PCB 5936.01 Autumn 2004
EXERCISE 6: MULTIPLE REGRESSION
The data (ex6aut04.xls) is drawn from Bumpus’ famous study of phenotypic selection in birds. The data are measurements of 6 phenotypic characters and an estimate of relative fitness for 88 individual birds. Your task, in general, is to use multiple linear regression analyses to find the best predictors of relative fitness. This is a standard method whose appeal lays in the ability to interpret the partial regression coefficients as parameters in a general model of multivariate character evolution (for background, see Roff, 1997, Evolutionary Quantitative Genetics, Chapman and Hall, New York). For example, the partial linear regression coefficients of relative fitness on a character may be interpreted as the gradient of directional selection on that character - the return in relative fitness for a unit change in the character.
The Data
TL: total length of the body in millimeters (from tip of beak to tip of tail along the spine, not counting wings or legs)
WL: length of the wing in millimeters (from junction with torso to tip)
MASS: wet mass in grams
LBH: length of the head and beak in millimeters
LH: length of the humerus in millimeters (bone in wing corresponding to humans' "upper arm")
LF: length of the femur in millimeters (bone in thigh, for you non-aficionados of vertebrate anatomy)
W: relative fitness (scaled such that average absolute fitness has a relative fitness of 1.00)
The Actual Assignment
Use the tool of multiple linear regression to find which combination of characters offers the best predictive relationship with relative fitness. For this assignment, let's look only for evidence of directional selection, so you will be squarely in the world of multiple linear regression on characters. You may have to subject the data to transformation to meet the assumptions of multiple linear regression. If you cannot find a transformation that places the data perfectly within the assumptions of the model, use whichever transformation offers the best fit or at least the least objectionable fit.
In your report on this assignment, you must describe whether you transformed the data and, if so, how that transformation affects your interpretation of the partial regression coefficient. You must also tell me how you went about choosing candidate regressions for consideration and which criteria you employed to decide which regression relationship was best. YOU NEED NOT PRESENT DETAILED PRINTOUTS FOR EACH AND EVERY REGRESSION RELATIONSHIP YOU EXAMINED. Finally, of course, offer your own diagnosis about whether you find the final, so-called best regression relationship believable and, more importantly, how much insight you feel it offers into the process and outcome of phenotypic selection in these birds.
Hints
Remember that multiple regression analysis is easy when the predictors are uncorrelated and difficult otherwise, so it's worthwhile to understand the correlation structure of the predictors before you embark upon the actual regression analysis. Keep in mind also that the humerus is a component of the overall wing length and that "LBH" is a component of the total length. Think hard about the statistical issues and how they relate to the biological issues. Finally, it's worth wondering if your search for the best predictors would be facilitated if you knew the agents of selection (if so, tell me how and why).