I am returning to the September thread that dealt with Ancova and its homogeneity of slopes assumption.
In Excel I wrote a “power” program to determine the likelihood an experiment with one experiment factor set up in a completely randomized design with one covariate and with a set number of replications would find preplanned treatment contrasts significant. The program also determines the likelihood that the assumption of homogeneity of slopes would be violated. I find that as the number of replications increases, so does the likelihood that the assumption would be violated – with 20 reps about 90% of the simulated data sets yield a p value for homogeneity of slopes less than 0.05.
Is this to be expected? If it is not, perhaps the short summary of my methodology will (hopefully) give someone an idea of where I am going wrong.
The program simulates the covariate values by picking random values from a normal distribution with a defined mean and standard deviation. For example, in a feeding study with 4 diets and 20 replications, the random number generator generates 80 values which are in turn randomly “assigned” to treatment groups. The response variable for each individual in each diet – weight gain/day -- is also generated from a normal distribution, with each diet’s mean and standard deviation selected by the experiment planner. For each treatment (diet) the covariates and responses are arranged so the highest covariate value is matched with the highest response. The reason for this arrangement is that one would expect, say, the largest animal to have the highest daily gain.
These data are then analyzed with an Ancova and the p value for homogeneity of slopes is recorded.
This process is repeated 999 more times. The number of times the p value is less than 0.05 is divided by 1000 to give the likelihood an experiment using subjects from similar populations would violate Ancova’s assumption of equal slopes.
Thanks,
Richard Baldy
College of Agriculture
California State University
Chico, CA 95929
