4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
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1 4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected by the expermenter and nferences are to be made only about those populatons, then the model s called a fxed effects model If nstead the populatons are representatve of other populatons that are not sampled, and the expermenter wshes to nfer propertes of all populatons, then the model s called a random effects model 52 Fxed Effects ANOVA 521 The Model Cell Means Model Y j = θ + ɛ j = 1, 2,, k, j = 1, 2,, n where θ are unknown populaton parameters, ɛ j are random errors, k s the number of dstnct populatons, and n s the sample sze n the th populaton Note, the sample szes may not be equal Note, f we assume E[ɛ j ] = 0, then the expected value of data s E[Y j ] = θ, j = 1, 2,, n We conclude that the θ are the populaton means, so θ s the populaton mean of populaton If only these k populatons are of nterest, then the θ are vewed as unknown constants n the fxed effects models If these k populatons are only representatve of a larger collecton of populatons, then the θ are vewed as k randomly sampled means, e θ are random varables n the random effects model For example, f treatment 1, 2, and 3 are appled to three random groups of patents, then θ 1, θ 2, and θ 3 are the unknown mean treatment responses and are fxed effects In contrast, f you sample college students and study ther grade pont average as a functon of the number of school days mssed n grades 1-12, then you can separate them nto populatons based on the number of days mssed, but the populaton means, θ for days mssed, are random varables representatve of a whole collecton {θ 0, θ 1,, θ M }, where M s the maxmum number of school days that can be mssed Clearly, n any reasonable sample sze, we don t expect to observe students from all possble populatons Instead, we observe students wth several values of and use statstcal nference to extend to the whole populaton Henceforth we focus on the fxed effects model Alternatve Parameterzaton Then, Often, you wll see another parameterzaton of the one-way ANOVA model Y j = µ + τ + ɛ j E[Y j ] = µ + τ where µ s the grand mean and τ s the unque effect of treatment Note, there are k + 1 parameters n ths model formulaton and ths leads to dentfablty problems
2 Data The data assocated wth ANOVA mght be summarzed n a table of the followng form: Treatment k y 11 y 21 y 31 y k1 y 12 y 22 y 32 y k2 y 2n2 y 3n3 y 1n1 y knk where Treatment s a label to descrbe the populatons and s borrowed from the common ANOVA applcaton of determnng whether dfferent treatments lead to better prognoss n medcal applcatons Example: heghts of sngers n a chor Suppose, for example, that you are studyng the heghts of sngers n a chor Your data table s below [Fnd the orgnal data at the Data & Story Lbrary] Soprano Alto Tenor Bass The treatments are snger types (soprano, alto, tenor, bass) In workng wth ths data, you mght be nterested n determnng whether the mean heghts of all treatments are the same Specfcally, you mght expect a sgnfcant dfference n the mean heghts of basses and sopranos, because most, f not all, of the former are male, and the latter are female 522 Identfablty Identfable Recall our overall framework We have a populaton and assocated wth t are unknown populaton parameter(s) θ We assume there s some probablty model that descrbes data X sampled from ths populaton The probablty model defnes the pdf f θ (x) (or pmf for dscrete outcomes) for the data 2
3 Defnton: dentfable A populaton parameter θ s dentfable f dstnct θ correspond to dstnct pdfs (or pmfs for dscrete random varables) That s, f θ θ, then the pdf of the data f θ (x) f θ (x) are dstnct functons For example, f µ 1 µ 2, then the correspondng normal pdfs are not the same: f µ1 (x) = 1 [ exp (x µ ] 1) 1 [ exp (x µ ] 1) = f µ2 (x) 2πσ 2σ 2πσ 2σ ndcatng that populaton mean of normally dstrbuted random varables s dentfable 1 dentfablty s a property of the model (not the estmates of the populaton parameter), so solvng dentfablty problems nvolves changng the model 2 f a model s not dentfable, then estmaton of or nference on ts populaton parameters s not possble Alternatve Parameterzaton s Overparameterzed In the alternatve formulaton, there are k + 1 parameters and k sample means avalable from the data The extra degree of freedom n the data ndcates that the model s undentfable More than one choce of (µ, τ 1,, τ k ) can lead to the same data One restrcton on the parameters must be added to make the model dentfable There are multple choces for that restrcton that change the way the parameters are nterpretted τ = 0 means that we can nterpret the τ as devatons from the overall mean attrbutable to each populaton τ 1 = 0 mght be useful f populaton 1 s the control group and we want to nterpret the τ, > 1 as devatons from no treatment 523 ANOVA Framework Assumptons 1 E[ɛ j ] = 0, Var(ɛ j ) = σ 2 < for all, j, Cov(ɛ j, ɛ kl ) = 0 for all, j, k, l wth j or k l 2 ɛ j N(0, σ 2 ) ndependent 3 Homoscedastcty: σ 2 = σ2 Comments: Assumpton 2 s requred for hypothess testng and confdence ntervals Wthout assumpton 2, we are lmted to do estmaton Wth assumpton 1 about varance, we can fnd the estmate wth mnmum varance Non-normalty can lead to dffcultes, but there are solutons for other knds of dstrbutons We wll not dscuss much here We can use CLT to get normalty on populaton means f n s large enough and the real dstrbuton s farly symmetrc Robustness of ANOVA to volatons of 2 depends on the extent to whch 3 s true For ths reason, people wll often transform the Y random varables to acheve 3 so that they do not need to worry so much about normalty of ther data 3
4 Classc ANOVA Hypothess H 0 : θ 1 = θ 2 = = θ k H A : θ θ j for some j Ths hypothess s not often so nterestng Take the example of comparng several treatments One may often nclude a control as a treatment to make sure that the experment runs as planned One knows before even collectng data that the control should have a dfferent outcome compared to the rest, whch means ths classc H 0 wll always be rejected We mght stll llke to know f θ 2 θ 3 Contrast Defnton: Let t = (t 1,, t k ) be a vector of random varables, ther realzatons, parameters, or statstcs Let a = (a 1,, a k ) be constants, then a t s a lnear combnaton of t s If a = 0, then the lnear combnaton s called a contrast Classcal ANOVA Hypothess n Terms of Contrasts Theorem: θ 1 = = θ k f and only f a θ = 0 for all a A, where A = {a = (a 1,, a k ) : a = 0} Proof: The forward mplcaton s obvous a θ = θ a = 0 The reverse mplcaton s also qute easy Consder a (1) = (1, 1, 0,, 0) A Ths one shows θ 1 = θ 2 Smlarly, a (2) = (0, 1, 1, 0,, 0) shows θ 2 = θ 3 In general, the set a (1), a (2),, a (k 1) spans the space A Therefore, all possble equaltes encoded n θ 1 = = θ k are mpled by combnng these vectors approprately 524 Inference on Contrasts Inference on Contrasts Under the ANOVA assumptons, we have Y j N(θ, σ 2 ) Defne the populaton sample means and note by the CLT Also, for any a, Ȳ = 1 n n Y j Ȳ N(θ, σ 2 /n ) a Ȳ N(, ) 4
5 wth mean and varance [ E a Ȳ ] = a θ ( Var a Ȳ ) = σ 2 a 2 n Z test for Generc Contrast Gven the above, the statstc Z = a Ȳ a θ N(0, 1) a σ 2 n t-test for Generc Contrast But of course, we don t usually know σ 2 Instead, we use S = 1 n 1 n ( Yj Ȳ ) 2 whch s unbased for σ 2 (σ 2 wth heteroscedastcty) and also has dstrbuton (n 1)S 2 σ 2 χ 2 n 1 If assumpton 3 of homoscedastcty apples, then we can pool sample varances to get a better estmate of σ 2 Namely, wth N = n, we use the pooled sample varance S 2 p = 1 N k (n 1)S 2 = 1 N k n ( Yj Ȳ ) 2 j=1 Because the S 2 are ndependent, we also have (N k)s 2 p σ 2 χ 2 N k Also, because S 2 p s ndependent of Ȳ, we have that statstc a Ȳ a θ S p a 2 n whch allows confdence ntervals of the usual form t N k a Ȳ t N k,α/2 S p a 2 n a θ a Ȳ + t N k,α/2 S p a 2 n 525 Classcal ANOVA Parttonng Varance 5
6 Often, ANOVA s presented as a way of parttonng the varance The total varablty can be summarzed as the total sum of squares n SS tot = (Y j Ȳ ) 2 j=1 Note, ths s just (N 1 tmes the combned sample varance) By addng and subtractng the sample means Ȳ, we can partton the total varance nto parts n [ ( Ȳ Ȳ ) 2 + (Y j Ȳ ) 2] j=1 Expand the quadratc and recognze the cross-term becomes 0 because to fnd n j=1 n (Y j Ȳ ) 2 = j=1 Interprettng each part of ths sum, we have (Y j Ȳ ) = n Ȳ n Ȳ = 0 n (Ȳ Ȳ ) 2 + SS tot = SS treatment + SS E n (Y j Ȳ ) 2 j=1 where SS treatment s the sum-of-squares due to treatments (e between treatments) and SS E s the sum-ofsquares due to error (e wthn treatments) There are N observatons, so there are N 1 df for SS tot There are k treatments, so there are k 1 df for SS treatment Wthn the th treatment, there are n 1 df for a total of (n 1) = N k df wthn treatments for the SS E Estmates of σ 2 Under the ANOVA assumptons, SS E N k uses all the data to estmate populaton varance σ2 (t s the pooled sample varance) E [ SSE N k ] = 1 k N k E[(Y j Ȳ ) 2 ] = 1 k N k E[(n 1)S 2] = 1 k N k (n 1)σ 2 constant varance assumpton = σ 2 Under the classc ANOVA null hypothess H 0 : θ 1 = = θ k = θ Ȳ N(θ, σ 2 /n ) or n Ȳ N ( n θ, σ 2) Intutvely, snce Ȳ provdes a sample estmate of θ, we have [ ] [ ] SStreatment E = 1 k 1 k 1 E ( n Ȳ n Ȳ ) 2 [ ] = 1 k 1 E n (Ȳ Ȳ ) 2 = σ 2 [If not convnced, you can work out the detals by movng the expectaton nto the sum and usng model assumptons] so Fnally, also under H 0, [ ] SStot E = 1 N 1 N 1 Y j N(θ, σ 2 ) n (Y j Ȳ ) 2 = σ 2 j=1 s the tradtonal sample varance, whch estmates the populaton varance 6
7 F Test for Testng Classcal ANOVA Hypothess These estmates of σ 2 provde the bass of the F test for the classc hypothess Under H 0, recall that SStot σ 2 χ 2 N 1 Theorem: Cochran s Theorem Let Z N(0, 1) for = 1,, ν and ν Z 2 = Q 1 + Q Q s wth s ν Then, Q 1, Q 2,, Q s are ndependent χ 2 random varables wth ν 1, ν 2,, ν s df, respectvely f and only f ν = ν 1 + ν ν s Proof: Omtted Snce (N k) + (k 1) = N 1, Cochran s theorem mples Therefore, SS treatment k 1 SS E N k F = SS treatment/(k 1) SS E /(N k) = χ 2 k 1 = χ 2 N k F k 1,N k If the alternatve hypothess s correct, then we expect SS treatment /(k 1) to overestmate the populaton varance, so large values of ths statstc wll ndcate problems wth H 0, thus rejecton s accordng to a one-taled test when F > F k 1,N k,α The ANOVA Table The one-way ANOVA analyss s summarzed n the ANOVA table Source of Varaton Sum of Squares Degrees of Freedom Mean Square F Between treatments SS treatment k 1 MS treatment = SS treatment k 1 Wthn treatments SS E N k MS E = SS E N k Total SS tot N 1 F = MS treatment MS E 7
The experimental unit of a study is the object on which measurements are taken.
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