Linear predictor function

Abstract: Generalized linear mixed models provide a flexible framework for modeling a range of data, although with non-Gaussian response variables the likelihood cannot be obtained in closed form. Markov chain Monte Carlo methods solve this problem by sampling from a series of simpler conditional distributions that can be evaluated. The R package MCMCglmm implements such an algorithm for a range of model fitting problems. More than one response variable can be analyzed simultaneously, and these variables are allowed to follow Gaussian, Poisson, multi(bi)nominal, exponential, zero-inflated and censored distributions. A range of variance structures are permitted for the random effects, including interactions with categorical or continuous variables (i.e., random regression), and more complicated variance structures that arise through shared ancestry, either through a pedigree or through a phylogeny. Missing values are permitted in the response variable(s) and data can be known up to some level of measurement error as in meta-analysis. All simu- lation is done in C/ C++ using the CSparse library for sparse linear systems.

Abstract: Part I: Foundations of Multiple Regression Analysis. Overview. Simple Linear Regression and Correlation. Regression Diagnostics. Computers and Computer Programs. Elements of Multiple Regression Analysis: Two Independent Variables. General Method of Multiple Regression Analysis: Matrix Operations. Statistical Control: Partial and Semi-Partial Correlation. Prediction. Part II: Multiple Regression Analysis. Variance Partitioning. Analysis of Effects. A Categorical Independent Variable: Dummy, Effect, And Orthogonal Coding. Multiple Categorical Independent Variables and Factorial Designs. Curvilinear Regression Analysis. Continuous and Categorical Independent Variables I: Attribute-Treatment Interaction, Comparing Regression Equations. Continuous and Categorical Independent Variables II: Analysis of Covariance. Elements of Multilevel Analysis. Categorical Dependent Variable: Logistic Regression. Part III: Structural Equation Models. Structural Equation Models with Observed Variables: Path Analysis. Structural Equation Models with Latent Variables. Part IV: Multivariate Analysis. Regression, Discriminant, And Multivariate Analysis of Variance: Two Groups. Canonical, Discriminant, And Multivariate Analysis of Variance: Extensions. Appendices.

Abstract: Summary 1. Linear regression models are an important statistical tool in evolutionary and ecological studies. Unfortunately, these models often yield some uninterpretable estimates and hypothesis tests, especially when models contain interactions or polynomial terms. Furthermore, the standard errors for treatment groups, although often of interest for including in a publication, are not directly available in a standard linear model. 2. Centring and standardization of input variables are simple means to improve the interpretability of regression coefficients. Further, refitting the model with a slightly modified model structure allows extracting the appropriate standard errors for treatment groups directly from the model. 3. Centring will make main effects biologically interpretable even when involved in interactions and thus avoids the potential misinterpretation of main effects. This also applies to the estimation of linear effects in the presence of polynomials. Categorical input variables can also be centred and this sometimes assists interpretation. 4. Standardization (z-transformation) of input variables results in the estimation of standardized slopes or standardized partial regression coefficients. Standardized slopes are comparable in magnitude within models as well as between studies. They have some advantages over partial correlation coefficients and are often the more interesting standardized effect size. 5. The thoughtful removal of intercepts or main effects allows extracting treatment means or treatment slopes and their appropriate standard errors directly from a linear model. This provides a simple alternative to the more complicated calculation of standard errors from contrasts and main effects. 6. The simple methods presented here put the focus on parameter estimation (point estimates as well as confidence intervals) rather than on significance thresholds. They allow fitting complex, but meaningful models that can be concisely presented and interpreted. The presented methods can also be applied to generalised linear models (GLM) and linear mixed models.