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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?