Fixed vs. Random Effects Fixed effect –we are interested in the effects of the treatments (or blocks) per se –if the experiment were repeated, the levels would be the same –conclusions apply to the treatment (or block) levels that were tested –treatment (or block) effects sum to zero Random effect –represents a sample from a larger reference population –the specific levels used are not of particular interest –conclusions apply to the reference population inference space may be broad (all possible random effects) or narrow (just the random effects in the experiment) –goal is generally to estimate the variance among treatments (or other groups) Need to know which effects are fixed or random to determine appropriate F tests in ANOVA

Fixed or Random? lambs born from common parents (same ram and ewe) are given different formulations of a vitamin supplement comparison of new herbicides for potential licensing comparison of herbicides used in different decades (1980’s, 1990’s, 2000’s) nitrogen fertilizer treatments at rates of 0, 50, 100, and 150 kg N/ha years of evaluation of new canola varieties (2008, 2009, 2010) location of a crop rotation experiment that is conducted on three farmers’ fields in the Willamette valley (Junction City, Albany, Woodburn) species of trees in an old growth forest

Fixed and random models for the CRD Fixed Model (Model I) Random Model (Model II) Y ij = µ +  i +  ij Expected Source df Mean Square Treatment t-1 Error tr-t 2 T 2 e r  2 e  Expected  Te  r 22  Source df Mean Square Treatment t -1 Error tr - t 2 e  variance among fixed treatment effects

Models for the RBD Fixed Model Random Model Y ij = µ +  i +  j +  ij Mixed Model

RBD Mixed Model Analyses with SAS Mixed Models - contain both random and fixed effects Note that PROC GLM will only handle LM! PROC GLIMMIX can handle all of the situations above Distribution Treatments Fixed Blocks Fixed Treatments Fixed Blocks Random Normal (continuous) (PROC GLM) Linear Model (LM) (PROC MIXED) Linear Mixed Model (LMM) Non-normal (categories or counts) (PROC GENMOD) Generalized Linear Model (GLM) (PROC GLIMMIX) Generalized Linear Mixed Model (GLMM)

Generalized Linear Models An alternative to data transformations Principle is to make the model fit the data, rather than changing the data to fit the model Models include link functions that allow heterogeneous variances and nonlinearity Analysis and estimation are based on maximum likelihood methods Becoming more widely used - recommended by the experts Need some understanding of the underlying theory to implement properly Notes adapted from ASA GLMM Workshop, Long Beach, CA, 2010

Generalized Linear Models ANOVA/Regression model is fit to a non-normal data set Three elements: 1. Random component – a probability distribution for Y i from the exponential family of distributions 2. Systematic component – represent the linear predictors (X variables) in the model 3. Link function – links the random and systematic elements Form is mean + trt effect No error term

Generalized Linear Models ANOVA/Regression model is fit to a non-normal data set Three elements: 1. Random component – a probability distribution for Y i from the exponential family of distributions (this is known) 2. Systematic component – represent the linear predictors (X variables) in the model 3. Link function – links the random and systematic elements Form is mean + trt effect No error term

Log of Distribution = “Log-Likelihood” Binary responses (0 or 1) Probability of success follows a binomial distribution “canonical parameter” Takes the form Y * function of P

Example – logit link µ can only vary from 0 to 1  can take on any value Use an inverse function to convert means to the original scale

Some Common Distributions & Link(s)

Linear Models for an RBD in SAS Treatments fixed, Blocks fixed –PROC GLM (normal) or PROC GENMOD (non-normal) –all effects appear in model statement Model Response = Block Treatment; Treatments fixed, Blocks random –PROC MIXED (normal) or PROC GLIMMIX (non-normal) –Only fixed effects appear in model statement Model Response = Treatment; Random Block;

GLIMMIX basic syntax for an RBD fixed effects go in the model statement random effects go in the random statement default means and standard errors from lsmeans statement are on a log scale ilink option gives back-transformed means on original scale and estimates standard errors on original scale diff option requests significant tests between all possible pairs of treatments in the trial, proc glimmix; class treatment block; model response = treatment / link=log s dist=poisson; random block; lsmeans treatment/ilink diff;

Estimation in LMM, GLM, and GLMM Does not use Least Squares estimation Does not calculate Sums of Squares or Mean Squares Estimates are by Maximum Likelihood Output includes Source of variation degrees of freedom F tests and p-values Treatment means and standard errors Comparisons of means and standard errors