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: F would be smaller here where diffs are less clear: The F-test

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

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

ANOVA: Model Group: IIIII I Grand Mean Yij Model:

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

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: Y ij = μ + α i + ε ij where μ & α i (of greatest interest) are structural components, and the ε ij 's are random components. assumptions on ε ij 's (& in effect on Y ij 's): l) ε ij 's mutually indep (i.e., randomly assign subjects to groups & one subject's score doesn't affect another's) 2) ε ij 's normally distributed with mean=0 (i.e., errors cancel each other) in each population. 3) homogeneity of variances: σ 2 1 =σ 2 2 =...=σ 2 ε <--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 Brief Explanation of Logic of F-test

Logic of F-test Y ij 's -- population of scores - vary around group mean because of ε ij 's: draw sample size n, compute stats like μ 's & MS A 's repeatedly draw such samples, compile distribution of stats (Keppel pp.94-96): means of the corresponding theoretical distributions are the "expected values“ E(MS S/A ) = σ 2 ε UE of error variance E(MS A ) = σ 2 ε + [nΣ(α i ) 2 ]/(a-1) not UE of error variance, but also in combo w. treatment effects F = MS A /MS S/A compare their E'd values:

Logic of F-test, cont’d if Ho : μ 1 =μ 2 =...=μ a were true, then nΣ(α i ) 2 /(a-1)=0 Ho above states no group diffs. This is equivalent to Ho: α 1 =α 2 =...=α a =0, again stating no group diffs. all groups would have mean μ+α i =μ+0=μ; no treatment effects. if (Ho were true and therefore) all α i 's=0, then (α i ) 2 =0, so nΣ(α i ) 2 /(a-1) would equal nΣ0/(a-1)=0. SO! under H 0 then, F would be: 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 (α i ) 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.)