PCB 5936
Autumn 2004
EXERCISE 4: RANDOM EFFECTS AND MIXED MODELS AND MORE THOUGHT
A scientist was interested in whether birth rates responded to density variation in different ways
in different populations. More specifically, the scientist thought that in populations with a history of high
densities, birth rates would be lower at low densities and higher at high densities when compared to the
birth rates at low and high density in populations with a history of low densities. The scientist designed
an experiment to investigate whether this was the case.
The scientist constructed experimental “populations” in mesocosms (artificial environments);
from this point onward, each experimental unit will be called a “replicate.” The mesocosms are large
enough to capture typical natural rates of individual movement and interactions among individuals and
they have their own renewable food resources for the animals under study. The scientist had extensive
background data on density distributions in many natural populations of his/her focal organism, so the
experiment was not designed in a vacuum.
The experiment used stocks of animals drawn from two types of natural populations, “low density
history” populations and “high density history” populations. The scientist used a nested design in which
stocks from eight different natural populations were “nested” within each level of “history” (i.e. four “low
density” populations and four “high density” populations). The scientist examined the birth rates of these
stocks at three experimental densities, 16, 32, and 64 females per replicate. The scientist had five
replicates at each combination of treatment density and original population. Hence there were 120
replicates, 40 per experimental density; 20 of the replicates were drawn from “low density history”
populations, 5 replicates from each of 4 individual natural populations nested within that history, and 20
drawn from “high density history” populations with the same type of nesting. The scientist measured the
birth rates of females in each replicate (number of offspring produced per 60-day measurement period)
and averaged those rates among all females in a replicate so as to have one number per replicate.
But once the data are collected, the scientist gets conflicting advice about the analysis.
a) One consultant tells the scientist to consider each of the three effects (history, population within
history, treatment) to be qualitative random effects. Write the expected mean square for each effect in the
analysis of variance (treatment, history, population within history, the interaction of population within
history and treatment, the interaction of history with treatment). Remember the basic principles of
finding EMS and work from there; don’t forget to multiply through by the actual numbers (five replicates
per combination, 4 populations nested within each history level, etc.).
b) Another consultant tells the scientist that the treatment effect is a qualitative fixed effect. Without
passing judgment on the advice, write the expected mean square for each effect if treatment is fixed but
the others are random, as they were in part a).
c) A third consultant tells the scientist that “history” and “treatment” should be fixed effects and that only
the effect of population nested within history is truly random. Again, without passing judgment on the
advice, write the expected mean square for each effect under this scenario.
d) Now pass judgment on the advice: tell the scientist what you think the proper classification should be
and why.