8 The independence problem

Transcription

1 Noparam Stat 46/55 Jame Kwo 8 The depedece problem 8.. Example (Tua qualty) ## Hollader & Wolfe (973), p. 87f. ## Aemet of tua qualty. We compare the Huter L meaure of ## lghte to the average of coumer pael core (recoded a ## teger value from to 6 ad averaged over 8 uch value) ## 9 lot of caed tua. x <- c(44.4, 45.9, 4.9, 53.3, 44.7, 44., 5.7, 45., 6.) y <- c(.6, 3.,.5, 5., 3.6, 4., 5.,.8, 3.8) ## The alteratve hypothe of teret that the ## Huter L value potvely aocated wth the pael core. plot(x,y) cor.tet(x, y, method "kedall", alteratve "greater") ## > p.597 cor.tet(x, y, method "kedall", alteratve "greater", exact FALSE) # ug large ample approxmato ## > p.4765 ## Compare th to cor.tet(x, y, method "pearm", alteratve "g") cor.tet(x, y, alteratve "g") What the queto? What the frt thg to do? Kedall tau.4444 What level.9 tet? Reject f K> 4 Data: we obta bvarate obervato (X,Y) (X,,Y), oe obervato o each of ubject. Aumpto: They are a radom ample from a cotuou bvarate populato. 8. A dtrbuto free tet for depedece baed o g H: X ad Y radom varaable are depedet,.e. FX, Y ( x, y) FX ( x) FY ( y) for all (x,y).

2 Noparam Stat 47/55 Jame Kwo the Kedall populato correlato coeffcet {( Y Y )( X ) > } τ X f X ad Y are depedet (why?) P Doe τ mply depedece? 8.. Procedure Calculate the value of (-)/ pared g tattc Q (( X, Y ),( X j, Y j )) for < j, Q ( a, b),( c, d) ( d b)( c a) > ( d b)( c a) < where ( ) [ ] [ ] The Kedall tattc the K j + Q(( X, Y ),( X j, Y j )) Oe-ded upper tal tet: To tet H v the alteratve H: X ad Y are potvely correlated,.e. τ>, at the α-level of gfcace, Reject H f K k α (ue Table A.3) Iterpretato of τ >? Oe-ded lower tal tet: To tet H v the alteratve H: X ad Y are egatvely correlated,.e. τ<, at the α-level of gfcace, Reject H f K -k α (ue Table A.3) Two-ded tet: To tet H v the alteratve H3: X ad Y are depedet,.e. τ, at the α- level of gfcace, Reject H f K k α/ (ue Table A.3) 8.. Large-ample approxmato E K ( )( + 5) var K 8 ad the tadardzed vero * K EK K ~N(,) approxmately a ted to fty. var K / { } The ormal theory approxmato of the procedure follow accordgly. (how?)

3 Noparam Stat 48/55 Jame Kwo * K # of cocordat par K # of dcordat par K K -K * R output gve T K. 6 for the example. How to compute K from T? N(-)/ 9*8/ 36. K. K6-6. * To compute K, t eough to kow oly rak of X ad Y. (why?) * What the rage of K? Maxmum ad mmum? Whe do they occur? * Power ad ample ze reult: t there. Complcated. 8.3 Etmator (Kedall tau) The etmator of the Kedall populato correlato coeffcet τ K ˆ τ ( ) called Kedall ample rak correlato coeffcet * ˆ τ Stattc 8.3. Computer mplemetato R ad Mtab Well mplemeted. 8.4 A aymptotcally dtrbuto-free cofdece terval baed o the Kedall tattc Skp. 8.5 A dtrbuto free tet for depedece baed o rak (Spearma) Order the X value ad Y value ad call tehr rak R ad S each. The Spearma rak correlato coeffcet gve by

4 Noparam Stat 49/55 Jame Kwo r where D + + R 6 S ( ) ( S R. D ) Oe-ded upper tal tet: To tet H v the alteratve H: X ad Y are potvely aocated, at the α-level of gfcace, Reject H f r (ue Table A.3) r, α Iterpretato of τ >? Oe-ded lower tal tet: To tet H v the alteratve H: X ad Y are egatvely aocated, at the α-level of gfcace, Rject H f r r, α Two-ded tet: To tet H v the alteratve H3: X ad Y are depedet,.e. τ, at the α- level of gfcace, Rject H f r r, α 8.5. Large-ample approxmato E ( r ) var ( r ) ad the tadardzed vero * r E ( r ) / r ( ) r / ~N(,) approxmately a ted to fty. var r { ( )} The ormal theory approxmato of the procedure follow accordgly. (how?) * Motvato, term of D?? * Pearo product momet ample correlato coeffcet. * the Spearma rak corr. Coeff. I mply the Pearo corr. Coeff. Appled to the rak vector.

5 Noparam Stat 5/55 Jame Kwo 9 Regreo Skp! Comparg two ucce probablte.. Breat cacer data Lver Sca Patet Ye No Total Black 4 8 Whte Total I there a gfcace dfferece betwee the chace of whte patet recevg a ca ad the chace of a black patet recevg a ca?.. Death pealty ad gu regtrato Lver Sca Gu Regtrato Ye No Total Favor Oppoe Total I there a aocato betwee the atttude toward gu regtrato ad atttude toward the death pealty? What the dfferece betwee the two amplg cheme? P Pr(a black patet receve a lver ca) P Pr(a whte patet receve a lver ca) H: p >p. phat, phat, phat, Sdhat A. α.5 level tet- accept? P-value? Cofrm: χ4.85 ~ A

6 Noparam Stat 5/55 Jame Kwo..3 Data We oberve the outcome of. ad. depedet repeated Beroull tral, each wth ucce probablty p ad p repectvely. Succee Falure Total Sample O O. Sample O O. Total Aumpto A. O the umber of uccee oberved. depedet Beroull tral, each wth ucce probablty p. A. O the umber of uccee oberved. depedet Beroull tral, each wth ucce probablty p. A3. The two ample are depedet. Wat to tet: H p p : p p ot pecfed.. Approxmate tet ad cofdece terval for p -p (Pearo) Let D pˆ ˆ p where O O p ˆ ˆ, p.. It SD etmated by ˆ pˆ( pˆ) pˆ( pˆ) SD +.. where O + O pˆ (why?) +.. Uder H, the tadardzed vero of D pˆ pˆ A ~ N(,) approxmately a both ample ze goe to fty. ˆ SD Approxmate oe-ded upper-tal, lower-tal, two-ded tet

7 Noparam Stat 5/55 Jame Kwo To tet H veru H: p p >, <, at the approxmate α level of gfcace, Reject H f A z, A z /,. A zα, α α Approxmate CI for p -p gve by ~ p pˆ ± z SD where ˆ α / ~ pˆ ( pˆ ˆ ˆ ) p ( p ) SD + Why??.... x ch-quared tet of homogeety The equvalet of the large-ample two-ded tet. If H true, the bet etmator of p. p ˆ ad the expected value of the radom quatte O,..,O are etmated by E.. ˆ. p.... ad E, E, E? A meaure of dcrepacy betwee the oberved frequece ad the ther etmator uder H, the ch-quared tattc gve by ( O ) j Ej χ Iterpretato? Motvato?, j, E j It ca be how that A χ. A mple formula gve by ( χ O O O O.. ).... The two ded tet equvalet to the tet whch Reject H f χ χ α, (why?).. The x ch-quared tet of depedece Data: Nether of the four um. etc are fxed but each obervato from a geeral populato cro-clafed o the ba of two charactertc. C Not C Total D O O.

8 Noparam Stat 53/55 Jame Kwo Not D O O. Total.... p P(C ad D) p P(ot C ad D) p P(C ad ot D) p P(ot C ad ot D) H: the occurrece of the two charactertc are depedet. (More formally, H: p j p. xp.j ) The we have the ame procedure a above! * Dfferet amplg cheme. For homogeety, row colum are fxed. For depedece (cro-ectoal amplg), oly the total fxed. Yate correcto for cotuty Sample ze determato (p.467) The degree of aocato meaured by, e.g. the odd rato..3 Computer Implemetato R: ch.q Mtab:.3 A exact tet for the dfferece betwee two ucce probablte (Fher) The codtoal dtrbuto:.. x. x P ( O x.,.,.,. )... where max(,. _... ) x m(.,. ). Hypergeometrc dtrbuto. Fher exact tet judge whether O gfcatly mall or gfcatly large wth repect to the codtoal dtrbuto defed by th equato. To tet H v H : p <p, Reject H f O q α where q α choe o that Pr(O <q α.,.,.,.) α.