Sociology 5811: Lecture 14: ANOVA 2 Copyright © 2005 by Evan Schofer Do not copy or distribute without permission

Announcements Midterm next class Today: Bring a Calculator Bring Pencil/Eraser Today: Wrap up ANOVA Midterm review activities

Review: ANOVA ANOVA = “ANalysis Of VAriance” “Oneway ANOVA” : The simplest form ANOVA lets us test to see if any group mean differs from the mean of all groups combined Answers: “Are all groups equal or not?” H0: All groups have the same population mean m1 = m2 = m3 = m4 H1: One or more groups differ But, doesn’t distinguish which specific group(s) differ.

ANOVA: Concepts & Definitions The grand mean is the mean of all groups ex: mean of all entry-level workers = $8.70/hour The group mean is the mean of a particular sub-group of the population The effect (a) of a group is the difference between that group’s mean from the grand mean The dependent variable is our variable of interest We explore whether its value “depends” on a case’s group

Review: Sum of Squared Deviation The total deviation can partitioned into aj and eij components: The total variance (SStotal) is made up of: aj : between group variance (SSbetween) eij : within group variance (SSwithin) SStotal = SSbetween + SSwithin

Review: Sum of Squared Variance The sum of squares grows as N gets larger. To derive a more comparable measure, we “average” it, just as with the variance: i.e, (divide) by N-1 It is desirable, for similar reasons, to “average” the Sum of Squares between/within Result the “Mean Square” variance MSbetween and MSwithin

Sum of Squared Variance Choosing relevant denominators we get:

Mean Squares and Group Differences MSbetween > MSwithin: MSbetween < MSwithin:

Mean Squares and Group Differences Question: Which suggests that group means are quite different: MSbetween > MSwithin or MSbetween < MSwithin Answer: If between group variance is greater than within, the groups are quite distinct It is unlikely that they came from a population with the same mean But, if within is greater than between, the groups aren’t very different – they overlap a lot It is plausible that m1 = m2 = m3 = m4

The F Ratio The ratio of MSbetween to MSwithin is referred to as the F ratio: If MSbetween > MSwithin then F > 1 If MSbetween < MSwithin then F < 1 Higher F indicates that groups are more separate

The F Ratio The F ratio has a sampling distribution That is, estimates of F vary depending on exactly which sample you draw Again, this sampling distribution has known properties that can be looked up in a table The “F-distribution” Different from z & t! Statisticians have determined how much area falls under the curve for a given value of F… So, we can test hypotheses.

The F Ratio Assumptions required for hypothesis testing using an F-statistic 1. J groups are drawn from a normally distributed population 2. Population variances of groups are equal If these assumptions hold, the F statistic can be looked up in an F-distribution table Much like T distributions But, there are 2 degrees of freedom: J-1 and N-J One for number of groups, one for N.

The F Ratio Example: Looking for wage discrimination within a firm The company has workers of three ethnic groups: Whites, African-Americans, Asian-Americans You observe in a sample of 200 employees: Y-barWhite = $8.78 / hour Y-barAfAm = $8.52 / hour Y-barAsianAm = $8.91 / hour

The F Ratio Suppose you calculate the following from your sample: Recall that N = 200, J = 3 Degrees of Freedom: J-1 = 2, N-J = 197 If a = .05, the critical F value for 2, 197 is 3.00 See Knoke, p. 514 The observed F easily exceeds the critical value Thus, we can reject H0 We can conclude that the groups do not all have the same population mean.

Critical a area: Reject H0 The F Ratio: Visually Rough sketch of F-distribution N = 200, J = 3; Degrees of Freedom: J-1 = 2, N-J = 197 If a = .05, the critical F value for 2, 197 is 3.00 0 1 2 3 4 5 Critical a area: Reject H0

Post-Hoc Tests Limitation: ANOVA doesn’t tell you which group(s) differ from each other Solution: “Post-Hoc Tests”: e.g., Bonferroni, Scheffe, Tukey tests Tests provide pair-wise comparison of all group Ex: Bonferroni Corrects (reduces) a for pair-wise tests, to avoid Type I error Which one should you use? Bonferroni & Scheffe are very conservative For most purposes, they all produce identical results Consult an advanced stats text to learn the subtle differences.

ANOVA: Example Example: GSS data: Schooling & Race Oneway ANOVA, plus descriptives and Scheffe Post-hoc test Note: N = 2752, grand mean = 13.36

ANOVA: Example Example: GSS data: Schooling & Race Sum of Squares: Between, within, and total Mean square variance: between and within Compare F-value to “critical f” in F-table. Or compare p-value (“Sig.”) to a We reject H0 because F is big, p-value is small.

ANOVA Example: Scheffe Test Tests compare every group Compare p-value (“Sig.”) to a Note: White/Black difference is significant; White/Other difference is not

Comparison with T-Test T-test strategy: Determine the width of the sampling distribution of the difference in means… Use that info to assess probability that groups have same mean (difference in means = 0) ANOVA strategy Compute F-ratio, which indicates what kind of deviation is larger: “between” vs. “within” group High F-value indicates groups are separate Note: For two groups, ANOVA and T-test produce identical results.

Bivariate Analyses Up until now, we have focused on a single variable: Y Even in T-test for difference in means & ANOVA, we just talked about Y – but for multiple groups… Alternately, we can think of these as simple bivariate analyses Where group type is a “variable” Ex: Seeing if girls differ from boys on a test… … is equivalent to examining whether gender (a first variable) affects test score (a second variable).

2 Groups = Bivariate Analysis Group 1: Boys Case Score 1 57 2 64 3 48 Case Gender Score 1 57 2 64 3 48 4 53 5 87 6 73 Group 2: Girls Case Score 1 53 2 87 3 73 2 Groups = Bivariate analysis of Gender and Test Score

T-test, ANOVA, and Regression Both T-test and ANOVA illustrate fundamental concepts needed to understand “Regression” Relevant ANOVA concepts The idea of a “model” Partitioning variance A dependent variable Relevant T-test concepts Using the t-distribution for hypothesis tests Note: For many applications, regression will supersede T-test, ANOVA But in some cases, they are still useful…