PCB 5936.01
Autumn 2004
REMARKS ON EXERCISE FIVE
Part A: All Effects Random
Source of Variation
EMS
History
e 2 + 5 P(H)D 2 + 20 HD 2 + 15 P(H) 2 + 60 H 2
Population (History)
e 2 + 5 P(H)D 2 + 15 P(H) 2
Density Treatment
e 2 + 5 P(H)D 2 + 20 HD 2 + 40 D 2
History x Density Treatment
e 2 + 5 P(H)D 2 + 20 HD 2
Population (History) x Density Trt
e 2 + 5 P(H)D 2
Residual
e 2
Notes:
1. Notice that in this design, the effect of history is not readily testable. If one assumes that the
interaction of history and density is negligible, then the effect of history can be placed over the
effect of population within history. But the very nature of the scientist’s idea was that the form of
the density response would depend upon evolutionary history, that is, there would be an
interaction between history and density. Thus one can only test the effect of history if the very
pattern that was expected, and for which the experiment was designed, does not exist. Now if the
scientist doesn’t really care about an additive, overall effect of history, then this situation is not a
predicament. Of course, if the interaction of population within history and density is not
significant and if the main effect of population within history is not significant, then the effect of
history is readily testable by placing it over the interaction of history and density. So if the
scientist really is interested in the overall effect of history, he/she has to hope that either the very
effect the experiment was designed to investigate is insignificant or two other effects are
insignificant. This would hardly be a shining example of good experimental design for the
overall effect of history.
2. The likely power of this design depends the effect in which the scientist is most interested.
For the History x Density interaction, the test of significance will face the critical F-value for 2,
12 degrees of freedom. The main effect of Density will face a critical F-value for 2,2 degrees of
freedom, which is not very promising. The test for any main effect of History is not very
promising however one might hope to perform it. All in all, this is probably as good a design as
one can devise for testing the basic idea, that history affects the density response (the history x
density interaction will be significant), but the cost is that the design is not well-suited for other
possible effects of history.
Part B: History and Population Within History Random but Density Treatment Fixed
Source of Variation
EMS
History
e 2 + 15 P(H) 2 + 60 H 2
Population (History)
e 2 + 15 P(H) 2
Density Treatment
e 2 + 5 P(H)D 2 + 20 HD 2 + 40 D
History x Density Treatment
e 2 + 5 P(H)D 2 + 20 HD 2
Population (History) x Density Trt
e 2 + 5 P(H)D 2
Residual
e 2
Notes:
1. The EMS and the likely power of the test for the history-density interaction is unaltered by
considering density to be a fixed effect. This reinforces the notion that this design is as good as
one can devise for testing the history x density interaction because it is robust to an argument
about whether density should be considered fixed or random. Conceptually, the overall design
makes sense in that one wouldn’t want to conclude that history matters unless its influence is
visible above and beyond the differences among individual populations with a common history.
2. The EMS are much simpler and lead to a ready test for the overall main effect of history. The
test may not have much power, facing a critical F-value with 1,6 degrees of freedom, but the
situation is a certain improvement over any scenario for testing the main effect of history in the
previous interpretation of the design. The test for population with history will have more power,
placing the effect of population within history over the residual instead of over the population
within history by density interaction.
Part C: History and Density Treatment Fixed and only Population Within History Random
Source of Variation
EMS
History
e 2 + 15 P(H) 2 + 60 H
Population (History)
e 2 + 15 P(H) 2
Density Treatment
e 2 + 5 P(H)D 2 + 40 D
History x Density Treatment
e 2 + 5 P(H)D 2 + 20 HD
Population (History) x Density Trt
e 2 + 5 P(H)D 2
Residual
e 2
Notes:
1. This design offers no real change in testing the interaction of history and density treatment,
which was the main object of the experiment, so, again, it seems that this overall plan is quite
robust and reliable for testing that interaction.
2. There is an important substantive change in this interpretation that revolves around how the
effect of the density treatment would be tested. Here one would test density treatment over the
interaction of population, nested within history, and density (as opposed to the history-density
interaction dictated by the other interpretations). This is likely to be a more powerful test, all
things considered, because the degrees of freedom in the denominator will be larger than in the
other interpretations.
General Comments
Obviously one can’t decide whether any effect is a fixed or random effect based on how
conveniently the EMS fall out. But the different levels of convenience surely constitute a
temptation for one decision over another. An ancillary lesson here follows from the fact that the
main object of the experiment is robust to the argument about whether density is fixed or random
but other effects are not: experiments designed for one goal may not be very good for subsidiary
goals, even though it might seem easy to include those goals in the overall design. One should
keep this in mind in one’s own work as well as in evaluating papers that draw data from
elsewhere when the data were not collected with the focal paper’s goals in mind.
Now, as to which is the best interpretation, there is some room for argument. No one
would seriously argue that population nested within history is anything but random. One could
argue fairly that density treatment is a fixed effect here; while one might have used other
densities, it’s probably not true that all densities are equally likely to be selected because,
typically, higher densities are multiples of the lowest density chosen and the density choices
usually also include some pragmatic considerations. As to “history,” one could argue that
populations have a continuum of histories and that “low” and “high” are arbitrarily sampled
selections from that continuum (which makes history a random effect). One might also argue that
history represents a broad dichotomy between clear extremes on a continuum and is therefore best
considered as a fixed effect.