Estimation of regression models under linear restrictions

The variance-covariance matrices of restricted regression and mixed regression estimators are compared and the consequences of introducing variability in the restrictions are examined.     

A general linear model for the genotypic covariance between relatives under assortative mating

A linear model for the genotypic covariance between relatives under assortative mating comprising the classical linear model and the model of selective assortative mating is proposed. The general conditions on the genetical and developmental mechanisms of quantitative characters, as well as on selection and the mating system, on which the model is based, are explicitly stated and discussed. A classification of different relationships is presented and it is shown that these conditions are sufficient to obtain the genotypic covariance between relatives only if the relationship is a combination of descendant-ancestor, full sib, Type 1 and Nth uncle-niece relationships. All the traditional relationships, i.e., those for which the covariances of the relatives have been obtained in the literature, fall into this category. These conditions also ensure that the regression of the individual's genotypic value on the genotypic value or phenotype of any of its ancestors is always linear.Paper No. 6619 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh, North Carolina. This investigation was supported in part by NIH Research Grant No. GM 11546 from the National Institute of General Medical Sciences     

Dynamic conditionally linear mixed models for longitudinal data

We develop a new class of models, dynamic conditionally linear mixed models, for longitudinal data by decomposing the within-subject covariance matrix using a special Cholesky decomposition. Here 'dynamic' means using past responses as covariates and 'conditional linearity' means that parameters entering the model linearly may be random, but nonlinear parameters are nonrandom. This setup offers several advantages and is surprisingly similar to models obtained from the first-order linearization method applied to nonlinear mixed models. First, it allows for flexible and computationally tractable models that include a wide array of covariance structures; these structures may depend on covariates and hence may differ across subjects. This class of models includes, e.g., all standard linear mixed models, antedependence models, and Vonesh-Carter models. Second, it guarantees the fitted marginal covariance matrix of the data is positive definite. We develop methods for Bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited.     

A general approach to mixed effects modeling of residual variances in generalized linear mixed models

We propose a general Bayesian approach to heteroskedastic error modeling for generalized linear mixed models (GLMM) in which linked functions of conditional means and residual variances are specified as separate linear combinations of fixed and random effects. We focus on the linear mixed model (LMM) analysis of birth weight (BW) and the cumulative probit mixed model (CPMM) analysis of calving ease (CE). The deviance information criterion (DIC) was demonstrated to be useful in correctly choosing between homoskedastic and heteroskedastic error GLMM for both traits when data was generated according to a mixed model specification for both location parameters and residual variances. Heteroskedastic error LMM and CPMM were fitted, respectively, to BW and CE data on 8847 Italian Piemontese first parity dams in which residual variances were modeled as functions of fixed calf sex and random herd effects. The posterior mean residual variance for male calves was over 40% greater than that for female calves for both traits. Also, the posterior means of the standard deviation of the herd-specific variance ratios (relative to a unitary baseline) were estimated to be 0.60 ± 0.09 for BW and 0.74 ± 0.14 for CE. For both traits, the heteroskedastic error LMM and CPMM were chosen over their homoskedastic error counterparts based on DIC values.     

Nearest neighbour adjustment and linear variance models in plant breeding trials

This paper reviews methods for nearest neighbour analysis that adjust for local trend in one dimension. Such methods are commonly used in plant breeding and variety testing. The focus is on simple differencing methods, including first differences and the Papadakis method. We discuss mixed model representations of these methods on the scale of the observed data. Modelling observed data has a number of practical advantages compared to differencing, for example the facility to conveniently compute adjusted cultivar means. Most models considered involve a linear variance-covariance structure and can be represented as state-space models. The reviewed methods and models are exemplified using three datasets.     

The Effects of Misspecification of a Regression Model

The consequences of the misspecification of a regression model are considered. For small effects of covariates a proportional consistency theorem is derived. The consistent estimation of the covariance matrix of the estimates is discussed.     

Estimation of regression parameters and the hazard function in transformed linear survival models

An estimator of the regression parameters in a semiparametric transformed linear survival model is examined. This estimator consists of a single Newton-like update of the solution to a rank-based estimating equation from an initial consistent estimator. An automated penalized likelihood algorithm is proposed for estimating the optimal weight function for the estimating equations and the error hazard function that is needed in the variance estimator. In simulations, the estimated optimal weights are found to give reasonably efficient estimators of the regression parameters, and the variance estimators are found to perform well. The methodology is applied to an analysis of prognostic factors in non-Hodgkin's lymphoma.     

Circularity in linear programming models of optimal diet

The linear programming model of optimal diet for herbivores has been criticized for being biologically unrealistic, for being too successful given statistical realities, and for being circular. I try to clarify the issue of circularity. Circularity arises if constraint lines are estimated from average values for governing parameters, when the assumed constraints are not effective. This may occur (1) under benign season conditions when consumers do not maximize their food intake, because of costs associated with food processing and storage, or (2) when an unidentified constraint limits intake. To evaluate hypotheses about the factors controlling diet composition, it must be shown that (1) consumers respond appropriately to variation in the parameters controlling constraint settings, and (2) the assumed constraints are close to their maximum (or minimum) settings.     

The effect of miss-specified baseline characteristics on inference for longitudinal trends in linear mixed models

The main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences among subjects at the start of the study (baseline characteristics, cross-sectional effects). Often, especially in observational studies, longitudinal trends are studied after correction for many potentially important baseline differences between subjects. We show that, in the context of linear mixed models, inference for longitudinal trends is in general biased if a wrong model for the baseline characteristics is used. However, we will argue that this bias is small in most practical situations and completely vanishes in the special case of a growth curve model for complete balanced data. In the latter case, inference for longitudinal trends is completely independent of additional baseline covariates that might have been omitted from the model.