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The F-test

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: μ 1 = μ 2 = ... = μ a ) when MSbetw is large relative to MSw /in e.g., F would be big here, where group diffs are clear:

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The F-test

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  1. 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:

  2. F-test Intuitively:Variance Between vs. Variance Within price: high mdm low 1 sales, p(buy), F= Var Betw Grps ------------------- Var W/in Grps 2 3 4

  3. sales, p(buy) $Price: low medium high low medium high low medium high low medium high A) B) C) D) F-test Intuitively:Variance Between vs. Variance Within F= Var Betw Grps ------------------- Var W/in Grps

  4. ANOVA: Model Group: III II I Yij Grand Mean Model:

  5. Another take on intuition follows, more math-y, less visual

  6. 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'sare random components. assumptions on ε ij's(& in effect on Yij's): l) εij'smutually indep (i.e., randomly assign subjects to groups & one subject's score doesn't affect another's) 2) εij'snormally 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

  7. 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) = σ2εUE 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:

  8. 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: • that is, F would be a ratio of 2 independent estimates of error variance, so F should be "near" 1.

  9. 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,  will be >0 • and F will be : much >1 (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.)

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