Analysis of Variance

Comparing Several Groups Section: M 8-10 M 12-2 T 9-11 W 8-10 R 9-11 Exam 1 Mean: 76.6% 82.5% 83.7% 79.5% 82.9% Do the group means differ? Naive approach Independent-samples t-tests of all pairs Each test doesn't use all data  Less power 10 total tests  Greater chance of Type I error Analysis of Variance (ANOVA) Single test for any group differences Null hypothesis: All means are equal Works using variance of the sample means Also based on separating explained and unexplained variance

Variance of Sample Means Say we have k groups of n subjects each Want to test whether mi are all equal for i = 1,…,k Even if all population means are equal, sample means will vary Central limit theorem tells how much: Compare actual variance of sample means to amount expected by chance or Estimate s2 using MSresidual Test statistic: If F is large enough Sample means vary more than expected by chance Reject null hypothesis m M1 M2 M3 Mk

Residual Mean Square Best estimate of population variance, s2 Based on remaining variability after removing group differences Extends MS for independent-samples t-test 2 groups: k groups:

Example: Lab Sections Group means: Variance of group means: Var(M) expected by chance: Instead: Estimate of s2: Ratio of between-group variance to amount expected by chance: Fcrit = 2.46 F = .96 P(F4, 105) F = .96 p = .44

Partitioning Sum of Squares General approach to ANOVA uses sums of squares Works with unequal sample sizes and other complications Breaks total variability into explained and unexplained parts SStotal Sum of squares for all data, treated as a single sample Based on differences from grand mean, SSresidual (SSwithin) Variability within groups, just like with independent-samples t-test Not explainable by group differences SStreatment (SSbetween) Variability explainable by difference among groups Treat each datum as its group mean and take difference from grand mean

Magic of Squares M1 SStotal = S(X – M)2 ×5 M M3 ×5 SSresidual = Si(S(Xi – Mi)2) ×6 ×4 M4 M2 SStreatment = Si(ni(Mi – M)2)

Test Statistic for ANOVA Goal: Determine whether group differences account for more variability than expected by chance Same approach as with regression Calculate Mean Square: H0: MStreatment ≈ s2 Estimate s2 using Test statistic:

Lab Sections Revisited Two approaches Variance of section means Mean Square for treatment (lab section) Variance of group means Var(76.6 82.5 83.7 79.5 82.9) = 8.69 Expected by chance: s2/22 Var(M) vs. s2/22, or 22*var(M) vs. s2 Mean Square for treatment 22 ` mean(M)

Two Views of ANOVA Simple: Equal sample sizes General: Using sums of squares m M1 M2 M3 Mk If sample sizes equal:

Degrees of Freedom dftreatment = k – 1 SStotal dftotal = Sni – 1 dfresidual = S(ni – 1) = Sni – k .