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12. Comparing groups (ANOVA)

# 12. Comparing groups (ANOVA) - 12 Comparing Groups Analysis...

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12. Comparing Groups: Analysis of Variance (ANOVA) Methods Response y Explanatory x var’s Method Categorical Categorical Contingency tables (Ch. 8,15) (chi-squared, etc., logistic regr.) Quantitative Quantitative Regression and correlation (Ch 9 bivariate, 11 multiple regr.) Quantitative Categorical ANOVA (Ch. 12) (Where does Ch. 7 on comparing 2 means or 2 proportions fit into this?) Ch. 12 compares the mean of y for the groups corresponding to the categories of the categorical explanatory var’s (factors). Examples: y = mental impairment, x’s = treatment type, gender, marital status y = income, x’s = race, education (<HS, HS, college), type of job

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Comparing means across categories of one classification (1-way ANOVA) Let g = number of groups We’re interested in inference about the population means μ 1 , μ 2 , ... , μ g The analysis of variance (ANOVA) is an F test of H 0 : μ 1 = μ 2 = ⋅ ⋅ ⋅ = μ g H a : The means are not all identical The test analyzes whether the differences observed among the sample means could have reasonably occurred by chance, if H 0 were true (due to R. A. Fisher).
One-way analysis of variance Assumptions for the F significance test : The g population dist’s for the response variable are normal The population standard dev’s are equal for the g groups ( σ ) Randomization, such that samples from the g populations can be treated as independent random samples (separate methods used for dependent samples)

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Variability between and within groups (Picture of two possible cases for comparing means of 3 groups; which gives more evidence against H 0 ?) The F test statistic is large (and P- value is small) if variability between groups is large relative to variability within groups Both estimates unbiased when H 0 is true (then F tends to fluctuate around 1 according to F dist.) Between-groups estimate tends to overestimate variance when H 0 false (then F is large, P- value = right-tail prob. small) 2 2 (between-groups estimate of variance ) (within-groups estimate of variance ) F σ σ =
Detailed formulas later, but basically Each estimate is a ratio of a sum of squares (SS) divided by a df value, giving a mean square (MS). The F test statistic is a ratio of the mean squares. P- value = right-tail probability from F distribution (almost always the case for F and chi-squared tests). Software reports an “ANOVA table” that reports the SS values, df values, MS values, F test statistic, P- value. (Looks like ANOVA table for regression.)

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Exercise 12.12: Does number of good friends depend on happiness? (GSS data) Very happy Pretty happy Not too happy Mean 10.4 7.4 8.3 Std. dev. 17.8 13.6 15.6 n 276 468 87 Do you think the population distributions are normal? A different measure of location, such as the median, may be more relevant. Keeping this in mind, we use these data to illustrate one-way ANOVA.
ANOVA table Sum of Mean Source squares df square F Sig Between-groups 1627 2 813 3.47 0.032 Within-groups 193901 828 234 Total 195528 830 The mean squares are 1627/2 = 813 and 193901/828 = 234.

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12. Comparing groups (ANOVA) - 12 Comparing Groups Analysis...

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