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The F-test why? later in gory detail now? brief explanation of logic of F-test for now -- intuitive level: F is big (i.e., reject Ho: μ1 = μ2 = ... = μa) when MSbetw is large relative to MSw/in e.g., F would be big here, where group diffs are clear: F would be smaller here where diffs are less clear: F-test Intuitively: Variance Between vs. Variance Within x1 x2 price: high mdm x3 1 F= Var Betw Grps ------------------Var W/in Grps 2 3 4 low sales, p(buy), F= Var Betw Grps ------------------Var W/in Grps F-test Intuitively: Variance Between vs. Variance Within sales, p(buy) $Price: low medium high A) low medium high B) low medium high C) low medium high D) ANOVA: Model Group: x1 III x2 II I x3 Grand Mean Model: Yij i ij Yij Another take on intuition follows, more mathy, less visual Brief Explanation of Logic of F-test For the simple design we've been working with (l factor, complete randomization; subjects randomly assigned to l of a group--no blocking or repeated measures factors, etc.), model is: Yij = μ + αi + εij where μ & αi (of greatest interest) are structural components, and the εij's are random components. assumptions on ε ij's (& in effect on Yij's): l) εij's mutually indep (i.e., randomly assign subjects to groups & one subject's score doesn't affect another's) 2) εij's normally distributed with mean=0 (i.e., errors cancel each other) in each population. 3) homogeneity of variances: σ21=σ22=...=σ2ε <--error variance Use these assumptions to learn more about what went into ANOVA table. In particular -- test statistic F Later - general rules to generate F tests in diff designs Logic of F-test • Yij's -- population of scores - vary around group mean because of εij's: • draw sample size n, compute stats like μ's & MSA's repeatedly draw such samples, compile distribution of stats (Keppel pp.94-96): • means of the corresponding theoretical distributions are the "expected values“ • E(MSS/A) = σ2ε UE of error variance • E(MSA) = σ2 ε + [nΣ(αi)2]/(a-1) not UE of error variance, but also in combo w. treatment effects • F = MSA/MSS/A compare their E'd values: 2 n ( ) i 2 + E( MS A ) a -1 = E( MS S/A ) 2 Logic of F-test, cont’d • if Ho : μ1=μ2=...=μa were true, then nΣ(αi)2/(a-1)=0 • Ho above states no group diffs. This is equivalent to Ho: α1=α2=...=αa=0, again stating no group diffs. • all groups would have mean μ+αi=μ+0=μ; no treatment effects. • if (Ho were true and therefore) all αi's=0, then (αi)2=0, so nΣ(αi)2/(a-1) would equal nΣ0/(a-1)=0. • SO! under H0 then, F would be: 2 +0 ˆ error 2 ˆ error • that is, F would be a ratio of 2 independent estimates of error variance, so F should be "near" 1. Logic of F-test, cont’d • When F is large, reject Ho as not plausible, because: • when Ho is not true, each (αi)2 will be > or = 0, n( i )2 will be >0 a -1 • and F will be : 2 + som ething ˆ error much >1 2 ˆ error (for more on the intuition underlying the F-test, see Keppel pp.26-28; and for more on expected mean squares, see Keppel p.95.)