Latent Class Regression

Transcription

1 Latent Class Regresson Karen Bandeen-Roche October 8, 06

2 Objectves For you to leave here knowng What s the LCR model and ts underlyng assumptons? How are LCR parameters nterpreted? How does one check the assumptons of an LCR model? Latent class regresson analogs to LCA for fttng and dentfablty

3 Motvatng Eample: Fralty Latent trat (IRT) assumes t s contnuous. Latent class model assumes t s dscrete Densty % Robust 80 Intermedate Fral 5 Fral Fralty 3

4 Motvatng Eample: Fralty Is fralty assocated wth age, educaton, and/or dsease burden? Do age, educaton, and/or dsease burden predct heghtened rsk for membershp n some fralty classes as opposed to others? 4

5 Part I: Model

6 Latent Class Regresson (LCR) Recall the standard latent class model : Dscrete latent varables & dscrete ndcator varables Indcators measure dscrete subpopulatons rather than underlyng contnuous scores Patterns of responses are thought to contan nformaton above and beyond aggregaton of responses The predcton goal s clusterng ndvduals rather than contnuous response varables We add structural pece to model where covarates eplan class membershp 6

7 Latent Class Regresson Model Structural Model 7

8 Analyss of underlyng subpopulatons Latent class regresson POPULATION POPULATION U X P P (X) P J P J (X) M J JM M J JM Y Y M Y Y Y M Y M Y Y M Goodman, 974 Dayton & Macready, 988

9 Latent Varable Models Latent Class Regresson (LCR) Model Model: Structural model: f Y A latent polytomous logstc regresson J ( y ) P (, ) π ( π j j M m y mj m y m [ U ] Pr{ U j } P (, ) mj ) ( ) ep J + ep k j j ( ) k Measurement model: π Pr Y U π s MJ { j} mj m [ ] Y U condtonal probabltes 9

10 Structural Model Wth two classes, the latent varable (class membershp) s dchotomous 0

11 Parameter Interpretaton Consder smplest case: classes ( vs. ) Pr( C ) log Pr( C ) or equvalently, + 0 Pr( C log Pr( C s a log odds rato. ) ) + 0 Ths s a latent logstc regresson.

12 Parameter Interpretaton Consder smplest case: classes ( vs. ) log Pr( C Pr( C ) ) + 0 where s a log odds rato. Eample: ep( ) and f female, 0 f male Women have twce the odds of beng n class versus class than men, holdng all else constant

13 3 Solvng for P j ( ) Pr(C j ) 0 ), Pr( ), Pr( log ) ( ) ( log C C p p + + Usng the fact: p ( ) + p ( ), we obtan ), Pr( ) ( ), Pr( ) ( e C p e e C p

14 Parameter Interpretaton General Case (J-)*(p+) s where p number of covarates J-: one class s reference class so all of ts coeffcents are techncally zero p+: for each class (ecept the reference), there s one for each covarate plus another for the ntercept. 4

15 Parameter Interpretaton General Case Need more than one equaton Choose a reference class (e.g. class ) log Pr( C Pr( C,, ) ) e e 3 Pr( C log Pr( C e / e e 3 3 3,, ) ) OR for class versus class for females versus males OR for class 3 versus class for females versus males 3 OR for class 3 versus class for 3 females versus males 5

16 6 Solvng for P j ( ) Pr(C ) 0 ), Pr( ), Pr( log ) ( ) ( log C C P P Where we assume that ), Pr( ) ( j j j j e e e e e C P (Snce class s the reference class)

17 Assumptons Condtonal Independence: gven an ndvdual s class, hs symptoms are ndependent Pr(y m, y r C ) Pr(y m C )* Pr(y r C ) Non-dfferental Measurement: gven an ndvdual s class, covarates are not assocated wth symptoms Pr(y m, C ) Pr(y m C ) 7

18 Condtonal Dependence 8

19 Dfferental Measurement 9

20 Part II: Fttng

21 Model Buldng Step : Get the measurement part rght f y Ft standard latent class model frst. Works: margnalzaton property J [ ] M ( ) y J M ( ) y y y m π π P * π π ( y ) P ( ) dg( ) m m j mj mj j j m j m Bandeen-Roche et al., J Am Statst Assoc., 997 Step : Model buldng as n multple logstc regresson m mj mj

22 Mamum lkelhood estmator Latent Class Regresson Lkelhood Pr( Y y j j m EM Algorthm E-step as before Model Estmaton ) Pr( Y J y,..., ( y M-step polytomous logstc regresson wth posteror probabltes as outcomes, M Y ym p ( ) π mj ( π where p j ( ) y J e j Y mj j e M ) j m y ) M ) Bandeen-Roche et al., J Am Statst Assoc., 997

23 Eample: Fralty of older adults Step. Measurement model Crteron -Class Model 3-Class Model CLASS CLASS NON-FRAIL FRAIL CLASS ROBUST CLASS INTERMEDIATE CLASS 3 FRAIL Weght Loss Weakness Slowness Low Physcal Actvty Ehauston Class Prevalence (%) (Bandeen-Roche et al. 006) 3

24 Step. Structural Model To evaluate the assocatons between fralty and age, educaton, and dsease burden Run latent class regresson whle fng the number of latent classes derved from the LCA Assumng condtonal ndependence and non-dfferental measurement 4

25 Fralty: Structural Model Reference group Non-fral ep(coeff) Varable Coeff OR Coeff SE Coeff Z Coeff+/-.96SE Coeff CI ep(coeff CI) OR CI Age, (0.04,0.33) (.04,.6) Educaton, (-0.466,-0.58) (0.63,0.77) Dseases (0.6,.58) (.86,3.9) Intercept (-.80,-.478) (0.06,0.3) 4 Centered at means Years 3 Number (count) 4 Odds (among the non-fral; rather than odds ratos) 5

26 Fralty: Structural Model Reference group Non-fral Varable Coeff OR Coeff SE Coeff Z Coeff CI OR CI Age, (0.04,0.33) (.04,.6) Educaton, (-0.466,-0.58) (0.63,0.77) Dseases (0.6,.58) (.86,3.9) Intercept (-.80,-.478) (0.06,0.3) 4 Prob odds/(+odds) 0.8/ estmated fral prevalence among those wth mean age and educaton & no dseases 6

27 Fralty: Structural Model Reference group Non-fral Varable Coeff OR Coeff SE Coeff Z Coeff CI OR CI Age, (0.04,0.33) (.04,.6) Educaton, (-0.466,-0.58) (0.63,0.77) Dseases (0.6,.58) (.86,3.9) Intercept (-.80,-.478) (0.06,0.3) 4 we estmate that the odds of beng fral (vs. nonfral) ncreases.4-fold wth each added dsease 7

28 Part III: Evaluatng Ft

29 Model Checkng Analog resdual checkng n lnear regresson IT S CRITICALLY IMPORTANT! Can gve msleadng fndngs f measurement model assumptons are unwarranted Phlosophcal opnon: we learn prmarly by specfyng how smple models fal to ft, not by observng that comple models happen to ft Two types of checkng Whether the model fts (e.g. observed vs. epected) How a model may fal to ft (ASSUMPTIONS) 9

30 Checkng Whether the Model Fts Means ) Do Y s aggregate as epected gven the model? Ø Check the measurement model (LCA) Ø Check whether the measurement model s comparable wthout (LCA) and wth (LCR) covarates 30

31 Checkng Whether the Model Fts ) Do Y s relate to the X s as epected gven the model Idea: focus on one tem at a tme J M ym ( y y,..., YM ym ) Pj ( ) π mj ( mj ) j m P( Y π m ) v If nterested n tem m, gnore ( margnalze over ) other tems: ym P( Y y ) P ( ) π ( π m m J j j mj mj ) ( y m ) 3

32 Comparng Ftted to Observed v Construct the predcted curve by plottng ths probablty versus any gven Add a smooth splne (e.g. use lowess n STATA) to reveal systematc trend (sold lne) v Supermpose t wth an observed tem response curve by Plot tem response (0 or ) by Add smooth splne to reveal systematc trend (dashed lne) 3

33 Checkng How the Model Fals to Ft Check Assumptons condtonal ndependence non-dfferental measurement 33

34 Checkng How the Model Fals to Ft Basc deas: Suppose the model s true If we knew persons latent class membershps, we would check drectly: v Stratfy nto classes, then, wthn classes: v Check correlatons or parwse odds ratos among the tem responses (Condtonal Independence) v Regress tem responses on covarates (non-dfferental measurement) v Regress class membershps on covarates, hope for v Smlar fndngs re regresson coeffcents v No strong effects of outlers v Identfy strongly nonlnear covarates effects 34

35 Checkng How the Model Fals to Ft But n realty, we don t know the true latent class membershp! Latent class membershps must be estmated Randomze people nto pseudo classes usng ther posteror probabltes or assgn to most lkely class correspondng to the hghest posteror probablty Posteror probablty s defned as Pr( C Pr( y C j) Pr( C j ) j, y ) J Pr( y C j) Pr( C j ) j Analyze as descrbed before, ecept usng pseudo class membershp rather than true ones Bandeen-Roche et al., J Am Statst Assoc.,

36 Utlty of Model Checkng May modfy nterpretaton to ncorporate lack of ft/volaton of assumpton May help elucdate a transformaton that that would be more approprate (e.g. log(age) versus age) May suggest how to mprove measurement (e.g. better survey nstrument) May lead to beleve that LCR s not approprate 36

37 Part IV: Identfablty / Estmablty

38 Identfablty Latent class (bnary Y) regresson Latent class analyss (measurement only) ~ multnomal, dm Y ~ ( p ) ( p) M Unconstraned J-class model: M(J-) parameters Need M M( J ) Latent class regresson As above + full-rank X Bandeen-Roche et al, JASA, 997; Huang & Bandeen-Roche, Psychometrka, 004

39 Identfablty To best assure dentfcaton Incorporate a pror theory as much as possble v Set πs to 0 or where t makes sense to do so v Set πs equal to each other If program fals to converge v Run the program longer v Re-ntlze n very dfferent places v Add constrants (e.g. set πs to 0 or where sensble) v Stop (attemptng to do too much wth one s data) 39

40 Objectves For you to leave here knowng What s the LCR model and ts underlyng assumptons? How are LCR parameters nterpreted? How does one check the assumptons of an LCR model? Latent class regresson analogs to LCA for fttng and dentfablty

41 Append Mplus codes 4

42 MPLUS fttng of LCA TITLE: Latent Class Analyss of Fralty Components Usng Combned WHAS I and II Data Age DATA: FILE IS "h:\teachng\40.658\007\lcr.dat"; VARIABLE: NAMES ARE based shrnk weak slow ehaust kcal sweght age educ dsease; USEVARIABLES ARE shrnk weak slow ehaust kcal; MISSING ARE ALL (999999); CATEGORICAL ARE shrnk-kcal; CLASSES fralty(); ANALYSIS: TYPE IS MIXTURE; MODEL: %OVERALL% Declare mssng value code Assgn label fralty to the latent class varable and specfy number of classes %fralty#% [shrnk$* weak$* slow$* ehaust$* kcal$*]; %fralty#% [shrnk$*- weak$*- slow$*- ehaust$*- kcal$*-]; OUTPUT: TECH0 TECH; SAVEDATA: FILE IS "h:\teachng\40.658\00\lcasave.out"; SAVECPROB; Assgn startng values for thresholds (optonal) TECH0: output of observed vs. estmated frequences of response patterns; TECH, TECH4: results of Lo-Mendell-Rubn test and bootstrapped lkelhood rato test for comparng models wth k vs. k- classes Save posteror class probabltes 4

43 Step. Structural Model TECHNICAL OUTPUT VUONG-LO-MENDELL-RUBIN LIKELIHOOD RATIO TEST FOR (H0) VERSUS 3 CLASSES H0 Loglkelhood Value Tmes the Loglkelhood Dfference.48 Dfference n the Number of Parameters 6 Mean Standard Devaton 6.56 P-Value 0.44 LO-MENDELL-RUBIN ADJUSTED LRT TEST Value.78 P-Value 0.50 TECHNICAL 4 OUTPUT PARAMETRIC BOOTSTRAPPED LIKELIHOOD RATIO TEST FOR (H0) VERSUS 3 CLASSES H0 Loglkelhood Value Tmes the Loglkelhood Dfference.48 Dfference n the Number of Parameters 6 Appromate P-Value Successful Bootstrap Draws 67 43

44 MPLUS fttng of LCR TITLE: Latent Class Regresson Analyss of Fralty Components Usng Combned WHAS I and II Data Age DATA: FILE IS "h:\whas\fral\paper\lcr.dat"; VARIABLE: NAMES ARE based shrnk weak slow ehaust kcal sweght age educ dsease; USEVARIABLES ARE shrnk weak slow ehaust kcal age educ dsease; CENTERING GRANDMEAN(age educ); MISSING ARE ALL (999999); CATEGORICAL ARE shrnk-kcal; CLASSES fralty(); Centerng predctors age and educaton for meanngful nterpretaton of ntercept ANALYSIS: TYPE IS MIXTURE; MODEL: %OVERALL% fralty# ON age educ dsease; %fralty#% [shrnk$*3 weak$*3 slow$*3 ehaust$*3 kcal$*3]; %fralty#% [shrnk$*- weak$*- slow$*- ehaust$*- kcal$*-]; Structural regresson model usng Class as the reference group Why have startng values? 44

45 MPLUS Output Categorcal Latent Varables Estmates S.E. Est./S.E. FRAILTY# ON AGE EDUC DISEASE Intercepts FRAILTY# Usng Class as the reference group ALTERNATIVE PARAMETERIZATIONS FOR THE CATEGORICAL LATENT VARIABLE REGRESSION Parameterzaton usng Reference Class FRAILTY# ON AGE EDUC DISEASE Intercepts FRAILTY#

46 Checkng Whether the Model Fts ) Do Y s aggregate as epected gven the model? v Compare observed pattern frequences to predcted pattern frequences 46

47 Fralty Eample: Observed versus Epected Response Patterns: Ignorng Covarates 5 most frequently observed patterns among the non-fral Crteron Weght loss Weak Slow Ehauston Low Actvty Pattern Frequences Observed Epected -class -class 3-class N N N N N N N Y N N N N N N Y N Y N N N N N Y N Y most frequently observed patterns among the fral (Bandeen-Roche et al. 006) N Y Y N Y N Y Y Y Y N N Y Y Y Y Y Y Y Y Y Y Y N Y Latent Class Model Ft statstcs Pearson Ch-Square 568 (p<.000) 4.4 (p.) 3. (p.5) AIC BIC

48 MPLUS Input TITLE: Weghted Latent Class Analyss of Fralty Components Usng Combned WHAS I and II Data Age DATA: FILE IS "C:\teachng\ \lcr.dat"; VARIABLE: NAMES ARE based shrnk weak slow ehaust kcal sweght age educ dsease; USEVARIABLES ARE shrnk weak slow ehaust kcal; MISSING ARE ALL (999999); CATEGORICAL ARE shrnk-kcal; CLASSES fralty(); ANALYSIS: TYPE IS MIXTURE; MODEL: %OVERALL% %fralty#% [shrnk$*- weak$*- slow$*0 ehaust$*- kcal$*-]; %fralty#% [shrnk$* weak$* slow$* ehaust$* kcal$*]; OUTPUT: TECH0 SAVEDATA: FILE IS "C:\teachng\ \lcasave.out"; SAVECPROB; Contans observed vs. epected frequences of response patterns Save estmated posteror probabltes of class membershp 48

49 MPLUS Output TECHNICAL 0 OUTPUT MODEL FIT INFORMATION FOR THE LATENT CLASS INDICATOR MODEL PART RESPONSE PATTERNS No. Pattern No. Pattern No. Pattern No. Pattern Order of response patterns? RESPONSE PATTERN FREQUENCIES AND CHI-SQUARE CONTRIBUTIONS Response Frequency Standardzed Ch-square Contrbuton Pattern Observed Estmated Resdual Pearson Loglkelhood Deleted (z-score) Recall: for a bvarate table, standardzed resdual (O E) / [(E) / *(-E/N) / ] 49

50 Emprcal Testng of Identfablty Must run model more than once usng dfferent startng values to check dentfablty! Mplus nput: ANALYSIS: TYPE IS MIXTURE; STARTS ; STITERATIONS0; Number of ntal stage random sets of startng values and the number of fnal stage optmzatons to use Mamum number of teraton allowng n the ntal stage 50

51 Parameter Constrants: Mplus Eample TITLE: ths s an eample of a LCA wth bnary latent class ndcators and parameter constrants DATA: FILE IS e7.3.dat; VARIABLE: NAMES ARE u-u4; CLASSES c (); CATEGORICAL u-u4; ANALYSIS: TYPE MIXTURE; MODEL: %OVERALL% %c#% [u$*-]; [u$-u3$*-] (); [u4$*-] (p); %c#% [u$@-5]; [u$-u3$*] (); [u4$*] (p); MODEL CONSTRAINT: p - p; OUTPUT: TECH TECH8; u and u3 have same π n class u has π equal to n class u and u3 have same π n class The threshold of u4 n class s equal to -*threshold of u4 n class (.e. same error rate) MPLUS User s Gude p. 53 5