BSC 5936.01
Autumn 2004
EXERCISE 7: ANALYSIS OF COVARIANCE
Background: Some Useful Circumstances for Analyses of Covariance
Female ectothermic vertebrates, and those of the poeciliid H. formosa are no exception, exhibit
indeterminate growth. As a result, a collection of females from a natural population will include
individuals of a wide range of body sizes. This range can reflect variation in some combination of female
age and size at maturity, growth mate after maturity, and reproductive history. The key observation is
that the distribution of body size can vary widely from one population to another. The fact that body size
can vary within and among populations is important for understanding population dynamic variation
among locations and the variation in life history among populations because vital rates and many lifehistory traits change with the size of the individual. For example, one might make collections of females
from four populations and use the analysis of variance to uncover variation among populations in brood
size. But such a finding need not indicate much life-history variation as much as it might indicate
variation in body size; females may be larger in some populations than others and the variation in brood
size, which often follows variation in body size, might be caused simply by the variation in body size
among the populations. So, as you might expect, the analysis of covariance is an important tool for
disentangling the sources of variation in life-history traits in fishes, amphibians, and reptiles.
A second useful circumstance for an analysis of covariance is the diagnosis of population
variation in "condition." The traditional interpretation of "condition" by field biologists is the overall
robustness of the animal, or some index of its mass relative to its length. Of two animals with the same
skeletal size (measured as length), the one in "better condition" would be heavier for its length. This
measure is useful in quantifying stressful environments (e.g. animals lose "condition" in habitats that
induce high demand metabolic rates or that have low food supplies) or stressful seasons (e.g. animals
often lose "condition" in winter). An old measure of condition was obtained by dividing the wet mass of
an animal (none of this accounting for fat storage or any such more high and mighty measure) by its
length; the higher the ratio, the better the "condition." Some biologists might define “condition”
differently, perhaps by examining not total mass but the mass of stored fat, reasoning that individuals in
better condition will be individuals who have managed to accrue a surplus of energy and are able to store
that surplus without compromising some other important functions.
The Data
The data are female traits and reproductive characters of individuals of the least killifish,
Heterandria formosa, from a collection made in four populations in June of 1994. Each datum represents
a single mature female. The variables, as listed on the data sheet, are
POP
SL
FAT
LEAN
TOTOFF
TREP
the population from which the females were collected
the standard length of the female
the total dry mass of the female, before dissection or any other treatment
the dry mass of the female after treatment to remove lipid material
the total number of embryos being carried by that female in her ovary
the total mass of reproductive tissue, ovary plus embryos, before lipid extraction
There are 20 entries per population. For a larger context for these data, see Leips and Travis, 1999, J.
Anim. Ecol. 68:595-616.
The Exercise
With that background, let's explore some biological questions. Your job is to interpret the
following biological questions in statistical terms as you deem most useful and use the analysis of
covariance, somehow, to answer them. For this exercise, stop at the diagnosis of significance and forego
multiple comparisons for post-hoc inference.
1. Do females from different populations vary in condition? First, obviously, you must decide on the
appropriate variable for analysis and justify it. Then present the results of your statistical analysis.
2. Do females from different populations vary in their potential fecundity? Again, justify your choice of
the appropriate variable for analysis and then present the results of your statistical analysis.
The Exercise Reconsidered
Now that you've answered the biological questions using analysis of covariance, let's do a few
mundane statistical analyses.
1. For the analysis of covariance that you performed to answer question 1 above, perform a one-way
analysis of variance on the covariate you employed.
2. Perform the following analyses.
a) Perform a simple regression analysis of your dependent variable on the covariate, ignoring
the fact that different populations are represented in the data. Note the sum of squares
for the regression and the residual.
b) Perform individual regression analyses within each population, noting the sums of squares for
each regression and each residual sum of square, and compile the total residual sum of
squares.
c) Perform a one-way analysis of variance of the original dependent variable, ignoring the fact
that some covariate might be important. Note the sum of squares for the effect of
"population" and the residual.
d) Create a series of analysis of variance tables, using only sums of squares, just as we did for the
analyses we performed on the Bumpus’ data in lecture. Verify the relationships among
residual sums of squares that we outlined in lecture to assure yourself that you really
understand the analysis of covariance and that you could do it yourself without a fancy
program; that is, you can reconstitute the appropriate sums of squares just from doing
regressions and analyses of variance. In other words, circle and connect related sums of
squares like we did in lecture and show where individual and pooled regression residuals
combine to generate the appropriate sums of squares.
e) Compare the “model” sums of squares you obtained in parts a) and b) to the sums of
squares for the effect of the covariate and the effect of population (the "adjusted mean"
test) in the presence of each other that you obtained when you performed your analysis of
covariance. Given the pattern, what do you conclude?