1 INFERENCE FOR CONTRASTS (Chapter 4 Recall: A contrast s a lnear combnaton of effects wth coeffcents summng to zero: " where " = 0. Specfc types of contrasts of nterest nclude: Dfferences n effects Dfferences n means A specal type of dfference n means often of nterest n an experment wth a control group: The dfference between the control group effect and the mean of the other treatment effects. It's unbased: E( = E( = E ( = (µ + " = " µ + " = ", snce " µ = % ( µ = 0. " ' Recall: The least squares estmator of the contrast " s ˆ " =
3 4 Recall two model assumptons: Y t = µ +! + " t. The " t are ndependent random arables. Ths mples that the Y t 's are ndependent. Snce each s a lnear combnaton of the Y t 's for the th treatment group only, t follows that the 's are ndependent. Thus Var( = c Var( " = = " c. Recall two model assumptons: Y t = µ +! + " t. For each and t, " t ~ N(0, These mply: Y t ~ N(µ +!, Snce the Y t 's are ndependent, each, as a lnear combnaton of ndependent normal random arables, s also normal. Snce the contrast estmator s a lnear combnaton of the ndependent normal random arables, t too must be normal. Summarzng: ~ N( ", " c.
5 6 Standardzng, (* ~ N(0,1 Usng the estmate mse for, we obtan the standard error for the contrast estmator : se( = mse". Replacng the standard deaton of the contrast by the standard erron (* ges, whch no longer has a normal dstrbuton because of the substtuton of for. The usual trck works: In terms of random arables: SE( Y c " = ". (** = As mentoned before, / = SSE/[(n- ] ~ (n-/(n-. It can be proed that the numerator and denomnaton (** are ndependent. Thus
7 8 ~ t(n-. We can use ths as a test statstc to do nference (confdence nterals and hypothess tests for contrasts. Example: In the battery experment, treatments 1 and were alkalne batteres, whle types 3 and 4 were heay duty. To compare the alkalne wth the heay duty, we consder the dfference of means contrast D = (1/(! 1 +! - (1/(! 3 +! 4. Fnd a 95% confdence nteral for the contrast. State precsely what the resultng confdence nteral means. Perform a hypothess test wth null hypothess: The means for the two types are equal. Comments: 1. For a two-sded test, we could also do an F-test wth test statstc t.. A ery smlar analyss shows: The standard error for the th treatment mean µ +! mse s r. The test statstc µ has a t-dstrbuton wth n - degrees of freedom. So we can do hypothess tests and form confdence nterals for a sngle mean.. We haen't done examples of fndng confdence nterals or hypothess tests for effect dfferences or for treatment means, snce n practce n ANOVA, one does not usually do just one test or confdence nteral, so modfed technques for multple comparsons are needed.
The Problem of Multple Comparsons Suppose we want to form confdence nterals for two means or for two effect dfferences. If we formed a 95% confdence nteral for, say,! 1 -!, and another 95% confdence nteral for! 3 -! 4, we would get two nterals, say (a,b and (c,d, respectely. These would mean: 1. We hae produced (a,b by a method whch, for 95% percent of all completely randomzed samples of the same sze wth the specfed numben each treatment, yelds an nteral contanng! 1 -!, and. We hae produced (c,d by a method whch, for 95% percent of all completely randomzed samples of the same sze wth the specfed numben each treatment, yelds an nteral contanng! 3 -! 4. But there s absolutely no reason to belee that the 95% of samples n (1 are the same as the 95% of samples n (. 9 If we let A be the eent that the confdence nteral for! 1 -! actually contans! 1 -!, and let B be the eent that the confdence nteral for! 3 -! 4 actually contans! 3 -! 4, the best we can say n general s the followng: P(obtanng a sample gng a confdence nteral for! 1 -! that actually contans! 1 -! and also gng a confdence nteral for! 3 -! 4 that actually contans! 3 -! 4 = P(A%B = 1 - P((A%B C = 1 - P(A C B C = 1 - [P(A C + P(B C - P(A C % B C ] = 1 - P(A C - P(B C + P(A C % B C! 1 - P(A C - P(B C = 1-0.05-0.05 = 0.90 Smlarly, f we were formng k 95% confdence nterals, our "confdence" that for all of them, the correspondng true effect dfference would le n the correspondng CI would be reduced to 1 -.05k. Thus, other technques are needed for such "smultaneous nference" (or "multple comparsons". 10