Ch. 8: Multicategory Logit Models

Transcription

1 _ 8 # 4 Chapter 8: Mlticategry Lgit Mdels 1 Ch. 8: Mlticategry Lgit Mdels Sppse respnse is nminal, bt has categries (cntrast with lgistic regressin, where has categries). This chapter examines extensins f lgistic regressin t handle sch mlticategry respnses. Sppse Lgit Mdels fr Nminal Respnse have categries (rder f listing f categries is irrelevant), and let Since "!$# %, we need nly specify &# Baseline-Categry Lgits Pair each respnse categry with a baseline categry, say the th. The baseline-categry lgits are then )+*-,/. 12% The Mdel The lgit mdel sing baseline-categry lgits with predictr 7 has the frm )+*-,. 98 ;:=< 7 >?4@ Mdel invlves 546 lgit eqatins, with separate parameters fr each. (Take 8 < 9A.) (' #. Chapter 8: Mlticategry Lgit Mdels 2 Nte. The baseline-categry lgit is the lgit fr the cnditinal dds that the respnse categry is given that it is ne f either r (the baseline categry): CB := B ;:D DFE either r G H IE either r G DFE either r G J4K FE either r H $ The mdel determines cnditinal lgits fr any ther pair f respnse categries: )+*-,L. NM PO )Q*3,L. RM B )Q*3,/. RM NO B S8 M :K< M 7 $4@S8 O :K< O 7 S8 M 4K8 O T UV W : <RM 4 <PO T UV W 7 intercept slpe )+*-,/. PO Can slve fr in terms f mdel parameters (with < 9A ): S8 [:K< 7 S8 [:K< 7 : \ ('!G# S8 \ :K< \ 7 \!$# S8 \ :=< \ 7 Chapter 8: Mlticategry Lgit Mdels 3 Alligatr Fd Chice Measred length (m) and primary fd chice (^% fish, invertebrates, `H ther) fr 59 alligatrs. acb de acb fcg acb fcg acb ecg acb ed I I I F I acb eh acb eh acb e3i acb hcd acb hch I O I I I acb jcf acb jch acb jch acb jck acb ig I O I F I acb ik acb ik acb ik acb kch acb kck I I O F I acb lck d3b gcf d3b gcf d-b f(a d3b f-a I F F F F d3b fcj d3b fcl d3b e(a d-b hcj d3b ji F F F O F d3b il d3b ke f-b dch f(b hcj f-b hk F F O F F f-b jck f-b ia f-b kcl O F F acb ed acb jcg acb if acb lcf d3b fcj d3b id f-b jcj Wish t mdel fd chice as a fnctin f length. Fitted baseline-categry lgit mdel with baseline m n (ther) is )Q*3,L.[ # Rp )Q*3,L.[ N p )Q*3,L.[ # N qrs4kat-a 7 vw q-xwyz4= ] {(q-v 7 F I O I F I F } qrf4=v] q3xwy- : ~ 4zA]t-Aw$4@4z ] {(q-v( 7 %4J{ AwyCx : ] n-v-v 7 S, e.g., cnditinal n fd chice being either fish r invertebrates, estimated dds that chice is fish increases with length (accrding t an S-shaped crve). Fr each 1-meter increase in length, the estimated dds that chice pƒ ƒ is fish rather than invertebrates increases by a factr f %A] v. Chapter 8: Mlticategry Lgit Mdels 4 The estimated prbabilities f the tcmes (Fish, Invertebrate, Other) are &# } qrs4kat-a 7 : 3 qrf4kat3a 7 : v] q3xwyz4d w {(q-v 7 vw q3xwyz4= ] {(q-v 7 : 3 qrf4kat3a 7 : v] q3xwyz4d w {(q-v 7 p : 3 qrf4kat3a 7 : v] q3xwyz4d w {(q-v 7 Fr an alligatr f length 7 9n] x meters, estimated prb. that the primary fd is ther eqals p : } qrf4kat3â n xw} : A 3n] vw q-xwyz4= w {(q(vsn x(š Finally, t test whether primary fd chice is independent f length (H FŒ < # <P 9A ), the LRT test is 4 SŽL J4KŽ # S = C&4D # 9x3 x 4Dy3v]t3 q r df {F4D 9 P-vale % A3A-A- (CATMOD give Wald test statistic f 8.94 (df, P-vale H A-v )). There is strng evidence f an effect f length n primary fd chice.

2 Chapter 8: Mlticategry Lgit Mdels 5 Chapter 8: Mlticategry Lgit Mdels 6 SAS Otpt: Baseline-Categry Lgits Mdel SAS fr Alligatr Fd Chice Example SAS Prgram data gatr; inpt length chice cards; 1.24 I 1.3 I 1.3 I 1.32 F 1.32 F 1.4 F 1.42 I 1.42 F 1.45 I 1.45 O 1.47 I 1.47 F 1.5 I 1.52 I 1.55 I 1.6 I 1.63 I 1.65 O 1.65 I 1.65 F 1.65 F 1.68 F 1.7 I 1.73 O 1.78 I 1.78 I 1.78 O 1.8 I 1.8 F 1.85 F 1.88 I 1.93 I 1.98 I 2.3 F 2.3 F 2.16 F 2.26 F 2.31 F 2.31 F 2.36 F 2.36 F 2.39 F 2.41 F 2.44 F 2.46 F 2.56 O 2.67 F 2.72 I 2.79 F 2.84 F 3.25 O 3.28 O 3.33 F 3.56 F 3.58 F 3.66 F 3.68 O 3.71 F 3.89 F ; /* BASELINE-CATEGORY LOGITS MODEL */ prc catmd; respnse lgits; direct length; mdel chice = length / pred=prb; /* INDEPENDENCE MODEL */ prc catmd; respnse lgits; pplatin length; mdel chice = / pred=prb; rn; Srce DF Chi-Sqare Prb INTERCEPT LENGTH LIKELIHOOD RATIO ANALYSIS OF MAXIMUM-LIKELIHOOD ESTIMATES Standard Chi- Effect Parameter Estimate Errr Sqare Prb INTERCEPT LENGTH SAS Otpt: Independence Mdel Srce DF Chi-Sqare Prb INTERCEPT LIKELIHOOD RATIO Chapter 8: Mlticategry Lgit Mdels 7 Example (Incme and Jb Satisfactin.). The fllwing table crss-classifies black respndents t the 1991 General Scial Srvey accrding t incme and jb satisfactin. Jb Satisfactin Very A little Mderately Very Incme Dissat. Dissat. Satis. Satis , , , The next few pages shw sme f the steps in extracting these data sing the web-based sftware at the Srvey Dcmentatin & Analysis (SDA) web site ( Fllwing that is a SAS prgram and its tpt fr varis analyses f these data. In these analyses we treat the explanatry variable incme as rdinal (qantitative) sing scres na]" A n-a. Nte that the baseline-categry lgit mdel wld therwise be satrated: n 5{ c parameters with nly n 5{% nnredndant (baseline-categry) sample lgits. Chapter 8: Mlticategry Lgit Mdels 8 rincme RESPONDENTS INCOME Text f this Qestin r Item Did y earn any incme frm (OCCUPATION DESCRIBED IN Q. 2) in [1973/74/75/76/77/79/8/82/83/84/85/86/87/ 88/89/9/92]? IF YES: In which f these grps did yr earnings frm (OCCUPATION IN Q. 2) fr last year- -[ ]--fall? That is, befre taxes r ther dedctins. Jst tell me the letter. % Valid % All N Vale Label LT $ ,345 2 $1 TO $3 TO $4 TO $5 TO $6 TO $7 TO ,275 8 $8 TO ,319 9 $ ,419 1 $ ,72 11 $ , $25 OR MORE REFUSED DK NA ,59. (N Data) ,284 Ttal Prperties Data type: nmeric Missing-data cdes:,98,99 Recrd/clmns: 1/

3 Chapter 8: Mlticategry Lgit Mdels 9 Chapter 8: Mlticategry Lgit Mdels 1 SDA 1.2: Tables General Scial Srvey Cmlative File Apr 9, 2 (Sn 1:29 PM PDT) SDA Tables Prgram (Selected Stdy: GSS Cmlative Datafile) Help: General / Recding Variables REQUIRED Variable names t specify Rw: rincme(r: 1-4 "<5"; 5-9 "5-15"; 1-11 "15-25"; 12 ">25") OPTIONAL Variable names t specify Clmn: satjb Cntrl: Selectin Filter(s): year(91) race(2) Example: age(18-5) gender(1) Weight: N Weight Percentaging: Clmn Rw Ttal Other ptins Statistics Sppress table Qestin text Clr cding Shw T-statistic Rn the Table Clear Fields Variables Rle Name Label Range MD Rw rincme(recded) RESPONDENTS INCOME 1-4 Clmn satjb JOB OR HOUSEWORK 1-4,8,9 Filter year(91) GSS YEAR FOR THIS RESPONDENT Filter race(2) RACE OF RESPONDENT(=BLACK) 1-3 Cells cntain: -N f cases 1 VERY SATISFIED Freqency Distribtin 2 MOD. SATISFIED satjb 3 A LITTLE DISSAT 4 VERY DISSATISFIED ROW TOTAL 1 < rincme > COL TOTAL Recde fr rincme Change nmber f decimal places t display Fr percents: 1 Fr statistics: 2 1 = 1-4 "<5"; 2 = 5-9 "5-15"; 3 = 1-11 "15-25"; 4 = 12 ">25" Allcatin f cases Valid cases 15 Cases exclded by filters 35,8 Cases with invalid cdes n rw r clmn variable 99 Ttal cases 35,284 CSM, UC Berkeley Chapter 8: Mlticategry Lgit Mdels 11 Chapter 8: Mlticategry Lgit Mdels 12 Fitting Mdels t Incme/Jb Satisfactin Data w/ SAS SAS Prgram data jbsatis; inpt incme satis cards; ; /* BASELINE-CATEGORY LOGIT MODEL */ prc catmd; weight cnt; respnse lgits; direct incme; mdel satis = incme / pred=freq; /* CUMULATIVE LOGIT (PROPORTIONAL ODDS) MODEL */ prc lgistic; weight cnt; mdel satis = incme; /* INDEPENDENCE AS BASELINE-CATEGORY LOGIT MODEL */ prc catmd; weight cnt; respnse lgits; pplatin incme; mdel satis = / pred=freq; /* INDEPENDENCE AS LOGLINEAR MODEL */ prc genmd; class incme satis; mdel cnt = incme satis / dist=pi link=lg; Fitted Baseline-Categry Lgit Mdel (PROC CATMOD) Srce DF Chi-Sqare Prb INTERCEPT INCOME LIKELIHOOD RATIO ANALYSIS OF MAXIMUM-LIKELIHOOD ESTIMATES Standard Chi- Effect Parameter Estimate Errr Sqare Prb INTERCEPT INCOME /* USUAL TEST OF INDEPENDENCE FOR TWO-WAY TABLES */ prc freq; weight cnt; tables incme*satis / expected chisq; rn;

4 Chapter 8: Mlticategry Lgit Mdels 13 Chapter 8: Mlticategry Lgit Mdels 14 ML PREDICTED VALUES FOR RESPONSE FUNCTIONS AND FREQUENCIES -----Observed Predicted---- Fnctin Standard Standard Sample Nmber Fnctin Errr Fnctin Errr F F F F F F F F F F F F Fitted Cmlative Lgit Mdel (PROC LOGISTIC) Mdel Fitting Infrmatin and Testing Glbal Nll Hypthesis BETA= Intercept Intercept and Criterin Only Cvariates Chi-Sqare fr Cvariates -2 LOG L with 1 DF (p=.34) Analysis f Maximm Likelihd Estimates Parameter Standard Wald Pr > Variable DF Estimate Errr Chi-Sqare Chi-Sqare INTERCP INTERCP INTERCP INCOME F F F F Chapter 8: Mlticategry Lgit Mdels 15 Chapter 8: Mlticategry Lgit Mdels 16 ML PREDICTED VALUES FOR RESPONSE FUNCTIONS AND FREQUENCIES -----Observed Predicted---- Fnctin Standard Standard Sample Nmber Fnctin Errr Fnctin Errr Independence as Baseline-Categ. Lgit Mdel (CATMOD) Srce DF Chi-Sqare Prb INTERCEPT LIKELIHOOD RATIO ANALYSIS OF MAXIMUM-LIKELIHOOD ESTIMATES Standard Chi- Effect Parameter Estimate Errr Sqare Prb INTERCEPT F F F F F F F F F F F F F F F F

5 Chapter 8: Mlticategry Lgit Mdels 17 Chapter 8: Mlticategry Lgit Mdels 18 Pissn Lglinear Mdel f Independence (GENMOD) Criteria Fr Assessing Gdness Of Fit Criterin DF Vale Vale/DF Deviance Pearsn Chi-Sqare Lg Likelihd Analysis Of Parameter Estimates Parameter DF Estimate Std Err ChiSqare Pr>Chi INTERCEPT INCOME INCOME INCOME INCOME SATIS SATIS SATIS SATIS Usal Tests fr Independence (PROC FREQ) STATISTICS FOR TABLE OF INCOME BY SATIS Statistic DF Vale Prb ---- Chi-Sqare Likelihd Rati Chi-Sqare Baseline Categry Lgits Mdel fr Jb Satis. The baseline categry lgits mdel (treating incme as rdinal) fits well ( { A-, df 9q, P-vale % qwy ). Als there is reasnably strng evidence f an effect f incme n jb satisfactin: 4 SŽ 4KŽ # S &4D # { n-ns4 { A( A n df 9qs4=n n P-vale % Aq (CATMOD als gives Wald test statistic, yw r3r (df n, P-vale H A{(r-v ).) Cntrast this test f independence with the sal tests which yield { n-n df 9x P-vale %t3- ] n-r( P-vale Htx3nw r Chapter 8: Mlticategry Lgit Mdels 19 Predictin eqatins are Nte. )Q*3,/. &# )Q*3,/. P )Q*3,/. p 8 #G: <&# 7 H q3a-n3xs4 A( -y 7 8 : < 7 H q3r-x{f4 A(yC{-r 7 8 p;: <Np 7 %3 r-q{(qs4 A(v]{ 7 Fr each lgit, the dds f being in the less satisfied categry (instead f cat. 4) decreases as 7 incme increases. There are 12 lgits mdeled (3 at each f 4 incme levels) and 6 parameters, s df 4=q 9q fr testing fit. LRT statistic fr gdness-f-fit is { A( (P-vale % qwy ), s mdel with linear trends fr incme effect fits well. This analysis ses the rdering f incme levels, bt ignres rdinality f jb satisfactin ( ). Chapter 8: Mlticategry Lgit Mdels 2 Lgit Mdels fr Ordinal Respnses If the respnse is rdinal with rdered categries, then the cmlative prbabilities are / # :DNL: := 12%3 Fr?46, the cmlative lgits are )Q*3, )+*-, J 4= )+*-, K )Q*3, # : C:= G# : C:= Fr a single predictr 7, the prprtinal dds mdel is )Q*3, 98 ;:=< 7 >2% i.e., a separate intercept 8 fr each cmlative lgit, bt the same slpe < fr each. P(Y j) 1..5 P(Y 3) P(Y 2) P(Y 1). x

6 ' ' ' Chapter 8: Mlticategry Lgit Mdels 21 Chapter 8: Mlticategry Lgit Mdels 22 Nte. is the mltiplicative effect n the dds that (each ) fr every 1-nit increase in 7. Similarly dds that at 7 7 eqal times dds at 7 7 #. Becase < des nt depend n, this factr is the same fr all 2%3?4@. Ths the name prprtinal dds mdel. Example (Jb Satisfactin and Incme). In SAS, PROC LOGISTIC fits rdinary lgistic regressin when 9, cmlative lgit mdel when. Here the predictin eqatin is )+*-, ] 8 4 A(vx-q 7 >3 ] n i.e., dds f being at lw end f the jb satisfactin scale decrease as 7 incme increases. Since 9 ' ƒ H x3{w w the estimated dds f jb satisfactin being belw an given level (instead f abve it) mltiplied by x3{( fr each 1-nit increase in 7. # Fr 7N 4 7 # %A, dds mltiply by ƒ % v-vw e.g., the dds f jb satisfactin being belw sme level (instead f abve it) at 7 9n-A are estimated t be v-v times the dds at 7 3A. T test H Œ < 9A (jb satis. indep. f incme), < Wald test stat. 4KA 9r] (yx3r < LRT stat. 9r] v(y df P-vale % A3A3{(A (Wald) 4F A-v3x3q A- 3A(y A3A-n{ (LRT) Cntrast with previs tests (thse that treated jb satisfactin ( ) as nminal, and thse that treated bth incme and jb satisfactin as nminal). Nte. PROC CATMOD will als fit cmlative lgit (prprtinal dds) mdels, bt ses weighted least sqares estimates f parameters, instead f MLE s.