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.

Predicting health programme participation: A gravity‐based, hierarchical modelling approach

Statistical analyses of health program participation seek to address a number of objectives compatible with the evaluation of demand for current resources. In this spirit, a spatial hierarchical model is developed for disentangling patterns in participation at the small area level, as a function of population-based demand and additional variation. For the former, a constrained gravity model is proposed to quantify factors associated with spatial choice and account for competition effects, for programs delivered by multiple clinics. The implications of gravity model misspecification within a mixed effects framework are also explored. The proposed model is applied to participation data from a no-fee mammography program in Brisbane, Australia. Attention is paid to the interpretation of various model outputs and their relevance for public health policy.

Steel–concrete–steel sandwich system in Arctic offshore structure: Materials, experiments, and design

The gravity based structure (GBS) with external Steel–Concrete–Steel (SCS) sandwich ice-resistant wall has been developed for the Arctic oil and gas drilling. This paper firstly reported the experimental studies on the mechanical properties of steel and concretes under Arctic low temperature. With the test data, design equations were developed to incorporate the influences of the low temperature on these mechanical properties. Two types of Arctic GBS structure with flower-conical SCS sandwich shell type and plate type of ice-resistant wall have been developed for the Arctic offshore structure. Besides the studies on the materials, two SCS sandwich prototype shells and plates were, respectively, prepared and tested under patch loading that simulated the localized ice-contact pressure. The structural behaviors of the SCS sandwich structure under patch loading were reported and discussions were made on the influences of different parameters on the structural behavior of the structure. Analytical models were developed to predict the punching shear resistances of the SCS sandwich structure through modifying the code provisions. The accuracies of the developed analytical models were checked through validations against 27 tests in the literature. Corresponding design procedures on resistances of SCS sandwich structure were recommended based on these discussions and validations.

Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

Gravity-induced coronal plane joint moments in adolescent idiopathic scoliosis

Background Adolescent Idiopathic Scoliosis is the most common type of spinal deformity, and whilst the risk of progression appears to be biomechanically mediated (larger deformities are more likely to progress), the detailed biomechanical mechanisms driving progression are not well understood. Gravitational forces in the upright position are the primary sustained loads experienced by the spine. In scoliosis they are asymmetrical, generating moments about the spinal joints which may promote asymmetrical growth and deformity progression. Using 3D imaging modalities to estimate segmental torso masses allows the gravitational loading on the scoliotic spine to be determined. The resulting distribution of joint moments aids understanding of the mechanics of scoliosis progression. Methods Existing low-dose CT scans were used to estimate torso segment masses and joint moments for 20 female scoliosis patients. Intervertebral joint moments at each vertebral level were found by summing the moments of each of the torso segment masses above the required joint. Results The patients’ mean age was 15.3 years (SD 2.3; range 11.9 – 22.3 years); mean thoracic major Cobb angle 52° (SD 5.9°; range 42°-63°) and mean weight 57.5 kg (SD 11.5 kg; range 41 – 84.7 kg). Joint moments of up to 7 Nm were estimated at the apical level. No significant correlation was found between the patients’ major Cobb angles and apical joint moments. Conclusions Patients with larger Cobb angles do not necessarily have higher joint moments, and curve shape is an important determinant of joint moment distribution. These findings may help to explain the variations in progression between individual patients. This study suggests that substantial corrective forces are required of either internal instrumentation or orthoses to effectively counter the gravity-induced moments acting to deform the spinal joints of idiopathic scoliosis patients.