Generalized linear latent variable models for repeated measures of spatially correlated multivariate data

Observations of multiple-response variables across space and over time occur often in environmental and ecological studies. Compared to purely spatial models for a single response variable in the exponential family of distributions, fewer statistical tools are available for multiple-response variables that are not necessarily Gaussian. An exception is a common-factor model developed for multivariate spatial data by Wang and Wall (2003, Biostatistics 4, 569-582). The purpose of this article is to extend this multivariate space-only model and develop a flexible class of generalized linear latent variable models for multivariate spatial-temporal data. For statistical inference, maximum likelihood estimates and their standard deviations are obtained using a Monte Carlo EM algorithm. We also use a novel way to automatically adjust the Monte Carlo sample size, which facilitates the convergence of the Monte Carlo EM algorithm. The methodology is illustrated by an ecological study of red pine trees in response to bark beetle challenges in a forest stand of Wisconsin.     

A test for a threshold in an ordered sequence of correlated proportions

A test procedure using chi-square statistics is proposed for determining a threshold in an ordered sequence of correlated proportions. The procedure is based on the multivariate Bernoulli model. It is applied to the problem of ascertaining when visual acuity has stabilized in a group of patients with regular follow-up after a vision-reducing acute abnormality.     

Estimating linear temporal trends from aggregated environmental monitoring data

Trend estimates are often used as part of environmental monitoring programs. These trends inform managers (e.g., are desired species increasing or undesired species decreasing?). Data collected from environmental monitoring programs is often aggregated (i.e., averaged), which confounds sampling and process variation. State-space models allow sampling variation and process variations to be separated. We used simulated time-series to compare linear trend estimations from three state-space models, a simple linear regression model, and an auto-regressive model. We also compared the performance of these five models to estimate trends from a long term monitoring program. We specifically estimated trends for two species of fish and four species of aquatic vegetation from the Upper Mississippi River system. We found that the simple linear regression had the best performance of all the given models because it was best able to recover parameters and had consistent numerical convergence. Conversely, the simple linear regression did the worst job estimating populations in a given year. The state-space models did not estimate trends well, but estimated population sizes best when the models converged. We found that a simple linear regression performed better than more complex autoregression and state-space models when used to analyze aggregated environmental monitoring data.     

Testing random effects in linear mixed‐effects models with serially correlated errors

In linear mixed‐effects models, random effects are used to capture the heterogeneity and variability between individuals due to unmeasured covariates or unknown biological differences. Testing for the need of random effects is a nonstandard problem because it requires testing on the boundary of parameter space where the asymptotic chi‐squared distribution of the classical tests such as likelihood ratio and score tests is incorrect. In the literature several tests have been proposed to overcome this difficulty, however all of these tests rely on the restrictive assumption of i.i.d. measurement errors. The presence of correlated errors, which often happens in practice, makes testing random effects much more difficult. In this paper, we propose a permutation test for random effects in the presence of serially correlated errors. The proposed test not only avoids issues with the boundary of parameter space, but also can be used for testing multiple random effects and any subset of them. Our permutation procedure includes the permutation procedure in Drikvandi, Verbeke, Khodadadi, and Partovi Nia (2013) as a special case when errors are i.i.d., though the test statistics are different. We use simulations and a real data analysis to evaluate the performance of the proposed permutation test. We have found that random slopes for linear and quadratic time effects may not be significant when measurement errors are serially correlated.     

Testing against umbrella or tree orderings for binomial proportions with an adaptation of an insect resistance case

Alternative hypotheses for order restrictions, such as umbrella or inverse umbrella (a.k.a tree) orderings, have been studied extensively in the literature, although less so when the studied response for each individual is the presence or absence of the event of interest. Two families of test statistics for solving the problem of testing against an umbrella or a tree ordering when the responses are binomial proportions are studied in this work and their asymptotic distributions are derived. A simulation study is conducted to compare the empirical power of some members of the derived families of test statistics with competing approaches. The methodology developed here was driven by an applied problem arising in stored products research where despite universal mortality in the case of doses of 1000 ppm of the insecticide phosphine, unexpected survival was noted at higher doses.     

Nonparametric regression using local kernel estimating equations for correlated failure time data

We study nonparametric regression for correlated failure timedata. Kernel estimating equations are used to estimate nonparametriccovariate effects. Independent and weighted-kernel estimatingequations are studied. The derivative of the nonparametric functionis first estimated and the nonparametric function is then estimatedby integrating the derivative estimator. We show that the nonparametrickernel estimator is consistent for any arbitrary working correlationmatrix and that its asymptotic variance is minimized by assumingworking independence. We evaluate the performance of the proposedkernel estimator using simulation studies, and apply the proposedmethod to the western Kenya parasitaemia data.     

Joint regression analysis of correlated data using Gaussian copulas

Summary .  This article concerns a new joint modeling approach for correlated data analysis. Utilizing Gaussian copulas, we present a unified and flexible machinery to integrate separate one-dimensional generalized linear models (GLMs) into a joint regression analysis of continuous, discrete, and mixed correlated outcomes. This essentially leads to a multivariate analogue of the univariate GLM theory and hence an efficiency gain in the estimation of regression coefficients. The availability of joint probability models enables us to develop a full maximum likelihood inference. Numerical illustrations are focused on regression models for discrete correlated data, including multidimensional logistic regression models and a joint model for mixed normal and binary outcomes. In the simulation studies, the proposed copula-based joint model is compared to the popular generalized estimating equations, which is a moment-based estimating equation method to join univariate GLMs. Two real-world data examples are used in the illustration.     

A simple approximation for calculating sample sizes for detecting linear trend in proportions

A simple approximate formula for sample sizes for detecting a linear trend in proportions is derived. The formulas for both the uncorrected and corrected Cochran-Armitage test are given. For two binomial proportions these reduce to those given by Casagrande, Pike, and Smith (1978, Biometrics 34, 483-486). Some numerical results of a power study for small sample sizes show that the nominal power corresponding to the approximate sample size is a reasonably good approximation to the actual power.     

Analyzing temporally correlated dolphin sightings data using generalized estimating equations

Many of the statistical techniques commonly used in ecology assume independence among responses. However, there are many marine mammal survey techniques, such as those involving time series or subgroups, which result in correlations within the data. Generalized estimating equations (GEEs) take such correlations into account and are an extension of generalized linear models. This study demonstrates the application of GEEs by modeling temporal variation in bottlenose dolphin presence from sightings data. Since dolphins could remain in the study area for several hours resulting in temporal autocorrelation, an autoregressive correlation structure was used within the GEE, each cluster representing hours within a day of survey effort. The results of the GEE model showed that there was significant diel, tidal, and interannual variation in the presence of dolphins. Dolphins were most likely to be seen in the early morning and during the summer months. Dolphin presence generally peaked during low tide, but this varied among years. There was a significantly lower probability of dolphins being present in 2003 than 2004, but not between 2004 and the other years (1991, 1992, and 2002). GEE‐model fitting packages are now readily available, making this a valuable, versatile tool for marine mammal biologists.