I231B QUANTITATIVE METHODS Analysis of Variance (ANOVA)

Syllabus Changes Thursday April 24 th, Regression April 29: Multivariate Regression May 1: Regression Diagnostics May 6 th : Logistic Regression May 8 th : Display of some advanced topics; Course Review 2

Analysis of Variance 3 In its simplest form, it is used to compare means for three or more categories.  Example:  Income (metric) and Marital Status (many categories) Relies on the F-distribution  Just like the t-distribution and chi-square distribution, there are several sampling distributions for each possible value of df.

What is ANOVA? 4 If we have a categorical variable with 3+ categories and a metric/scale variable, we could just run 3 t- tests.  The problem is that the 3 tests would not be independent of each other (i.e., all of the information is known). A better approach: compare the variability between groups (treatment variance + error) to the variability within the groups (error)

The F-ratio MS = mean square bg = between groups wg = within groups 5 df = # of categories – 1 (k-1)

Interpreting the F-ratio 6 Generally, an f-ratio is a measure of how different the means are relative to the variability within each sample Larger values of ‘f’  greater likelihood that the difference between means are not just due to chance alone

Null Hypothesis in ANOVA If there is no difference between the means, then the between-group sum of squares should = the within-group sum of squares. 7

Visual ANOVA and f-ratio 8

F-distribution 9 A right-skewed distribution It is a ratio of two chi-square distributions

F-distribution 10 F-test is always a one-tailed test.  Why?

Relationship to t-test 11 Why not just run many t-tests between all possible combinations?  As number of comparisons grow, likelihood of some differences are expected– but do not necessarily indicate an overall difference. Still, t-tests become important after an ANOVA so that we can find out which pairs are significantly different. Certain ‘corrections’ can be applied to such post-hoc t-tests so that we account for multiple comparisons (e.g., Bonferroni correction, which divides p-value by the number of comparisons being made)

Logic of the ANOVA 12 Conceptual Intro to ANOVA Class Example: anova.do and sm96_compressed.dta