, mixed models, and generalised estimating equations approaches

Statistical methods are often used to analyse commercial catch and effort data to provide standardised fishing effort and/or a relative index of fish abundance for input into stock assessment models. Achieving reliable results has proved difficult in Australia's Northern Prawn Fishery (NPF), due to a combination of such factors as the biological characteristics of the animals, some aspects of the fleet dynamics, and the changes in fishing technology. For this set of data, we compared four modelling approaches (linear models, mixed models, generalised estimating equations, and generalised linear models) with respect to the outcomes of the standardised fishing effort or the relative index of abundance. We also varied the number and form of vessel covariates in the models. Within a subset of data from this fishery, modelling correlation structures did not alter the conclusions from simpler statistical models. The random-effects models also yielded similar results. This is because the estimators are all consistent even if the correlation structure is mis-specified, and the data set is very large. However, the standard errors from different models differed, suggesting that different methods have different statistical efficiency. We suggest that there is value in modelling the variance function and the correlation structure, to make valid and efficient statistical inferences and gain insight into the data. We found that fishing power was separable from the indices of prawn abundance only when we offset the impact of vessel characteristics at assumed values from external sources. This may be due to the large degree of confounding within the data, and the extreme temporal changes in certain aspects of individual vessels, the fleet and the fleet dynamics.

Working correlation structure misspecification, estimation and covariate design: Implications for generalised estimating equations performance

The method of generalised estimating equations for regression modelling of clustered outcomes allows for specification of a working matrix that is intended to approximate the true correlation matrix of the observations. We investigate the asymptotic relative efficiency of the generalised estimating equation for the mean parameters when the correlation parameters are estimated by various methods. The asymptotic relative efficiency depends on three-features of the analysis, namely (i) the discrepancy between the working correlation structure and the unobservable true correlation structure, (ii) the method by which the correlation parameters are estimated and (iii) the 'design', by which we refer to both the structures of the predictor matrices within clusters and distribution of cluster sizes. Analytical and numerical studies of realistic data-analysis scenarios show that choice of working covariance model has a substantial impact on regression estimator efficiency. Protection against avoidable loss of efficiency associated with covariance misspecification is obtained when a 'Gaussian estimation' pseudolikelihood procedure is used with an AR(1) structure.

A generalized estimating equations approach for analysis of the impact of new technology on a trawl fishery

The article describes a generalized estimating equations approach that was used to investigate the impact of technology on vessel performance in a trawl fishery during 1988-96, while accounting for spatial and temporal correlations in the catch-effort data. Robust estimation of parameters in the presence of several levels of clustering depended more on the choice of cluster definition than on the choice of correlation structure within the cluster. Models with smaller cluster sizes produced stable results, while models with larger cluster sizes, that may have had complex within-cluster correlation structures and that had within-cluster covariates, produced estimates sensitive to the correlation structure. The preferred model arising from this dataset assumed that catches from a vessel were correlated in the same years and the same areas, but independent in different years and areas. The model that assumed catches from a vessel were correlated in all years and areas, equivalent to a random effects term for vessel, produced spurious results. This was an unexpected finding that highlighted the need to adopt a systematic strategy for modelling. The article proposes a modelling strategy of selecting the best cluster definition first, and the working correlation structure (within clusters) second. The article discusses the selection and interpretation of the model in the light of background knowledge of the data and utility of the model, and the potential for this modelling approach to apply in similar statistical situations.

Estimating equations with nonignorably missing response data

Troxel, Lipsitz, and Brennan (1997, Biometrics 53, 857-869) considered parameter estimation from survey data with nonignorable nonresponse and proposed weighted estimating equations to remove the biases in the complete-case analysis that ignores missing observations. This paper suggests two alternative modifications for unbiased estimation of regression parameters when a binary outcome is potentially observed at successive time points. The weighting approach of Robins, Rotnitzky, and Zhao (1995, Journal of the American Statistical Association 90, 106-121) is also modified to obtain unbiased estimating functions. The suggested estimating functions are unbiased only when the missingness probability is correctly specified, and misspecification of the missingness model will result in biases in the estimates. Simulation studies are carried out to assess the performance of different methods when the covariate is binary or normal. For the simulation models used, the relative efficiency of the two new methods to the weighting methods is about 3.0 for the slope parameter and about 2.0 for the intercept parameter when the covariate is continuous and the missingness probability is correctly specified. All methods produce substantial biases in the estimates when the missingness model is misspecified or underspecified. Analysis of data from a medical survey illustrates the use and possible differences of these estimating functions.

Estimating equations for parameters in stochastic growth models from tag-recapture data

James (1991, Biometrics 47, 1519-1530) constructed unbiased estimating functions for estimating the two parameters in the von Bertalanffy growth curve from tag-recapture data. This paper provides unbiased estimating functions for a class of growth models that incorporate stochastic components and explanatory variables. a simulation study using seasonal growth models indicates that the proposed method works well while the least-squares methods that are commonly used in the literature may produce substantially biased estimates. The proposed model and method are also applied to real data from tagged rack lobsters to assess the possible seasonal effect on growth.

Estimating equations for removal data analysis

We consider the problem of estimating a population size from successive catches taken during a removal experiment and propose two estimating functions approaches, the traditional quasi-likelihood (TQL) approach for dependent observations and the conditional quasi-likelihood (CQL) approach using the conditional mean and conditional variance of the catch given previous catches. Asymptotic covariance of the estimates and the relationship between the two methods are derived. Simulation results and application to the catch data from smallmouth bass show that the proposed estimating functions perform better than other existing methods, especially in the presence of overdispersion.

Estimation of growth parameters from multiple-recapture data

This article develops a method for analysis of growth data with multiple recaptures when the initial ages for all individuals are unknown. The existing approaches either impute the initial ages or model them as random effects. Assumptions about the initial age are not verifiable because all the initial ages are unknown. We present an alternative approach that treats all the lengths including the length at first capture as correlated repeated measures for each individual. Optimal estimating equations are developed using the generalized estimating equations approach that only requires the first two moment assumptions. Explicit expressions for estimation of both mean growth parameters and variance components are given to minimize the computational complexity. Simulation studies indicate that the proposed method works well. Two real data sets are analyzed for illustration, one from whelks (Dicathais aegaota) and the other from southern rock lobster (Jasus edwardsii) in South Australia.

Application of an appearance-based intervention to improve sun protection outcomes of outdoor workers in Queensland, Australia

Outdoor workers are exposed to high levels of ultraviolet radiation (UVR) and may thus be at greater risk to experience UVR-related health effects such as skin cancer, sun burn, and cataracts. A number of intervention trials (n=14) have aimed to improve outdoor workers’ work-related sun protection cognitions and behaviours. Only one study however has reported the use of UV-photography as part of a multi-component intervention. This study was performed in the USA and showed long-term (12 months) improvements in work-related sun protection behaviours. Intervention effects of the other studies have varied greatly, depending on the population studied, intervention applied, and measurement of effect. Previous studies have not assessed whether: - Interventions are similarly effective for workers in stringent and less stringent policy organisations; - Policy effect is translated into workers’ leisure time protection; - Implemented interventions are effective in the long-term; - The facial UV-photograph technique is effective in Australian male outdoor workers without a large additional intervention package, and; - Such interventions will also affect workers’ leisure time sun-related cognitions and behaviours. Therefore, the present Protection of Outdoor Workers from Environmental Radiation [POWER]-study aimed to fill these gaps and had the objectives of: a) assessing outdoor workers’ sun-related cognitions and behaviours at work and during leisure time in stringent and less stringent sun protection policy environments; b) assessing the effect of an appearance-based intervention on workers’ risk perceptions, intentions and behaviours over time; c) assessing whether the intervention was equally effective within the two policy settings; and d) assessing the immediate post-intervention effect. Effectiveness was described in terms of changes in sun-related risk perceptions and intentions (as these factors were shown to be main precursors of behaviour change in many health promotion theories) and behaviour. The study purposefully selected and recruited two organisations with a large outdoor worker contingent in Queensland, Australia within a 40 kilometre radius of Brisbane. The two organisations differed in the stringency of implementation and reinforcement of their organisational sun protection policy. Data were collected from 154 male predominantly Australian born outdoor workers with an average age of 37 years and predominantly medium to fair skin (83%). Sun-related cognitions and behaviours of workers were assessed using self-report questionnaires at baseline and six to twelve months later. Variation in follow-up time was due to a time difference in the recruitment of the two organisations. Participants within each organisation were assigned to an intervention or control group. The intervention group participants received a one-off personalised Skin Cancer Risk Assessment Tool [SCRAT]-letter and a facial UV-photograph with detailed verbal information. This was followed by an immediate post-intervention questionnaire within three months of the start of the study. The control group only received the baseline and follow-up questionnaire. Data were analysed using a variety of techniques including: descriptive analyses, parametric and non-parametric tests, and generalised estimating equations. A 15% proportional difference observed was deemed of clinical significance, with the addition of reported statistical significance (p<0.05) where applicable. Objective 1: Assess and compare the current sun-related risk perceptions, intentions, behaviours, and policy awareness of outdoor workers in stringent and less stringent sun protection policy settings. Workers within the two organisations (stringent n=89 and less stringent n=65) were similar in their knowledge about skin cancer, self efficacy, attitudes, and social norms regarding sun protection at work and during leisure time. Participants were predominantly in favour of sun protection. Results highlighted that compared to workers in a less stringent policy organisation working for an organisation with stringent sun protection policies and practices resulted in more desirable sun protection intentions (less willing to tan p=0.03) ; actual behaviours at work (sufficient use of upper and lower body protection, headgear, and sunglasses (p<0.001 for all comparisons), and greater policy awareness (awareness of repercussions if Personal Protective Equipment (PPE) was not used, p<0.001)). However the effect of the work-related sun protection policy was found not to extend to leisure time sun protection. Objective 2: Compare changes in sun-related risk perceptions, intentions, and behaviours between the intervention and control group. The effect of the intervention was minimal and mainly resulted in a clinically significant reduction in work-related self-perceived risk of developing skin cancer in the intervention compared to the control group (16% and 32% for intervention and control group, respectively estimated their risk higher compared to other outdoor workers: , p=0.11). No other clinical significant effects were observed at 12 months follow-up. Objective 3: Assess whether the intervention was equally effective in the stringent sun protection policy organisation and the less stringent sun protection policy organisation. The appearance-based intervention resulted in a clinically significant improvement in the stringent policy intervention group participants’ intention to protect from the sun at work (workplace*time interaction, p=0.01). In addition to a reduction in their willingness to tan both at work (will tan at baseline: 17% and 61%, p=0.06, at follow-up: 54% and 33%, p=0.07, stringent and less stringent policy intervention group respectively. The workplace*time interaction was significant p<0.001) and during leisure time (will tan at baseline: 42% and 78%, p=0.01, at follow-up: 50% and 63%, p=0.43, stringent and less stringent policy intervention group respectively. The workplace*time interaction was significant p=0.01) over the course of the study compared to the less stringent policy intervention group. However, no changes in actual sun protection behaviours were found. Objective 4: Examine the effect of the intervention on level of alarm and concern regarding the health of the skin as well as sun protection behaviours in both organisations. The immediate post-intervention results showed that the stringent policy organisation participants indicated to be less alarmed (p=0.04) and concerned (p<0.01) about the health of their skin and less likely to show the facial UV-photograph to others (family p=0.03) compared to the less stringent policy participants. A clinically significantly larger proportion of participants from the stringent policy organisation reported they worried more about skin cancer (65%) and skin freckling (43%) compared to those in the less stringent policy organisation (46%,and 23% respectively , after seeing the UV-photograph). In summary the results of this study suggest that the having a stringent work-related sun protection policy was significantly related to for work-time sun protection practices, but did not extend to leisure time sun protection. This could reflect the insufficient level of sun protection found in the general Australian population during leisure time. Alternatively, reactance caused by being restricted in personal decisions through work-time policy could have contributed to lower leisure time sun protection. Finally, other factors could have also contributed to the less than optimal leisure time sun protection behaviours reported, such as unmeasured personal or cultural barriers. All these factors combined may have lead to reduced willingness to take proper preventive action during leisure time exposure. The intervention did not result in any measurable difference between the intervention and control groups in sun protection behaviours in this population, potentially due to the long lag time between the implementation of the intervention and assessment at 12-months follow-up. In addition, high levels of sun protection behaviours were found at baseline (ceiling effect) which left little room for improvement. Further, this study did not assess sunscreen use, which was the predominant behaviour assessed in previous effective appearance-based interventions trials. Additionally, previous trials were mainly conducted in female populations, whilst the POWER-study’s population was all male. The observed immediate post-intervention result could be due to more emphasis being placed on sun protection and risks related to sun exposure in the stringent policy organisation. Therefore participants from the stringent policy organisation could have been more aware of harmful effects of UVR and hence, by knowing that they usually protect adequately, not be as alarmed or concerned as the participants from the less stringent policy organisation. In conclusion, a facial UV-photograph and SCRAT-letter information alone may not achieve large changes in sun-related cognitions and behaviour, especially of assessed 6-12 months after the intervention was implemented and in workers who are already quite well protected. Differences found between workers in the present study appear to be more attributable to organisational policy. However, against a background of organisational policy, this intervention may be a useful addition to sun-related workplace health and safety programs. The study findings have been interpreted while respecting a number of limitations. These have included non-random allocation of participants due to pre-organised allocation of participants to study group in one organisation and difficulty in separating participants from either study group. Due to the transient nature of the outdoor worker population, only 105 of 154 workers available at baseline could be reached for follow-up. (attrition rate=32%). In addition the discrepancy in the time to follow-up assessment between the two organisations was a limitation of the current study. Given the caveats of this research, the following recommendations were made for future research: - Consensus should be reached to define "outdoor worker" in terms of time spent outside at work as well as in the way sun protection behaviours are measured and reported. - Future studies should implement and assess the value of the facial UV-photographs in a wide range of outdoor worker organisations and countries. - More timely and frequent follow-up assessments should be implemented in intervention studies to determine the intervention effect and to identify the best timing of booster sessions to optimise results. - Future research should continue to aim to target outdoor workers’ leisure time cognitions and behaviours and improve these if possible. Overall, policy appears to be an important factor in workers’ compliance with work-time use of sun protection. Given the evidence generated by this research, organisations employing outdoor workers should consider stringent implementation and reinforcement of a sun protection policy. Finally, more research is needed to improve ways to generate desirable behaviour in this population during leisure time.

Model selection with misspecified spatial covariance structure

Spatial data analysis has become more and more important in the studies of ecology and economics during the last decade. One focus of spatial data analysis is how to select predictors, variance functions and correlation functions. However, in general, the true covariance function is unknown and the working covariance structure is often misspecified. In this paper, our target is to find a good strategy to identify the best model from the candidate set using model selection criteria. This paper is to evaluate the ability of some information criteria (corrected Akaike information criterion, Bayesian information criterion (BIC) and residual information criterion (RIC)) for choosing the optimal model when the working correlation function, the working variance function and the working mean function are correct or misspecified. Simulations are carried out for small to moderate sample sizes. Four candidate covariance functions (exponential, Gaussian, Matern and rational quadratic) are used in simulation studies. With the summary in simulation results, we find that the misspecified working correlation structure can still capture some spatial correlation information in model fitting. When the sample size is large enough, BIC and RIC perform well even if the the working covariance is misspecified. Moreover, the performance of these information criteria is related to the average level of model fitting which can be indicated by the average adjusted R square ( [GRAPHICS] ), and overall RIC performs well.