Categorical linear models

Suppose that we have a continuous random variable Y and a categorical variable C with levels c 1 • Are all of these random variable means the same? In other words, is it true that: µ 1 (Y) = µ 2 (Y) = · · · = µ (Y)? • For which pairs of levels c i , c j are the associated random variable means µ i (Y), µ j (Y) equal? • For each pair of levels c i , c j , what is the difference µ j (Y) − µ i (Y) between the associated random variable means? Examining the data Fitting the model Sampling Variability Assumptions normality, equal variances, independence what if variances aren't equal? if variances depend on mean monoton-ically and data are positive, try log or square root transformations. otherwise , if sample sizes are about equal, know that the result will be inflation of the p value. EXAMPLE: HEIGHTS OF SINGERS BY SECTION. Choir singers are often divided into four sections by voice range. From highest voice range to lowest, these sections are: soprano, alto, tenor, and bass. Are the average heights among these groups different? If so, by how much? (Intuition would suggest some differences, even if only related to the fact that sopranos and altos are usually women, while tenors and basses are usually men.) From the statement of the problem, we determine three non-statistical formulate the questions of interest questions of interest: 1. Are the average heights of all of the sections equal? 2. If not (or for additional information), for which pairs of sections do the average heights differ? 3. Also if not (or for additional information), by how much do the average heights differ within each pair of sections? We translate these into corresponding statistical questions of interest. We state these questions in terms of the variables of interest: section, a categorical variable with levels S (soprano), A (alto), T (tenor), and B (bass); and height, a random variable that gives the height of a randomly chosen chorister from a suitable population of choirs. We also use the notation that sectionS is an indicator variable for the condition that section = S, and similarly for sectionA, sectionT, and sectionB. Also, we use β A , β T , and β B to denote the coefficients of sectionA, sectionT, and sectionB in the (unknown) true linear model of …