# c i. INFERENCE FOR CONTRASTS (Chapter 4) It's unbiased: Recall: A contrast is a linear combination of effects with coefficients summing to zero:
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1 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 " =
2 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.
3 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
4 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.
5 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 = = 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 k. Thus, other technques are needed for such "smultaneous nference" (or "multple comparsons". 10
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