17 Paired t-test. 18 Correlation and Regression

Transcription

1 7 Pare t-test Sometmes we have to eal wth ata that come lke pars. For example tal weght a fal weght for a grwoth harmoe. We have observatos (x,y ). They are very hgly correlate. So f we wat to test that the harmoe has o effect, or equvaletly that meas of {x } a {y } are the same, we caot affor to assume that {x } a {y } are epeet. O the other ha we ca form the ffereces = x y a test for the mea of { } to be 0. Ths s ow a smple t test. 8 Correlato a Regresso Ofte we have two relate varables X a Y a oly oe of them say X s observe. We woul lke to use the observe value of X to prect Y. If we kow the jot strbuto of (X, Y ), we ca eterme the cotoal strbuto of Y gve X a use t as a gue to our precto. The expectato of the cotoal strbuto or the cotoal expectato s a reasoable guess regarg what we mght expect for Y. The cotoal expectato has the property that t mmzes E(Y f(x)) over all fuctos f(x) thatepeolyox. I practce oe ofte mmzes E(Y f(x)) over a specfe lmte class of fuctos f( ). I lear regreeso oe lmts the choce of f to lear fuctos of X of the form a + bx where a a b are costats. The mmzato f E (Y a bx) a,b ca be explctly carre out. Dfferetatg wth repect to a a b we get the equatos or E (Y a bx) =0 E (Y a bx)x =0 E Y = a + be X E XY = ae X + be X 36

2 Solvg the equatos for a a b we get E XY E X E Y ˆb = E X (E X ) â = E Y ˆbE X We ca therefore wrte the regresso le as = Var X Y E Y = Var X X E X For ay two raom varables X a Y, the covarace s efe by =EXY EXEY. Note that Cov XX=EX (EX) = Var X. Oe ca ecompose Y EY as Y EY =ˆb X EX + Y â ˆbX Because the cross term vashes, we get Var Y = ( Cov XY ) Var X Var Y Var Y + ( ) Var X Var Y Var Y The lear correlato coeffcet ρ betwee X a Y s efe as Cov XY ρ = Var X Var Y We ca rewrte the earler relato as Var Y = ρ Var Y +( ρ )Var Y The frst term s the amout of reucto the varace ue to prectable a the seco term s the resual varace. ρ s the proporto of the varace that s reuce a ( ρ ) s the resual proporto. From Schwartz s equalty t s clear that ρ. If we have observatos (x,y ) from a bvarate populato, we ca estmate the meas, varaces a covaraces by ther correspog sample 37

3 values x = x ȳ = y s x = x x s y = y ȳ c x,y = x y xȳ ˆb = c x,y s x r = c x,y s x s y These are clearly cosstet estmators of the correspog populato values. We ca terchage the roles of X a Y a there s the regresso le X E X = Var Y ( Y E Y ) Whle both les pass through ( E X,E Y ), they have geeral fferet slopes uless Cov XY Var X = Var Y Cov XY or ρ = whch correspos to a exact lear relato betwee X a Y. 9 Multvarate Normal Dstrbutos Just as the famly of Normal strbutos exe by ther meas a varaces play a mportat role the stuy of real value raom varables, the 38

4 multvarate ormal strbutos are cetral to the stuy of raom vectors. O R, a Normal strbuto s specfe by ts probablty esty f(x,...,x )=k exp Q(x a,...,x a ) where a = (a,...,a ) s a locato or ceterg parameter a Q(x) = Q(x,...,x ) s a postve efte quaratc form Q(x) =< x,cx >=,j c,jx x j eterme by the symmetrc matrx C = {c,j }. The ormalzg costat k s eterme so that f(x)x = R Clearly by traslato a orthogoal rotato the tegral ca be calculate as k exp R = λ y y =(π) Π = λ = a k =( π ) Π = λ =( π ) (Det C) Here λ are the egevlues of C so that Π = λ = Det C. Sce f(x) s symmetrc arou x = a, t s clear that x f(x)x = a + (x a )f(x)x = a R R a thus {a } are the meas of the compoets {x }. I orer to calculate the varaces a covaraces t s better to calculate the momet geeratg fucto exp< θ,x>f(x)x R = k exp R Q(x a + C θ) exp< a,θ>+ Q(C θ) =exp<a,θ>+ <C θ, θ > By fferetatg wth respect to θ we ca calculate E x = a E x x j = a a j + c,j Cov x x j = E x x j a a j = c,j 39

5 Where c,j s the, jth etry of the verse C of C = {c,j }. It s more atural to parametrze the Multvarate Normal Dstrbutos by ther meas a covaraces a the esty takes the form f(a, Σ,x)= {a } = {E x } Σ={σ,j } = {Cov x x j } ( π) Det Σ exp < Σ (x a), (x a) > We hace assume that Σ s postve efte. I geeral t oly ees to be postve efte. If Σ has rak r<the ormal strbuto s egeerate a lves o a hyperplae of meso r a the esty ca be wrtte ow relatve to a choce of r coorates o the hyperplae. If the rak s 0, the Σ = 0 a the Normal strbuto egerates to a pot mass of at the mea. If =, the covarace matrx Σ ca be wrtte as σ x ρσ x σ y ρσ x σ y σ y wth Σ gve by σx ρ ρ σ xσ y ρ σ xσ y σ y 0 Testg for Correlato. If we have epeet observatos from a bvarate ormal strbuto wth meas µ x,µ y, varaces σx,σ y a correlato coeffecet ρ oe mght wat to test that ρ = 0 The test aturally wll be base o the statstc r = = x y xȳ s x s y where s x a s y are the sample staar evatos of x a y. I orer to ece o the crtcal rego we ee to eterme the strbuto of r uer 40

6 the ull hypothess. Sce r s uchage by ay chage of org a/or scale of the observatos, we ca assume wth out loss of geeralty that x,...,x a y,...,y are two epeet sets of epeet observatos from the staar ormal strbuto wth mea 0 a varace. Actually we wll assume that y,...,y are just arbtrary costats a show that the strbuto of r r s t wth egrees of freeom o matter what these costats are. The as log as x s a y s are epeet the strbuto of wll be t.ifweeotebya = y ȳ s y the a =0a a =. r r r = a x sx Let us chage coorates by a orthogoal trasformato z = Sx wth S gve by... a a... a After the frst two rows that form a orthoaormal set of vectors the rest of the matrx s complete to be orthogoal by selectg the rows to form a complete orthoormal set. I terms of z, whch are aga epeet staar ormals, z r = z + + z a r = z r z z whch has a t strbuto wth egrees of freeom. Large Sample Tests for Correlato. Oe ca calculate the asymptotc rbuto of r for large, the geeral case of ρ 0. If we efe U = s x,u = s y a U 3 = x y xȳ a eote by a,a a a 3 ther populato values, aρ, { (U a )} 4

7 have a jot ormal strbuto. The covarace matrx s easly calculate to be ρ ρ A = ρ ρ ρ ρ +ρ From r = U 3 U U we see that (r ρ) s asymptotcally ormal wth varace <c,ac>= ( ρ ) wth c =( ρ, ρ r, )=(, U r U, r U 3 ) (,,ρ) If we coser z = r log wth z +r ρ = ρ log the (z z +ρ ρ ) s asymptotcally ormal wth mea 0 a varace. Cofece Itervals. A cofece terval at level α s a raom terval I such that P θ θ I α for all θ. For example f X,lots,X are epeet observatos from N(µ, σ ) a terval of the form x ks, x + ks cotasµ f x µ k. s The strbuto of x µ s sat wth egrees of freeom. We ca eterme k α, from th e tables so that P x µ k =α. The terval s x sk α,, x+sk α, works. Essetally the terval cossts of all the possble values of the parameter θ for whch the ull hypothess that the true value s θ s ot rejecte at α level of sgfcace. I large samples the cofece tervals look lke ˆθ kα σ, θ + kα σ, where k α s eterme from the ormal table a σ = σ(ˆθ) s the varace of the lmtg ormal strbuto of (ˆθ θ). 4