# Quote

I am reading from Quinn and Keough's book. In chapter Correlation and Regression, section Fixed X, at page 94, they say

Linear regressions analysis assumes that the $x_i$ are known constants, i.e. they are fixed values controlled or set by the investigator with no variance associated with them. A linear model in which the predictor variables are fixed is known as Model I or a fixed effects model. This will often be the case in designed experiments where the levels of X are treatments chosen specifically. In these circumstances, we would commonly have replicate Y-values for each $x_i$ and X may well be a qualitative variable, so analyses that compare mean values of treatment groups might be more appropriate [..]. The fixed X assumption is probably not met for most regression analyses in biology because X and Y are usually both random variables recorded from a bivariate distribution. For example, Peake and Quinn (1993) did not choose mussel clumps of fixed areas but took a haphazard sample of clumps from the shore [..].

# Question

I used to think of that variables that should be integrated as random effects are those for which measurements was made with some level of inaccuracy and I am not sure this is what this quote is referring to. My question is

Are random effects used for explanatory variables measured with inaccuracy or for explanatory variables designating a subset of groups that are sampled?

Note that I am a little bit confused in how to think of these concepts in both the contexts of linear regressions and ANOVAs.

I would like to clarify my misunderstandings via two examples.

Example 1

Alice scuba dived, looked for anemones and for each one of them she measured their size and their depth. The question is "Does depth affect anemonea size?".

I can think of two potential reasons why depth should be modelled as a random effect:

• Because the depth was measured with some inaccuracy (specific to the depth meter Alice used)

• Because Alice has not exhaustively (and in the right proportions) sampled all the possible depths of interest.

Example 2

Alice went on the sea shore and randomly sampled mussel clumps. For each clump sampled, she counted the number of individual mussels and the number of species of mussels. The question is "Is the number of species of mussels affecting the number of individuals".

The counts are perfectly known (no inaccuracy in the measurement). From the above quote, I can still think of a reason why Alice should model the explanatory variable as a random effect; Alice haphazardly sampled clumps and a future study could have sampled different clumps. If all the clumps were sampled, then Alice should use a fixed effect model.

• Please be brief and indicate specfic goal. Seems to be a some-what weird question. – Subhash C. Davar May 10 '18 at 4:07