Analysis of Variance: Inferences about 2 or More Means Chapter 13 Homework: 1, 2, 7, 8, 9

Analysis of Variance or ANOVA Procedure for testing hypotheses about 2 or more means simultaneously e.g., amount of sleep effects on test scores group 1: 0 hrs group 2: 4 hrs group 3: 8 hrs ~

ANOVA: Null Hypothesis Omnibus H0: all possible H0 H0: m1 = m2 = m3 Pairwise H0: compare each pair of means H0: m1 = m2 H0: m1 = m3 H0: m2 = m3 ANOVA: assume H0 true for all comparisons ~

ANOVA: Alternative Null Hypothesis Best way to state: the null hypothesis is false at least one of all the possible H0 is false Does not tell us which one is false Post hoc tests (Ch 14) ~

Experimentwise Error Why can’t we just use t tests? Type 1 error: incorrectly rejecting H0 each comparison a = .05 but we have multiple comparisons Experimentwise probability of type 1 error P (1 or more Type 1 errors) ANOVA: only one H0 ~

Experimentwise Error H0: m1 = m2 = m3 Approximate experimentwise error H0: m1 = m2 a = .05 H0: m1 = m3 a = .05 H0: m2 = m3 a = .05 experimentwise a » .15

ANOVA Notation Test scores 0 hrs 4 hrs 8 hrs 10 14 22 8 16 14 8 18 16 10 14 22 8 16 14 8 18 16 6 16 20 32 64 72

ANOVA Notation columns = groups jth group j = 2 = 2d column = group 2 (4hrs) k = total # groups (columns) k = 3 nj = # observations in group j n3 = # observations in group 3 ~

ANOVA Notation sj2 = variance of group j Xi = ith observation in group X4 = 4th observation in group Xij = ith observation in group j X31 = 3d observation in group 1 ~

ANOVA Notation subscript G = grand refers to all data points in all groups taken together Grand mean: SXij = sum of all Xi in all groups = 168 nG = n3 + n2 + n3 = 12 ~

Logic of ANOVA Assume all groups from same population with same m and s2 Comparing means are they far enough apart to reject H0? ask same question for ANOVA MORE THAN 2 MEANS ~

Logic of ANOVA ANOVA: 2 point estimates of s2 Between groups variance of means Within groups pooled variance of all individual scores s2pooled ~

Logic of ANOVA Are differences between groups (means) bigger than difference between individuals? If is H0 false then distance between groups should be larger We will work with groups of equal size n1 = n2 = n3 Unequal n different formulas same logic & overall method ~

Mean Square Between Groups also called MSB Mean Square Between Groups variance of the group means find deviations from grand mean

Mean Square Within Groups also MSW: Within Groups Variance Pooled variance pool variances of all groups similar to s2 pooled for t test formula for equal n only different formula for unequal n ~

F ratio F test Compare the 2 point estimates of s2

F ratio If H0 is true then MSB = MSW then F = 1 if means are far apart then MSB > MSW F > 1 Set criterion to reject H0 determine how much greater than 1 Test statistic: Fobs compare to FCV Table A.4 (p 478) ~

F ratio: degrees of freedom Required to determine FCV ~ df for numerator and denominator of F dfB = (k - 1) (number of groups) - 1 dfW = (nG - k) df1 + df2 + df3 +.... + dfk ~ ANOVA nondirectional even though shade only right tail F is always positive ~

TABLE A.4: Critical values of F (a = .05)

Partitioning Sums of Squares sum of squared deviations

Partitioning Sums of Squares Finding Mean Squares MS = variance

Partitioning Sums of Squares Calculating observed value of F

ANOVA Summary Table Output of most computer programs partitioned SS _________________________________ Source SS df MS F Between SSB dfB MSB Fobs Within SSW dfW MSW Total SST dfT