The Bumpus house sparrow data: A reanalysis using structural equation models

Summary We analysed the data of H.C. Bumpus on the survival of house sparrows (Passer domesticus) using structural equation modelling techniques. Using data on seven morphological variables measured by Bumpus, we tested and confirmed a three-factor model that characterized physical attributes for general size, leg size and head size. Although males were physically larger than females, we found no difference between males and females in the physical attributes as measured by the three factors. Survival increased significantly with increasing general size and was unrelated to leg size and head size. Wing length, independent of its relationship to the general size factor, was also significantly related to survival. Higher survival was found among birds with short wings. Males had a higher survival compared to females. Their higher survival was mediated, to a lesser extent indirectly, through greater size and, to a greater extent directly, through effects of unknown origin. We favour the use of structural equation modelling methods in studies of selection because of their ability to test and confirm or disconfirm hypotheses related to selection events.     

Generalized spatial structural equation models

It is common in public health research to have high-dimensional, multivariate, spatially referenced data representing summaries of geographic regions. Often, it is desirable to examine relationships among these variables both within and across regions. An existing modeling technique called spatial factor analysis has been used and assumes that a common spatial factor underlies all the variables and causes them to be related to one another. An extension of this technique considers that there may be more than one underlying factor, and that relationships among the underlying latent variables are of primary interest. However, due to the complicated nature of the covariance structure of this type of data, existing methods are not satisfactory. We thus propose a generalized spatial structural equation model. In the first level of the model, we assume that the observed variables are related to particular underlying factors. In the second level of the model, we use the structural equation method to model the relationship among the underlying factors and use parametric spatial distributions on the covariance structure of the underlying factors. We apply the model to county-level cancer mortality and census summary data for Minnesota, including socioeconomic status and access to public utilities.     

Residual-based diagnostics for structural equation models

Summary .  Classical diagnostics for structural equation models are based on aggregate forms of the data and are ill suited for checking distributional or linearity assumptions. We extend recently developed goodness-of-fit tests for correlated data based on subject-specific residuals to structural equation models with latent variables. The proposed tests lend themselves to graphical displays and are designed to detect misspecified distributional or linearity assumptions. To complement graphical displays, test statistics are defined; the null distributions of the test statistics are approximated using computationally efficient simulation techniques. The properties of the proposed tests are examined via simulation studies. We illustrate the methods using data from a study of in utero lead exposure.     

Optimal design of longitudinal data analysis using generalized estimating equation models

Longitudinal studies are often applied in biomedical research and clinical trials to evaluate the treatment effect. The association pattern within the subject must be considered in both sample size calculation and the analysis. One of the most important approaches to analyze such a study is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which “working correlation structure” is introduced and the association pattern within the subject depends on a vector of association parameters denoted by ρ. The explicit sample size formulas for two‐group comparison in linear and logistic regression models are obtained based on the GEE method by Liu and Liang. For cluster randomized trials (CRTs), researchers proposed the optimal sample sizes at both the cluster and individual level as a function of sampling costs and the intracluster correlation coefficient (ICC). In these approaches, the optimal sample sizes depend strongly on the ICC. However, the ICC is usually unknown for CRTs and multicenter trials. To overcome this shortcoming, Van Breukelen et al. consider a range of possible ICC values identified from literature reviews and present Maximin designs (MMDs) based on relative efficiency (RE) and efficiency under budget and cost constraints. In this paper, the optimal sample size and number of repeated measurements using GEE models with an exchangeable working correlation matrix is proposed under the considerations of fixed budget, where “optimal” refers to maximum power for a given sampling budget. The equations of sample size and number of repeated measurements for a known parameter value ρ are derived and a straightforward algorithm for unknown ρ is developed. Applications in practice are discussed. We also discuss the existence of the optimal design when an AR(1) working correlation matrix is assumed. Our proposed method can be extended under the scenarios when the true and working correlation matrix are different.     

Bayesian lasso for semiparametric structural equation models

There has been great interest in developing nonlinear structural equation models and associated statistical inference procedures, including estimation and model selection methods. In this paper a general semiparametric structural equation model (SSEM) is developed in which the structural equation is composed of nonparametric functions of exogenous latent variables and fixed covariates on a set of latent endogenous variables. A basis representation is used to approximate these nonparametric functions in the structural equation and the Bayesian Lasso method coupled with a Markov Chain Monte Carlo (MCMC) algorithm is used for simultaneous estimation and model selection. The proposed method is illustrated using a simulation study and data from the Affective Dynamics and Individual Differences (ADID) study. Results demonstrate that our method can accurately estimate the unknown parameters and correctly identify the true underlying model.     

Analysis of twin data using SAS

Summary .  Twin studies are essential for assessing disease inheritance. Data generated from twin studies are traditionally analyzed using specialized computational programs. For many researchers, especially those who are new to twin studies, understanding and using those specialized computational programs can be a daunting task. Given that SAS (Statistical Analysis Software) is the most popular software for statistical analysis, we suggest that the use of SAS procedures for twin data may be a helpful alternative and demonstrate that we can obtain similar results from SAS to those produced by specialized computational programs. This numerical validation is practically useful, because a natural concern with general statistical software is whether it can deal with data that are generated from special study designs such as twin studies and if it can test a particular hypothesis. We concluded through our extensive simulation that SAS procedures can be used easily as a very convenient alternative to specialized programs for twin data analysis.     

A semiparametric Bayesian approach for structural equation models

In the development of structural equation models (SEMs), observed variables are usually assumed to be normally distributed. However, this assumption is likely to be violated in many practical researches. As the non‐normality of observed variables in an SEM can be obtained from either non‐normal latent variables or non‐normal residuals or both, semiparametric modeling with unknown distribution of latent variables or unknown distribution of residuals is needed. In this article, we find that an SEM becomes nonidentifiable when both the latent variable distribution and the residual distribution are unknown. Hence, it is impossible to estimate reliably both the latent variable distribution and the residual distribution without parametric assumptions on one or the other. We also find that the residuals in the measurement equation are more sensitive to the normality assumption than the latent variables, and the negative impact on the estimation of parameters and distributions due to the non‐normality of residuals is more serious. Therefore, when there is no prior knowledge about parametric distributions for either the latent variables or the residuals, we recommend making parametric assumption on latent variables, and modeling residuals nonparametrically. We propose a semiparametric Bayesian approach using the truncated Dirichlet process with a stick breaking prior to tackle the non‐normality of residuals in the measurement equation. Simulation studies and a real data analysis demonstrate our findings, and reveal the empirical performance of the proposed methodology. A free WinBUGS code to perform the analysis is available in Supporting Information.     

Genomic approaches to identification of tumour-susceptibility genes using mouse models

Individual susceptibility to cancer in humans is determined by complex interactions between germline genetic variation and levels of exposure to environmental carcinogens or tumour promoters. Only a small fraction of cancer susceptibility is inherited in a Mendelian manner (high-penetrance familial cancer), and most tumours result from the combined effects of many gene-gene and gene-environment interactions. The sequencing of the mouse genome provides new approaches to one of the most challenging tasks of cancer genetics today.     

Estimation of the benchmark dose by structural equation models

While epidemiological data typically contain a multivariate response and often also multiple exposure parameters, current methods for safe dose calculations, including the widely used benchmark approach, rely on standard regression techniques. In practice, dose-response modeling and calculation of the exposure limit are often based on the seemingly most sensitive outcome. However, this procedure ignores other available data, is inefficient, and fails to account for multiple testing. Instead, risk assessment could be based on structural equation models, which can accommodate both a multivariate exposure and a multivariate response function. Furthermore, such models will allow for measurement error in the observed variables, which is a requirement for unbiased estimation of the benchmark dose. This methodology is illustrated with the data on neurobehavioral effects in children prenatally exposed to methylmercury, where results based on standard regression models cause an underestimation of the true risk.     

A framework for structural equation models in general pedigrees

Background/Aims: Structural Equation Modeling (SEM) is an analysis approach that accounts for both the causal relationships between variables and the errors associated with the measurement of these variables. In this paper, a framework for implementing structural equation models (SEMs) in family data is proposed. Methods: This framework includes both a latent measurement model and a structural model with covariates. It allows for a wide variety of models, including latent growth curve models. Environmental, polygenic and other genetic variance components can be included in the SEM. Kronecker notation makes it easy to separate the SEM process from a familial correlation model. A limited information method of model fitting is discussed. We show how missing data and ascertainment may be handled. We give several examples of how the framework may be used. Results: A simulation study shows that our method is computationally feasible, and has good statistical properties. Conclusion: Our framework may be used to build and compare causal models using family data without any genetic marker data. It also allows for a nearly endless array of genetic association and/or linkage tests. A preliminary Matlab program is available, and we are currently implementing a more complete and user-friendly R package.     

Using bivariate models to understand between- and within-cluster regression coefficients, with application to twin data

In the regression analysis of clustered data it is important to allow for the possibility of distinct between- and within-cluster exposure effects on the outcome measure, represented, respectively, by regression coefficients for the cluster mean and the deviation of the individual-level exposure value from this mean. In twin data, the within-pair regression effect represents association conditional on exposures shared within pairs, including any common genetic or environmental influences on the outcome measure. It has therefore been proposed that a comparison of the within-pair regression effects between monozygous (MZ) and dizygous (DZ) twins can be used to examine whether the association between exposure and outcome has a genetic origin. We address this issue by proposing a bivariate model for exposure and outcome measurements in twin-pair data. The between- and within-pair regression coefficients are shown to be weighted averages of ratios of the exposure and outcome variances and covariances, from which it is straightforward to determine the conditions under which the within-pair regression effect in MZ pairs will be different from that in DZ pairs. In particular, we show that a correlation structure in twin pairs for exposure and outcome that appears to be due to genetic factors will not necessarily be reflected in distinct MZ and DZ values for the within-pair regression coefficients. We illustrate these results in a study of female twin pairs from Australia and North America relating mammographic breast density to weight and body mass index.     

Inferences about population dynamics from count data using multistate models: a comparison to capture–recapture approaches

Wildlife populations consist of individuals that contribute disproportionately to growth and viability. Understanding a population's spatial and temporal dynamics requires estimates of abundance and demographic rates that account for this heterogeneity. Estimating these quantities can be difficult, requiring years of intensive data collection. Often, this is accomplished through the capture and recapture of individual animals, which is generally only feasible at a limited number of locations. In contrast, N‐mixture models allow for the estimation of abundance, and spatial variation in abundance, from count data alone. We extend recently developed multistate, open population N‐mixture models, which can additionally estimate demographic rates based on an organism's life history characteristics. In our extension, we develop an approach to account for the case where not all individuals can be assigned to a state during sampling. Using only state‐specific count data, we show how our model can be used to estimate local population abundance, as well as density‐dependent recruitment rates and state‐specific survival. We apply our model to a population of black‐throated blue warblers (Setophaga caerulescens) that have been surveyed for 25 years on their breeding grounds at the Hubbard Brook Experimental Forest in New Hampshire, USA. The intensive data collection efforts allow us to compare our estimates to estimates derived from capture–recapture data. Our model performed well in estimating population abundance and density‐dependent rates of annual recruitment/immigration. Estimates of local carrying capacity and per capita recruitment of yearlings were consistent with those published in other studies. However, our model moderately underestimated annual survival probability of yearling and adult females and severely underestimates survival probabilities for both of these male stages. The most accurate and precise estimates will necessarily require some amount of intensive data collection efforts (such as capture–recapture). Integrated population models that combine data from both intensive and extensive sources are likely to be the most efficient approach for estimating demographic rates at large spatial and temporal scales.     

Statistical inference in mixed models and analysis of twin and family data

Analysis of data from twin and family studies provides the foundation for studies of disease inheritance. The development of advanced theory and computational software for general linear models has generated considerable interest for using mixed-effect models to analyze twin and family data, as a computationally more convenient and theoretically more sound alternative to the classical structure equation modeling. Despite the long history of twin and family data analysis, some fundamental questions remain unanswered. We addressed two important issues. One is to determine the necessary and sufficient conditions for the identifiability in the mixed-effects models for twin and family data. The other is to derive the asymptotic distribution of the likelihood ratio test, which is novel due to the fact that the standard regularity conditions are not satisfied. We considered a series of specific yet important examples in which we demonstrated how to formulate mixed-effect models to appropriately reflect the data, and our key idea is the use of the Cholesky decomposition. Finally, we applied our method and theory to provide a more precise estimate of the heritability of two data sets than the previously reported estimate.