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Unformatted text preview: 12. Comparing Groups: Analysis of Variance (ANOVA) Methods Response y Explanatory x var’s Method Categorical Categorical Contingency tables Quantitative Quantitative Regression and correlation Quantitative Categorical ANOVA (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 variables. Examples: y = mental impairment, x’s = treatment type, gender, marital status y = income, x’s = race, education (<HS, HS, college), type of job Comparing means across categories of one classification (1way 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 : 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). Oneway 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) Variability between and within groups • (Picture of two possible cases for comparing means of 3 groups; which gives more evidence against H ?) • 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.) Betweengroups estimate tends to overestimate variance when H 0 false (then F is large, P value = righttail prob. small) 2 2 (betweengroups estimate of variance ) (withingroups 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 = righttail probability from F distribution (almost always the case for F and chisquared tests). • Software reports an “ANOVA table” that reports the SS values, df values, MS values, F test statistic, P value. 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 oneway ANOVA. ANOVA table Sum of Mean Source squares df square F Sig Betweengroups 1627 2 813 3.47 0.032 Withingroups 193901 828 234 Total 195528 830 The mean squares are 1627/2 = 813 and 193901/828 = 234....
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This note was uploaded on 07/14/2011 for the course STA 101 taught by Professor Alan during the Fall '10 term at University of Florida.
 Fall '10
 ALAN
 Correlation, Variance

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