Here is a how Quinn and Keough (2002 Cambridge University Press) address the distinction between random and fixed effects. _________________________________________________________ 8.1.1 Types of predictor variables (factors) There are two types of categorical predictor variables in linear models. The most common type is a fixed factor, where all the levels of the factor (i.e. all the groups or treatments) that are of interest are included in the analysis. We cannot extrapolate our statistical conclusions beyond these specific levels to other groups or treatments not in the study. If we repeated the study, we would usually use the same levels of the fixed factor again. Linear models based on fixed categorical predictor variables (fixed factors) are termed fixed effects models (or Model 1 ANOVAs). Fixed effect models are analogous to linear regression models where X is assumed to be fixed. The other type of factor is a random factor, where we are only using a random selection of all the possible levels (or groups) of the factor and we usually wish to make inferences about all the possible groups from our sample of groups. If we repeated the study, we would usually take another sample of groups from the population of possible groups. Linear models based on random categorical predictor variables (random factors) are termed random effects models (or Model 2 ANOVAs). ______________________________________________________
In the Grossman query (below) temperature, rainfall, and density would likely be fixed because they are of interest -- the contrasts would be of interest across the particular values of temperature, rainfall, and density. Inference would be only to the measured values and their contrasts. All three variables become fixed if fitted as a regression instead of as categorical variables. Temperature might be taken as a random variable over a small range, but would not be credible as a random variable over a wide range, given its profound effect on biological processes. Location would be either random or fixed, depending on whether the inference was to only those 3 sites at the stated dates of measurement (fixed), or to all possible sites in some stated area (random), or to the hypothetical population of a very large number of repetitions at those sites (random, as above). If the locations were known to differ in some salient biological way, such that they could be ordered as to expected effect, location could be legitimately treated as fixed. The choice of random versus fixed categorical variable lies with the judgement and knowledge of the biologist. A good statistician will demure on demands for hard and fast rules. A good statistician will instead probe the biologist as to the scope of inference, then help the biologist form the correctly nested (log) likelihood ratio (as in Quinn and Keogh or any of many texts). The likelihood ratio is key - in either a decision theoretic context (as in Quinn and Keough) or with inference from a prior to a posterior probability, if that is what you want to do. ~ David Schneider Quoting "Street, Garrett" <gms...@msstate.edu>: > There is also an excellent section on what constitutes a random or fixed > effect in Tom Hobbs and Mevin Hooten's "Bayesian Models: a Statistical Primer > for Ecologists" using fecundity of spotted owls (adapted from Clark's work on > the subject), and again using hypothetical sampling of aboveground biomass, > as examples. Both examples are accompanied by clear and concise explanations > of the implications for the underlying distributions and assumptions of the > model one might seek to fit, and for the ecology informing the models. > > Garrett Street > Assistant Professor > Wildlife, Fisheries, and Aquaculture > Mississippi State University > > On May 17, 2016, at 4:34 PM, Brian Church > <church...@gmail.com<mailto:church...@gmail.com>> wrote: > > There is a fairly detailed discussion of fixed vs. random effects on > CrossValidated here: > http://stats.stackexchange.com/questions/4700/what-is-the-difference-between-fixed-effect-random-effect-and-mixed-effect-mode > > Based on the discussion there, it seems like temperature, rainfall, and > density could all be considered to be random effects for the following > reasons: > 1. You are unlikely to sample the entire populations for those variables. > 2. They are not being controlled > 3. They are likely continuous and distributed in some way (e.g., normal) > rather than discrete values > 4. You are unlikely to be interested in responses at a specific temperature, > rainfall, and density; rather, it seems more interesting to understand > effects relating to the underlying distributions of those variables. > > Those commenting in the CrossValidated forum cite a few sources, though they > seem to be general/mathematical rather than ecology-specific. Hope that helps > some. > > -Brian Church > > > On Tue, May 17, 2016 at 11:12 AM, Gary Grossman > <gdgross...@gmail.com<mailto:gdgross...@gmail.com>> wrote: > I'm having a bit of difficulty getting a clear understanding of what should > be considered a fixed vs. a random effect in a linear mixed model analysis of > field data. Even the statisticians seem to say "it depends on who's defining > it" or "sometimes the same treatment/variable can be either". Some examples > may help, let's say I collected samples annually in three sites and wanted to > test for the effect of daily rainfall, daily temperature, and density, on > recruitment of individuals in the following year. Using the lmer function in > R which of these would be fixed effects and which would be random? A > reference or two would help. I really couldn't find much in a google search > on field studies, but I didn't go to anything like zoological abstracts. TIA, > g2 > > -- > Gary D. Grossman, PhD > Fellow, American Fisheries Soc. > > Professor of Animal Ecology > Warnell School of Forestry & Natural Resources > University of Georgia > Athens, GA, USA 30602 > > Website - Science, Art (G. Grossman Fine Art) and Music > www.garygrossman.net<http://www.garygrossman.net> > > Board of Editors - Animal Biodiversity and Conservation > Editorial Board - Freshwater Biology > Editorial Board - Ecology Freshwater Fish > > Hutson Gallery Provincetown, MA - > www.hutsongallery.net/artists.html<http://www.hutsongallery.net/artists.html> > > > >
