Quantitative Genomics and Genetics BTRY 4830/6830; PBSB

Transcription

1 Quntttv Gnomcs n Gntcs BTRY 4830/6830; BSB Lctur8: Logstc rgrsson II Json Mzy jgm45@cornll.u Aprl 0, 208 (T 8:40-9:55

2 Announcmnts Grng (homworks n mtrm rojct wll b ssgn NEXT WEEK (!! Schul for th rst of th smstr (s nt sl

3 Schul Aprl 2 Gnom-W Assocton Stus (GWAS IX: Hplotyp tstng, ltrntv tsts Aprl 7 rojct Assgn Mnmum GWAS nlyss 3 Aprl 9 Avnc topcs I: M Mols Aprl 24 Avnc topcs II: Multpl rgrsson (pstss n multvrt rgrsson 4 Aprl 26 MAING LOCI: BAYESIAN ANALYSIS Bysn nfrnc I: nfrnc bscs n lnr mols My Bysn nfrnc II: MCMC lgorthms 5 My 3 EDIGREE / INBRED LINE ANALYSIS Bscs of lnkg nlyss / Inbr ln nlyss My 8 CLASSIC QUANTITATIVE GENETICS rojct Du Atv gntc vrnc n hrtblty 6

4 Rvw: Cs / Control hnotyps Whl lnr rgrsson my prov rsonbl mol for mny phnotyps, w r commonly ntrst n nlyzng phnotyps whr ths s NOT goo mol As n mpl, w r oftn n stutons whr w r ntrst n ntfyng cusl polymorphsms (loc tht contrbut to th rsk for vlopng ss,.g. hrt ss, bts, tc. In ths cs, th phnotyp w r msurng s oftn hs ss or os not hv ss or mor prcsly cs or control Rcll tht such phnotyps r proprts of msur nvuls n thrfor lmnts of smpl spc, such tht w cn fn rnom vrbl such s Y(cs n Y(control 0

5 Rvw: lnr vs. logstc Rcll tht for lnr rgrsson, th rgrsson functon ws ln n th rror trm ccount for th ffrnc btwn ch pont n th pct vlu (th lnr rgrsson ln, whch w ssum follow norml For logstc rgrsson, w us th logstc functon n th rror trm mks up th vlu to thr 0 or : Y Y X X Rcll n both of ths css w r plottng vrsus just X (bcus th pctur s sr to unrstn but th rl rgrsson (lnr or logstc s multpl rgrsson so th rl pctur s Y vs th X n X vrbls

6 Rvw: clcultng th componnts of n nvul I Rcll tht n nvul wth phnotyp Y s scrb by th followng quton: Y X Y (Y X Y,,,, To unrstn how n nvul wth phnotyp Y n gnotyp X brks own n ths quton, w n to consr th pct (prct! prt n th rror trm (w wll o ths sprtly

7 Rvw: clcultng th componnts of n nvul II For mpl, sy w hv n nvul tht hs gnotyp AA n phnotyp Y 0 W know X - n X - Sy w lso know tht for th populton, th tru prmtrs (whch w wll not know n prctc! W n to nfr thm! r: W cn thn clcult th X n th rror trm for : Y,,,, 0 0.2( 2.2( ( 2.2(

8 Rvw: clcultng th componnts of n nvul III For mpl, sy w hv n nvul tht hs gnotyp AA n phnotyp Y W know X - n X - Sy w lso know tht for th populton, th tru prmtrs (whch w wll not know n prctc! W n to nfr thm! r: W cn thn clcult th X n th rror trm for : Y,,,, 0.2( 2.2( ( 2.2(

9 0.2(02.2(0.2 X A A X k T j logstc( kx,, ( [t] T X [ W] 0. (76 2 (y ( 2 (, B r(b k (, B0.2 r(bkk k 0.2(2.2( (56 (56 0. rfor tks on of two vlus, whch s th rnc btwn, 0.9 (77 ( ( r(b ( r(b, ( k k ( ( ( r(b ( r(b (69 k k gnotyp 0.2 t n on or zro (s clss nots for grm. (DD 0.2(2.2(,, r(b r(bkk, (57 0.9,, B Bkk (57, 0. (78 Dth rror r(z brn(p (70 D Rcll tht trm s thr th ngtv of X(79 whn rnom vrbl hs Brnoull strbuton. Not tht 0ths r( brn(p X X fd > 0 ( fd > 0 (58 whn Y s on: mn( B,, B Y s zro n X 2 2 mn(, B by prmtrz sngl 2prmtr: T2(y ( [TW] ( (Y 0 X (7 r( brn(p X X (80 D 0 fd <0 0 (59 fd < (59,, Y X mn( B mn( B0, (Y X (Y (72 X(8 22,,B ,,, A logstc( XX,, XX (60 A logstc( (60 Aj,, (Y X (82,, vrbl wll tk th vlu 0.4 th probblty tht th rnom For A thantr strbuton populton, rcll tht logstc( of th X X (6 ( p? followng logstc( X,,, X (6 tks, j th,, mtr Ths vlu:,,,, ( (73 0. vrbl wll tk th (62 Z X (62 X (83 th probblty tht th rnom vlu,,,, p logstc( X X (23 tr p? Ths tks th followng vlu: brn(p (63 ( r(z 0.6 brn(p (74 Rvw: th rror trm p X X (24 5 p logstc( X (23 0. For 0.4mpl: (75 4 wth probblty logstc( X logstc( X X (24 wth0.9 stc( Xp0. X X probblty logstc( X (76 or rnt pnng on th pct vlu of th phnotyp p 0. (25 r( brn(p X X 0.9 (77 spcfc t wth gnotyp.

10 A Aj X logstc( X (22 k kx,, ( ( ( 2.2( 0.2 2, B r(b ( (, Bkk0.2(2.2( r(b 0.2 kk (56 (56 0. rfor tks on of two vlus, whch s th rnc btwn 0.9r(B (77 ( ( r(bkk( ( r(b r(b 0.2(02.2(0.2 k ( 0( ( ( k t gnotypn on or zro (s 0.2 clss nots for grm. 0.2(2.2( (68 D r(b (57 0.9,, B k k D B r(b (57 0.2(02.2( (78 k k D D Rcll tht th rror trm sfd thr ofnot tht X(79 whn r(z brn(p rnom vrbl hs Brnoull strbuton. 0ths r( 0th ngtv brn(p X X > (58 0.2(2.2( fd > 0 (58 mn( B22n - whn Y s on: Y s zro BX by 2prmtr: mn(,,b prmtrz,sngl 2 (69 [t] T T 0.2(2.2( 0.2 (Y 0 X (80 r( brn(p X X ( [ W] (y D 0 fd<<0 0 (59 fd (59 Y X mn( B, B22 mn( (Y r(z brn(p (Y,, (70 22,,B 0 X X(8 A logstc( (60 AA logstc( XX (60 jt XX,,,, ( T,, (Y X (82,, tht vrbl [ probblty W] (y ( (7 tht th rnom wll tk th vlu th probblty th rnom vrbl wll tk th vlu,, For A thantr strbuton populton, rcll tht logstc( ofth X X (6 ( p? followng logstc( X,,, X (6 tks, j th,, vlu: mtr Ths r p? Ths tks th vlu:,,,,, followng, Z, X (62 X (62 (83, (72, ( p logstc( X, X,,, (23 p logstc( X (23 brn(p (63 r(z X brn(p (63 5 Rvw: th rror trm I p 5 X (24 ( (73 (24 p X 0.6 mpl: X logstc( For X 4 wth probblty logstc( X p (25 stc( X X wth 0.4probblty logstc( X(74 or rnt pnng on th pct vlu of th phnotyp p 0.6 ( (75 0. t wth spcfc gnotyp.

11 Rvw: Notton Rmmbr tht whl w r plottng ths vrsus just X, th tru plot s vrsus BOTH X n X (hrr to s wht s gong on For n ntr smpl, w cn us mtr notton s follows: X (X X X X E(y ( 2 6 4!,,,,. n, n, n, n, 3 7 5

12 Infrnc Rcll tht our gol wth usng logstc rgrsson ws to mol th probblty strbuton of cs / control phnotyp whn thr s cusl polymorphsm To us ths for GWAS, w n to tst th null hypothss tht gnotyp s not cusl polymorphsm (or mor ccurtly tht th gntc mrkr w r tstng s not n LD wth cusl polymorphsm!: c 0 0 H 0 : 0\ 0 To ssss ths null hypothss, w wll us th sm pproch s n lnr rgrsson,.. w wll construct LRT lklhoo rto tst (rcll tht n F-tst s n LRT! W wll n MLE for th prmtrs of th logstc rgrsson for th LRT

13 MLE of logstc rgrsson prmtrs Rcll tht n MLE s smply sttstc ( functon tht tks th smpl s n nput n outputs th stmt of th prmtrs! In ths cs, w wnt to construct th followng MLE: MLE( ˆ MLE( ˆ, ˆ, ˆ To o ths, w n to mmz th log-lklhoo functon for th logstc rgrsson, whch hs th followng form (smpl sz n: l( n y ln( (,, ( y ln( (,, Unlk th cs of lnr rgrsson, whr w h clos-form quton tht llows us to plug n th Y s n X s n rturns th bt vlus tht mmz th log-lklhoo, thr s no such smpl quton for logstc rgrsson W wll thrfor n n lgorthm to clcult th MLE

14 Algorthm Bscs lgorthm - squnc of nstructons for tkng n nput n proucng n output W oftn us lgorthms n stmton of prmtrs whr th structur of th stmton quton (.g., th log-lklhoo s so complct tht w cnnot Drv smpl (clos form quton for th stmtor Cnnot sly trmn th vlu th stmtor shoul tk by othr mns (.g., by grphcl vsulzton W wll us lgorthms to srch for th prmtr vlus tht corrspon to th stmtor of ntrst Algorthms r not gurnt to prouc th corrct vlu of th stmtor (!!, bcus th lgorthm my convrg (rturn th wrong nswr (.g., convrgs to locl mmum or os not convrg! n bcus th comput tm to convrg to ctly th sm nswr s mprctcl for pplctons

15 IRLS lgorthm I For logstc rgrsson (n GLM s n gnrl! w wll construct n lgorthm to fn th prmtrs tht corrspon to th mmum of th log-lklhoo: l( n y ln( (,, ( y ln( (,, For logstc rgrsson (n GLM s n gnrl! w wll construct n Itrtv R-wght Lst Squrs (IRLS lgorthm, whch hs th followng structur:. Choos strtng vlus for th s. Snc w hv vctor of thr s n our cs, w ssgn ths numbrs n cll th rsultng vctor [0]. 2. Usng th r-wghtng quton (scrb nt sl, upt th vctor. 3. At ch stp t>0 chck f [t] (.. f ths r ppromtly qul usng n pproprt functon. If th vlu s blow fn thrshol, stop. If not, rpt stps 2,3.

16 Stp : IRLS lgorthm Ths r smply vlus of th vctor tht w ssgn (!! In on sns, ths cn b nythng w wnt (!! lthough for lgorthms n gnrl thr r usully som rstrctons n / or crtn strtng vlus tht r bttr thn othrs n th sns tht th lgorthm wll convrg fstr, fn mor optml soluton tc. In our cs, w cn ssgn our strtng vlus s follows: [0]

17 [t] [T W] T (y (,, Y (,, X,,,, (,,, ( Stp 2: IRLS lgorthm, (, ( Y, (,, ( (Y X (8, (,,,, (8, ( Y,,,, 0.6,, lcton, n mtr ton, w cn r-wrt ths s: Lt s consr th qutons,,,, (82,,, w n for vlu 0.4 wht w choos 2, w wll upt (,prouc nwmttr of thvlus vctor usngbcus, stp,, osn t 0.4 At r(z brn(p th sp ths tn to stop!: convrg. W cou l(ˆ yth followng l( ˆ, ˆ, ˆquton (83 y (thn o ths gn ntht gn untl w 0 X, l(, (Y ˆ ˆ (Y ˆ y 2 l( ˆ3 (83 0 X,, y 7 T T ˆ0 y l(7ˆ, 0,[t] 6 l( 0 y (84 [ W] (y ( 2, l(2,ˆ07 (Y X 2 (Y X ˆ 6 7 y4 l(5, 0, 0 y ( ,.. 7,,,,, ,,.,, [0] 2, (,, Howvr, gnrl, w o hv to b crful t,, n (2,, (,,,,, trcks, r,.n,.. 2, 2, n r possbl n thr som to choos n, n lgorthms clss.... multplcton, n, n, n mtrton, w (( r-wrt tton, mtr cn ths s: n, n, ( upt g compct mtr notton: For stp (2, w wll th prmtr n quton: y,, W2 3 n (,, ( y(,, D7 y [t] or[t] T y6 ln 2 6 y2 7 W 2, 2, or[t] or[t] D 2 y ln 4 5 ( y (4. 7,,,, or[t] or[t], , or[t] ( w of th, componnt tht ll W hv, sn. Not yn n, ( n, (,,,, y n our 0.4 hv us y n lnr rgrsson q n y ln yn (squr or[t] ˆ ˆ or[t] ˆ y mtr tht h l(or[t] ˆ y W l(,n,n by y ln ( mtr s, or[t], or[t] ((Wor[t] 0 for 6 j n ch of th gonl j,, l(ˆ0 y l( ˆ, 0, 0 y t usng th followng compct mtr notton: (Y 0 n YX y X (5

18 Stp 3: IRLS lgorthm D 2 At stp 3, w chck to s f w shoul stop th lgorthm n, f w c not to stop, w go bck to stp 2 If w c to stop, w wll ssum th fnl vlus of th vctor r th MLE (t my not b ctly th tru MLE, but w wll ssum tht [t] t s clos f w o not stop th lgorthm to rly!,.g. Thr r mny stoppng ruls, usng chng n Dvnc s on wy to D 2 n construct rul (not th ssu wth ln(0!!: n y ln y ln D D [t ] D y ( or[t], or[t], or[t] y or[t] or[t] or[t],, or[t] or[t] or[t],, ( y ln ( y ln s smll. Wh 4D <0 6 y ( or[t], or[t], or[t] y or[t] or[t] or[t],, or[t] or[t] or[t],,

19 Logstc hypothss tstng I Rcll tht our null n ltrntv hypothss r: W wll us th LRT for th null (0 n ltrntv (: LRT 2ln 2ln L(ˆ 0 y L(ˆ y H 0 : 0\ 0 H A : 6 0[ 6 0 For our cs, w n th followng: l( ˆ y l( LRT 2ln 2l(ˆ y 2l(ˆ 0 y ˆ, ˆ, ˆ y l( ˆ 0 y l( ˆ, 0, 0 y

20 Logstc hypothss tstng II For th ltrntv, w us our MLE stmts of our logstc rgrsson prmtrs w gt from our IRLS lgorthm n plug ths nto th log-lk quton l(ˆ y l(ˆ 0 y n X h y ln( ( ˆ, ˆ, ˆ ( y ln( ( ˆ, ˆ, ˆ For th null, w plug n th followng prmtr stmts nto ths sm quton nx h y ln( (,,,,,, ( ˆ,0, 0, 0 ( y ln( ( ˆ,0, 0, 0 whr w us th sm IRLS lgorthm to prov stmts of by runnng th lgorthm EXACTLY th sm wth ˆ,0 EXCET w st ˆ 0, ˆ 0 n w o not upt ths!

21 Logstc hypothss tstng III To clcult our p-vlu, w n to know th strbuton of our LRT sttstc unr th null hypothss Thr s no smpl form for ths strbuton for ny gvn n (contrst wth F-sttstcs!! but w know tht s n gos to nfnt, w know th strbuton s.. ( : LRT 2ln 2l(ˆ y 2l(ˆ 0 y n ct s LRT! 2 f, tht pn Wht s mor, t s rsonbly goo ssumpton tht unr our (not ll!! null, ths LRT s (ppromtly! ch-squr strbuton wth 2 grs of from (.f. ssumng n s not too smll! n!

22 S you on Thurs.! Tht s t for toy