Logistic Regression I. HRP 261 2/10/ am
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1 Logstc Rgrsson I HRP 26 2/0/03 0- am
2 Outln Introducton/rvw Th smplst logstc rgrsson from a 2x2 tabl llustrats how th math works Stp-by-stp xampls to b contnud nxt tm Dummy varabls Confoundng and ntracton
3 Th Logt Modl P D / ln P D / Logt functon log odds Basln odds r, Lnar functon of rsk factors and covarats for ndvdual : x 2 x 2 3 x 3 4 x 4
4 Rlatng odds to probablts Rlatng odds to probablts, / / /, / /, /, / on manpulat algbrac, r, r r D P D P D P r, r D P D P r D P r D P odds lgbra robablty
5 Indvdual Probablty Functons Indvdual Probablty Functons Probablts assocatd wth ach ndvdual s outcom:,,,,, / ~ / : NOT dvlop dsas dd : dvlopd dsas r r r r r D P D P yars old;40 lbs PD/smoks; smok wght ag smok wght ag Exampl:
6 Th Lklhood Functon Th lklhood functon s an quaton for th jont probablty of th obsrvd vnts as a functon of Lklhood Functon : all cass P D all cass / r, r, all controls P D all controls 0 / r,
7 Maxmum Lklhood Estmats of Tak th log of th lklhood functon to lnarz t Maxmz th functon just basc calculus: Tak th drvatv of th log lklhood functon St th drvatv qual to 0 Solv for
8 Adjustd Odds Rato Intrprtaton OR odds of dsas for th xposd odds of dsas for th unxposd alcohol alcohol 0 smokng smokng alcohol smokng 0 alcohol smokng alcohol alcohol
9 Adjustd odds rato, contnuous prdctor OR odds of dsas for th xposd odds of dsas for th unxposd alcohol alcohol smokng smokng ag ag 29 9 smokng ag 29 alcohol 9 alcohol smokng ag ag ag 29 9 ag 0
10 Practcal Intrprtaton x ˆ rf OR rsk factor of ntrst Th odds of dsas ncras multplcatvly by ß for for vry on-unt ncras n th xposur, controllng for othr varabls n th modl.
11 Smpl Logstc Rgrsson
12 2x2 Tabl courtsy Hosmr and Lmshow Exposur Exposur0 Dsas E π π 0 Dsas 0 π π 0
13 Odds Rato for smpl 2x2 Tabl OR courtsy Hosmr and Lmshow
14 Exampl : CHD and Ag 2x2 from Hosmr and Lmshow >55 yrs <55 yars CHD Prsnt 2 22 CHD Absnt 6 5
15 Th Lklhood Th Lklhood x x x l
16 Th Log Lklhood Th Log Lklhood 5log 5log 22 log 22log 6log 6log 2log 2log log l
17 Th Log Lklhood, cont. Th Log Lklhood, cont. 5log 0 22 log 22 6log 0 2log 2 log log log log log l rcall
18 Drvatv of th log lklhood Drvatv of th log lklhood ] [log d l d d l d ] [log
19 Maxmz Maxmz
20 Maxmz Maxmz OR x x
21 Hypothss Tstng H 0 : 0 What s th Wald tst hr? 5x2 ln 6x What s th Lklhood Rato Tst? Full modl ncluds ag varabl Rducd modl ncluds only ntrcpt Maxmum lklhood ought to b x dos MLE yld ths?
22 Lklhood valu for rducd l x logl 43log 43 d logl d modl 0 57 margnal odds of CHD! l x x x0 57
23 Lklhood valu of full modl l x x x x 22 x x x0 26 5
24 Fnally th LR 2ln L rducd L full 2ln2.x0 30 [ 2ln2.43x0 26 ]
25 Exampl 2: >2 xposur lvls *dummy codng CHD status Wht Black Hspanc Othr Prsnt Absnt From Hosmr and Lmshow
26 SAS CODE data rac; nput chd rac_2 rac_3 rac_4 numbr; datalns; nd; run; proc logstc datarac dscndng; wght numbr; modl chd rac_2 rac_3 rac_4; run; Not th us of dummy varabls. Basln catgory s wht hr.
27 What s th lklhood hr? What s th lklhood hr? x othr wht othr wht othr wht hsp wht hsp wht hsp wht black wht black wht black wht wht wht wht x x x x x l
28 SAS OUTPUT modl ft Intrcpt Intrcpt and Crtron Only Covarats AIC SC Log L Tstng Global Null Hypothss: BETA0 Tst Ch-Squar DF Pr > ChSq Lklhood Rato Scor Wald
29 SAS OUTPUT rgrsson coffcnts Analyss of Maxmum Lklhood Estmats Standard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Intrcpt rac_ rac_ rac_
30 4x ncras n odds of CHD for othr vs. wht SAS output OR stmats Th LOGISTIC Procdur Odds Rato Estmats Pont 95% Wald Effct Estmat Confdnc Lmts rac_ rac_ rac_ Intrprtaton: 8x ncras n odds of CHD for black vs. wht 6x ncras n odds of CHD for hspanc vs. wht
31 Exampl 3: Prostrat Cancr Study Quston: Dos PSA lvl prdct tumor pntraton nto th prostatc capsul ys/no? Is ths assocaton confoundd by rac? Dos rac modfy ths assocaton ntracton?
32 . What s th rlatonshp btwn PSA contnuous varabl and capsul pntraton bnary?
33 Capsul ys/no vs. PSA mg/ml capsul.0 psa vs. capsul psa
34 Man PSA pr quntl vs. proporton capsulys S-shapd? proporton wth capsulys PSA mg/ml
35 logt plot of psa prdctng capsul, by quntls lnar n th logt? Est. logt psa
36 psa vs. proporton, by dcl proporton wth capsulys PSA mg/ml
37 m numr of vnts M numbr of cass logt vs. psa,, by dcl Estmatd logt plot of psa prdctng capsul n th data st krstn.psa Est. logt psa
38 modl: capsul psa Tstng Global Null Hypothss: BETA0 Tst Ch-Squar DF Pr > ChSq Lklhood Rato <.000 Scor <.000 Wald <.000 Analyss of Maxmum Lklhood Estmats Standard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Intrcpt <.000 psa <.000
39 Modl: capsul psa rac Analyss of Maxmum Lklhood Estmats Standard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Intrcpt psa <.000 rac No ndcaton of confoundng by rac snc th rgrsson coffcnt s not changd n magntud.
40 Modl: capsul psa rac psa*rac Standard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Intrcpt psa rac psa*rac Evdnc of ffct modfcaton by rac p.07.
41 STRATIFIED BY RACE: rac Standard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Intrcpt <.000 psa < rac Analyss of Maxmum Lklhood Estmats Standard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Intrcpt psa
42 How to calculat OR s s from modl wth ntracton trm Standard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Intrcpt psa rac psa*rac Incrasd odds for vry 5 mg/ml ncras n PSA: If wht rac0: 5* If black rac: 5*
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