Mixed Models for Longitudinal Ordinal and Nominal Outcomes. Don Hedeker Department of Public Health Sciences University of Chicago

Transcription

1 Mixed Models for Longitudinal Ordinal and Nominal Outcomes Don Hedeker Department of Public Health Sciences University of Chicago Hedeker, D. (2008). Multilevel models for ordinal and nominal variables. In J. de Leeuw & E. Meijer (Eds.), Handbook of Multilevel Analysis. Springer, New York. Hedeker, D. (2015). Methods for multilevel ordinal data in prevention research. Prevention Science. Hedeker, D. & Gibbons, R.D. (2006). Longitudinal Data Analysis, chapters 10 & 11. Wiley. 1

2 Mixed-effects ordinal logistic regression model (Hedeker & Gibbons, 1994, 1996) i = 1,... N level-2 units (clusters or subjects) j = 1,..., n i level-1 units (subjects or repeated observations) c = 1, 2,..., C response categories Y ij = ordinal response of level-2 unit i and level-1 unit j How was your day? (asked repeatedly each day for a week) 2

3 Mixed-effects ordinal logistic regression model λ ijc = log P ijc = γ c (x (1 P ijc ) ijβ + z ijυ i ) P ijc = Pr (Y ij c υ ; γ c, β, Σ υ ) = 1 1+exp( λ ijc ) p ijc = Pr (Y ij = c υ ; γ c, β, Σ υ ) = P ijc P ijc 1 C 1 strictly increasing model thresholds γ c x ij = p 1 covariate vector z ij = r 1 design vector for random effects β = p 1 fixed regression parameters υ i = r 1 random effects for level-2 unit i N(0, Σ υ ) 3

4 Model for Latent Continuous Responses Model with p covariates for the latent response strength y ij : y ij = x ijβ + υ 0i + ε ij where υ 0i N(0, σ 2 υ), and assuming ε ij standard normal (mean 0 and σ 2 = 1) leads to mixed-effects ordinal probit regression ε ij standard logistic (mean 0 and σ 2 = π 2 /3) leads to mixed-effects ordinal logistic regression 4

5 Underlying latent variable not an essential assumption of the model useful for obtaining intra-class correlation (r) and for design effect (d) r = σ2 υ σ 2 υ + σ 2 d = σ2 υ + σ 2 σ 2 = 1/(1 r) ratio of actual variance to the variance that would be obtained by simple random sampling (holding sample size constant) 5

6 Mixed-effects model β estimates from mixed-effects (random intercepts) model are larger (in abs. value) than from fixed-effects model by approximately because d = συ 2 + σ 2 σ 2 V (y) = σ 2 υ + σ 2 in mixed-effects (random intercepts) model V (y) = σ 2 in fixed-effects model difference depends on size of random-effects variance σ 2 υ more complex for models with multiple random effects 6

7 Treatment-Related Change Across Time Data from the NIMH Schizophrenia collaborative study on treatment related changes in overall severity. IMPS item 79, Severity of Illness, was scored as: 1 = normal or borderline mentally ill 2 = mildly or moderately ill 3 = markedly ill 4 = severely or among the most extremely ill The experimental design and corresponding sample sizes: Sample size at Week Group completers PLC (n=108) % DRUG (n=329) % Drug = Chlorpromazine, Fluphenazine, or Thioridazine Main question of interest: Was there differential improvement for the drug groups relative to the control group? 7

8 Stata code: schiz ordinal logits cd "u:\stata_long\" log using schiz_ordinal_logits.log, replace infile id imps79 imps79b imps79o inter tx week sweek txswk /// using SCHIZX1.DAT.txt, clear recode imps79 imps79b imps79o (-9=.) /* only use weeks 0, 1, 3 6 for these descriptives */ keep if week == 0 week == 1 week == 3 week == 6 tabulate imps79o, generate(imps79oi) collapse (mean) p1=imps79oi1 p2=imps79oi2 p3=imps79oi3 /// p4=imps79oi4, by(tx sweek) list 8

9 tx sweek p1 p2 p3 p

10 . gen codds1 = (p2 + p3 + p4) / p1 (1 missing value generated). gen codds2 = (p3 + p4) / (p1 + p2). gen codds3 = p4 / (p1 + p2 + p3). list tx sweek codds1 codds2 codds tx sweek codds1 codds2 codds

11 . gen clogit1 = log(codds1) (1 missing value generated). gen clogit2 = log(codds2). gen clogit3 = log(codds3). list tx sweek clogit1 clogit2 clogit tx sweek clogit1 clogit2 clogit

12 * graph cumulative logits graph twoway (connected clogit1 sweek if tx==0) /// (connected clogit1 sweek if tx==1), /// legend(label(1 "placebo") label(2 "drug")) /// yscale(range(0 4)) xlabel(0(.5)2.5) graph twoway (connected clogit2 sweek if tx==0) /// (connected clogit2 sweek if tx==1), /// legend(label(1 "placebo") label(2 "drug")) /// yscale(range( )) graph twoway (connected clogit3 sweek if tx==0) /// (connected clogit3 sweek if tx==1), /// legend(label(1 "placebo") label(2 "drug")) /// yscale(range(-3-1)) 12

13 clogit sweek placebo drug clogit sweek placebo drug clogit sweek placebo drug 13

14 Under SSI, Inc > SuperMix (English) or SuperMix (English) Student Under File click on Open Spreadsheet Open C:\SuperMixEn Examples\Workshop\Binary\SCHIZX1.ss3 (or C:\SuperMixEn Student Examples\Workshop\Binary\SCHIZX1.ss3) 14

15 C:\SuperMixEn Examples\Workshop\Binary\SCHIZX1.ss3 15

16 Select Imps79O column, then Edit > Set Missing Value 16

17 Within-Subjects / Between-Subjects components Within-subjects model - level 1 (j = 1,..., n i obs) λ ijc = γ c [b 0i + b 1i W eekj ] Between-subjects model - level 2 (i = 1,..., N subjects) b 0i = β 0 + β 2 Grp i + υ 0i b 1i = β 1 + β 3 Grp i + υ 1i υ i N ID(0, Σ υ ) 17

18 Under File click on Open Existing Model Setup Open C:\SuperMixEn Examples\Workshop\Ordinal\schizo2.mum (or C:\SuperMixEn Student Examples\Workshop\Ordinal\schizo2.mum) 18

19 Note that Dependent Variable Type is ordered 19

20 Note the lack of TxDrug as an explanatory variable 20

21 Make sure Optimization Method is set to adaptive quadrature 21

22 Note: Cumulative Logit link function 22

23 Category response indicators (IMPS79O1-IMPS79O4); results of fixed-effects model (to be ignored, or for comparison purposes) 23

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27 Model Fit of Observed Proportions 27

28 Stata code: schiz ordinal mataest /* random int and trend model with sweek + tx*sweek from Supermix: Population Average Estimates Standard Parameter Estimate Error z Value P Value Threshold Threshold Threshold SqrtWeek Tx*SWeek */ mata beta = ( \ ) xmat_0 = (0, 0 \ 1, 0 \ , 0 \ , 0) xmat_1 = (0, 0 \ 1, 1 \ , \ , ) xbeta_0 = xmat_0*beta xbeta_1 = xmat_1*beta cut1 = * J(4,1,1) cut2 = * J(4,1,1) cut3 = * J(4,1,1) 28

29 /* results for placebo group */ clogit1_0 = cut1 - xbeta_0 clogit2_0 = cut2 - xbeta_0 clogit3_0 = cut3 - xbeta_0 cprob1_0 = invlogit(clogit1_0) cprob2_0 = invlogit(clogit2_0) cprob3_0 = invlogit(clogit3_0) prob1_0 = cprob1_0 prob2_0 = cprob2_0 - cprob1_0 prob3_0 = cprob3_0 - cprob2_0 prob4_0 = J(4,1,1) - cprob3_0 prob1_0 prob2_0 prob3_0 prob4_0 29

30 : prob1_ : prob2_ : prob3_ : prob4_

31 /* results for drug group */ clogit1_1 = cut1 - xbeta_1 clogit2_1 = cut2 - xbeta_1 clogit3_1 = cut3 - xbeta_1 cprob1_1 = invlogit(clogit1_1) cprob2_1 = invlogit(clogit2_1) cprob3_1 = invlogit(clogit3_1) prob1_1 = cprob1_1 prob2_1 = cprob2_1 - cprob1_1 prob3_1 = cprob3_1 - cprob2_1 prob4_1 = J(4,1,1) - cprob3_1 prob1_1 prob2_1 prob3_1 prob4_1 end 31

32 : prob1_ : prob2_ : prob3_ : prob4_

33 SAS IML code: SCHZOFIT.SAS - computing marginal probabilities - ordinal model adapted from syntax at (Week 12) TITLE1 NIMH Schizophrenia Data - Estimated Marginal Probabilities ; PROC IML; /* Results from random intercept and trend model */; /* using Population Average Estimates */; x0 = { , , , }; x1 = { , , , }; beta = { , }; thresh = { , , }; 33

34 za0 = (thresh[1] - x0*beta) ; zb0 = (thresh[2] - x0*beta) ; zc0 = (thresh[3] - x0*beta) ; za1 = (thresh[1] - x1*beta) ; zb1 = (thresh[2] - x1*beta) ; zc1 = (thresh[3] - x1*beta) ; grp0a = 1 / ( 1 + EXP(- za0)); grp0b = 1 / ( 1 + EXP(- zb0)); grp0c = 1 / ( 1 + EXP(- zc0)); grp1a = 1 / ( 1 + EXP(- za1)); grp1b = 1 / ( 1 + EXP(- zb1)); grp1c = 1 / ( 1 + EXP(- zc1)); print Random intercept and trend model ; print using Population Average Estimates ; print marginal prob for group 0 - catg 1 grp0a [FORMAT=8.4]; print marginal prob for group 0 - catg 2 (grp0b-grp0a) [FORMAT=8.4]; print marginal prob for group 0 - catg 3 (grp0c-grp0b) [FORMAT=8.4]; print marginal prob for group 0 - catg 4 (1-grp0c) [FORMAT=8.4]; print marginal prob for group 1 - catg 1 grp1a [FORMAT=8.4]; print marginal prob for group 1 - catg 2 (grp1b-grp1a) [FORMAT=8.4]; print marginal prob for group 1 - catg 3 (grp1c-grp1b) [FORMAT=8.4]; print marginal prob for group 1 - catg 4 (1-grp1c) [FORMAT=8.4]; 34

35 Proportional and Non-proportional Odds Proportional Odds model log P (Y ij c) = γ c [ x 1 P (Y ij c) ijβ + z ] ijυ i with υ i N(0, Σ υ ) relationship between the explanatory variables and the cumulative logits does not depend on c effects of x variables DO NOT vary across the C 1 cumulative logits 35

36 Non-Proportional/Partial Proportional Odds model log P (Y ij c) = γ 1 P (Y ij c) 0c [ u ijγ c + x ijβ + z ] ijυ i u ij = h 1 vector for the set of h covariates for which proportional odds is not assumed effects of u variables DO vary across the C 1 cumulative logits more flexible model for ordinal response relations can be used to empirically test proportional odds assumption 36

37 Proportional Odds Assumption: covariate effects are the same across all cumulative logits Response group Absent Mild Severe total control cumulative odds = = 2.7 logit -1 1 treatment cumulative odds = = 4.6 logit group difference =.5 for both cumulative logits 37

38 Non-Proportional Odds: covariate effects vary across the cumulative logits Response group Absent Mild Severe total control cumulative odds = = 2.7 logit -1 1 treatment cumulative odds = = 7.3 logit UNEQUAL group difference across cumulative logits 38

39 Open C:\SuperMixEn Examples\Workshop\Ordinal\schizo2np.mum (or C:\SuperMixEn Student Examples\Workshop\Ordinal\schizo2np.mum) 39

40 Note that Dependent Variable Type is ordered 40

41 Two explanatory variables: SqrtWeek and Tx SWeek 41

42 Explanatory Variable Interactions - both are selected 42

43 Proportional Odds Assumption Accepted: χ 2 4 = =

44 Linear Transforms Fixed part of model: λ c = ˆγ 0c ˆβ1 SqrtWeek + ˆβ 2 Tx*SWeek + ˆγ 1c SqrtWeek + ˆγ 2c Tx*SWeek] cumulative logit variable 1 vs 2,3,4 1,2 vs 3,4 1,2,3, vs 4 SqrtWeek ˆβ1 ˆβ1 + ˆγ 12 ˆβ1 + ˆγ 13 Tx SWeek ˆβ2 ˆβ2 + ˆγ 22 ˆβ2 + ˆγ 23 H 0 : β 1 + γ 12 = 0; SqrtWeek effect is 0 on the 2nd cumulative logit z = ˆβ 1 + ˆγ 12 SE( ˆβ 1 + ˆγ 12 ) 44

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48 NIMH Schiz Study: Severity of Illness (N = 437) Ordinal LR Estimates (se) - random intercept and trend model Proportional Non-Proportional Odds Model Odds Model 1 vs 2,3,4 1,2 vs 3,4 1,2,3 vs 4 Time (sqrt week) (0.190) (0.331) (0.224) (0.206) Drug by Time (0.208) (0.288) (0.229) (0.237) 2 log L Proportional Odds accepted (χ 2 4 = = 1.35) 48

49 Stata code: schizo.do log using c:\stata_examples\schizo.log, replace infile id imps79 imps79b imps79o inter tx week sweek txswk /// using clear recode imps79 imps79b imps79o (-9=.) summarize meologit imps79o sweek tx txswk id: meologit imps79o sweek tx txswk id: sweek, /// covariance(unstructured) log close this code is for proportional odds mixed model; Stata doesn t appear to have a procedure for non-proportional odds mixed model (other than treating the outcome as nominal) 49

50 SAS code: schizo.sas FILENAME SchizDat HTTP " DATA one; INFILE SchizDat; INPUT id imps79 imps79b imps79o int tx week sweek txswk ; /* get rid of observations with missing values */ IF imps79 > -9; /* random trend ORDINAL logistic regression via GLIMMIX */ PROC GLIMMIX METHOD=QUAD(QPOINTS=11) NOCLPRINT; CLASS id; MODEL imps79o(desc) = tx sweek tx*sweek / S DIST=MULTINOMIAL LINK=CUMLOGIT; RANDOM INTERCEPT sweek / SUBJECT=id TYPE=UN GCORR; RUN; GLIMMIX only allows for proportional odds mixed model; can use NLMIXED for non-proportional odds mixed model 50

51 /* ordinal NON PROPORTIONAL ODDS random trend model via NLMIXED */ PROC NLMIXED QPOINTS=11; PARMS b1_tx=-.06 b1_sweek=-.77 b1_txswk=-1.21 b2_tx=-.06 b2_sweek=-.77 b2_txswk=-1.21 b3_tx=-.06 b3_sweek=-.77 b3_txswk=-1.21 v0=3.77 c01=0 v1=1 g1=-5.9 g2=-2.8 g3=-.71; z1 = b1_tx*tx + b1_sweek*sweek + b1_txswk*tx*sweek + u0 + u1*sweek; z2 = b2_tx*tx + b2_sweek*sweek + b2_txswk*tx*sweek + u0 + u1*sweek; z3 = b3_tx*tx + b3_sweek*sweek + b3_txswk*tx*sweek + u0 + u1*sweek; IF (imps79o=1) THEN p = 1 / (1 + EXP(-(g1-z1))); ELSE IF (imps79o=2) THEN p = (1/(1 + EXP(-(g2-z2)))) - (1/(1 + EXP(-(g1-z1)))); ELSE IF (imps79o=3) THEN p = (1/(1 + EXP(-(g3-z3)))) - (1/(1 + EXP(-(g2-z2)))); ELSE IF (imps79o=4) THEN p = 1 - (1 / (1 + EXP(-(g3-z3)))); loglik = LOG(p); MODEL imps79o ~ GENERAL(loglik); RANDOM u0 u1 ~ NORMAL([0,0], [v0,c01,v1]) SUBJECT=id; ESTIMATE re corr c01/sqrt(v0*v1); RUN; 51

52 San Diego Homeless Research Project (Hough) 361 mentally ill subjects who were homeless or at very high risk of becoming homeless 2 conditions: HUD Section 8 rental certificates (yes/no) baseline and 6, 12, and 24 month follow-ups Categorical outcome: housing status streets / shelters (Y = 0) community / institutions (Y = 1) independent (Y = 2) Question: Do Section 8 certificates influence housing status across time? 52

53 Stata code: sdhouse ordinal logits.do cd "u:\stata_long\" log using sd_ordinal_logits.log, replace infile id Housing Inter Section8 Time1 Time2 Time3 Sect8T1 Sect8T2 Sect8T3 /// NonSect8 LinTime S8LinT using sdhouse.dat.txt, clear recode Housing (999=.) gen time6mo = LinTime replace time6mo = 4 if (time6mo ==3) tabulate Housing, generate(housingi) collapse (mean) p1=housingi1 p2=housingi2 p3=housingi3, by(section8 time6mo) list gen codds1 = (p2 + p3) / p1 gen codds2 = p3 / (p1 + p2) list Section8 time6mo codds1 codds2 gen clogit1 = log(codds1) gen clogit2 = log(codds2) list Section8 time6mo clogit1 clogit2 53

54 * graph cumulative logits graph twoway (connected clogit1 time6mo if Section8==0) /// (connected clogit1 time6mo if Section8==1), /// legend(label(1 "Non Section8") label(2 "Section8")) /// yscale(range(-.5 3)) xlabel(0(.5)4) graph twoway (connected clogit2 time6mo if Section8==0) /// (connected clogit2 time6mo if Section8==1), /// legend(label(1 "Non Section8") label(2 "Section8")) /// yscale(range( )) log close 54

55 clogit1: independent + community vs street clogit time6mo Non Section8 Section8 clogit2: independent vs community + street clogit time6mo Non Section8 Section8 55

56 Under SSI, Inc > SuperMix (English) or SuperMix (English) Student Under File click on Open Spreadsheet Open C:\SuperMixEn Examples\Workshop\Nominal\SDHOUSE.ss3 (or C:\SuperMixEn Student Examples\Workshop\Nominal\SDHOUSE.ss3) 56

57 C:\SuperMixEn Examples\Workshop\Nominal\SDHOUSE.ss3 57

58 Select Housing column, then Edit > Set Missing Value 58

59 Under File click on Open Existing Model Setup Open C:\SuperMixEn Examples\Workshop\Ordinal\SDO1.mum (or C:\SuperMixEn Student Examples\Workshop\Ordinal\SDO1.mum) 59

60 Note that Dependent Variable Type is ordered 60

61 All explanatory variables are indicator (dummy) variables 61

62 Housing status across time: 1289 observations within 361 subjects Ordinal Mixed Regression Model estimates and standard errors (se) Proportional Odds Non-Proportional Odds Non-street 1 Independent 2 difference term estimate se estimate se estimate se estimate se threshold threshold t1 (6 month) t2 (12 month) t3 (24 month) section 8 (yes=1) section 8 by t section 8 by t section 8 by t subject var (ICC.4) 2 log L (χ 2 7 = 52.14) bold indicates p <.05 italic indicates.05 < p <.10 1 = independent + community vs street 2 = independent vs community + street 62

63 For Non-Proportional Odds model, under File click on Open Existing Model Setup Open C:\SuperMixEn Examples\Workshop\Ordinal\SDO2.mum (or C:\SuperMixEn Student Examples\Workshop\Ordinal\SDO2.mum) 63

64 Note that Dependent Variable Type is ordered 64

65 Note Explanatory Variable Interactions is set to 7 65

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67 Stata code: sdo.do log using c:\stata_examples\sdo.log, replace infile Id Housing Inter Section8 Time1 Time2 Time3 /// Sect8T1 Sect8T2 Sect8T3 NonSect8 LinTime S8LinT /// using clear recode Housing (999=.) summarize meologit Housing Section8 Time1 Time2 Time3 Sect8T1 Sect8T2 Sect8T3 Id: log close this code is for proportional odds mixed model; Stata doesn t appear to have a procedure for non-proportional odds mixed model (other than treating the outcome as nominal) 67

68 SAS code: sdo.sas FILENAME SDDat HTTP " DATA one; INFILE SDDat; INPUT Id Housing Int Section8 Time1 Time2 Time3 Sect8T1 Sect8T2 Sect8T3 NonSect8 LinTime S8LinT ; IF Housing = 999 then Housing =.; /* Random-intercept proportional odds model via GLIMMIX */ PROC GLIMMIX DATA=one METHOD=QUAD(QPOINTS=21) NOCLPRINT; CLASS id; MODEL Housing(DESC) = Time1 Time2 Time3 Section8 Sect8T1 Sect8T2 Sect8T3 / SOLUTION DIST=MULTINOMIAL LINK=CUMLOGIT; RANDOM INTERCEPT / SUBJECT=id; RUN; GLIMMIX only allows for proportional odds mixed model; can use NLMIXED for non-proportional odds mixed model 68

69 /* Random-intercept NON-proportional odds model via NLMIXED */ PROC NLMIXED DATA=one QPOINTS=21; PARMS b1_time1=1.4 b1_time2=1.8 b1_time3=2.0 b1_section8=.4 b1_sect8t1=.9 b1_sect8t2=.9 b1_sect8t3=.4 b2_time1=1.4 b2_time2=1.8 b2_time3=2.0 b2_section8=.4 b2_sect8t1=.9 b2_sect8t2=.9 b2_sect8t3=.4 g1=.2 g2=2.3 varu=1; z1 = b1_time1*time1 + b1_time2*time2 + b1_time3*time3 + b1_section8*section8 + b1_sect8t1*sect8t1 + b1_sect8t2*sect8t2 + b1_sect8t3*sect8t3+ u; z2 = b2_time1*time1 + b2_time2*time2 + b2_time3*time3 + b2_section8*section8 + b2_sect8t1*sect8t1 + b2_sect8t2*sect8t2 + b2_sect8t3*sect8t3+ u; IF (HOUSING=0) THEN p = 1 / (1 + EXP(-(g1-z1))); ELSE IF (HOUSING=1) THEN p = (1/(1 + EXP(-(g2-z2)))) - (1/(1 + EXP(-(g1-z1)))); ELSE IF (HOUSING=2) THEN p = 1 - (1 / (1 + EXP(-(g2-z2)))); LogLike = LOG(p); MODEL Housing ~ GENERAL(LogLike); RANDOM u ~ NORMAL(0,varu) SUBJECT=Id; ESTIMATE icc varu/((((atan(1)*4)**2)/3) + varu); RUN; 69

70 Mixed Multinomial Logistic Regression Model Y ij = nominal response of level-2 unit i and level-1 unit j Which member of The Polkaholics is your favorite? (asked before, during, and after a show) 70

71 Mixed-effects Multinomial Logistic Regression Model log p ijc p ij1 = u ijγ c + z ijυ ic C 1 contrasts to reference cell (c = 1) regression effects γ c vary across contrasts random-effects υ ic vary across contrasts independent correlated For example, with C = 3 c = 2, 3,... C contrast ordinal nominal c1 2 & 3 vs 1 2 vs 1 c2 3 vs 1 & 2 3 vs 1 71

72 Model in terms of the category probabilities p ijc = Pr(Y ij = c υ ic ) = exp(z ijc ) 1 + C h=2 exp(z ijh ) for c = 2, 3,..., C p ij1 = Pr(Y ij = 1 υ ic ) = C h=2 exp(z ijh ) where the multinomial logit z ijc = u ijγ c + z ijυ ic 72

73 Stata code: sdhouse nominal logits.do cd "u:\stata_long\" log using sd_nominal_logits.log, replace infile id Housing Inter Section8 Time1 Time2 Time3 Sect8T1 Sect8T2 Sect8T3 /// NonSect8 LinTime S8LinT using sdhouse.dat.txt, clear recode Housing (999=.) gen time6mo = LinTime replace time6mo = 4 if (time6mo ==3) tabulate Housing, generate(housingi) collapse (mean) p1=housingi1 p2=housingi2 p3=housingi3, by(section8 time6mo) list gen odds2 = p2 / p1 gen odds3 = p3 / p1 list Section8 time6mo odds2 odds3 gen logit2 = log(odds2) gen logit3 = log(odds3) list Section8 time6mo logit2 logit3 73

74 * graph cumulative logits graph twoway (connected logit2 time6mo if Section8==0) /// (connected logit2 time6mo if Section8==1), /// legend(label(1 "Non Section8") label(2 "Section8")) /// yscale(range(-2 2)) xlabel(0(.5)4) graph twoway (connected logit3 time6mo if Section8==0) /// (connected logit3 time6mo if Section8==1), /// legend(label(1 "Non Section8") label(2 "Section8")) /// yscale(range(-2 2)) log close 74

75 logit2: community vs street logit time6mo Non Section8 Section8 logit3: independent vs street logit time6mo Non Section8 Section8 75

76 Under File click on Open Existing Model Setup Open C:\SuperMixEn Examples\Workshop\Nominal\sdhouse.mum (or C:\SuperMixEn Student Examples\Workshop\Nominal\sdhouse.mum) 76

77 Note that Dependent Variable Type is nominal 77

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79 Can select first or last category as the reference cell 79

80 Try independent random effects first 80

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83 Now allow the random effects to be correlated 83

84 LR test comparing models: χ 2 1 = =

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86 STATA code: sdn.do log using u:\stata_examples\sdn.log, replace infile id housing inter section8 time1 time2 time3 /// sect8t1 sect8t2 sect8t3 nonsect8 lintime s8lint /// using clear recode housing (999=.) summarize local myx section8 time1 time2 time3 sect8t1 sect8t2 sect8t3 gsem (ib(first).housing <- myx mlogit gsem (1.housing <- myx M1[id]) (2.housing <- myx M2[id]), /// cov(m1[id]*m2[id]) mlogit log close 86

87 SAS code: sdn.sas FILENAME SDDat HTTP " DATA one; INFILE SDDat; INPUT Id Housing Int Section8 Time1 Time2 Time3 Sect8T1 Sect8T2 Sect8T3 NonSect8 LinTime S8LinT ; IF Housing = 999 then Housing =.; /* Nominal random intercept logistic via GLIMMIX - independent random effects */ PROC GLIMMIX DATA=one METHOD=QUAD(QPOINTS=11) NOCLPRINT; CLASS Id Housing; MODEL Housing(REF=FIRST) = Time1 Time2 Time3 Section8 Sect8T1 Sect8T2 Sect8T3 / SOLUTION DIST=MULTINOMIAL LINK=GLOGIT; RANDOM INTERCEPT / SUBJECT=Id GROUP=Housing; RUN; GLIMMIX assumes the random effects for the logits are independent; can use NLMIXED to allow for correlated random effects 87

88 /* Nominal random intercept logistic via NLMIXED - correlated random effects */ PROC NLMIXED DATA=one QPOINTS=11; PARMS b1_0=-.49 b1_t1=1.63 b1_t2=2.37 b1_t3=1.8 b1_sect8=.43 b1_sect8t1=-.46 b1_sect8t2=-2.12 b1_sect8t3=-1.1 var1=1 b2_0=-1.7 b2_t1=1.90 b2_t2=2.97 b2_t3=2.9 b2_sect8=.54 b2_sect8t1=1.13 b2_sect8t2=-.04 b2_sect8t3=-.07 var2=1 c12=0; z1 = b1_0 + b1_t1*time1 + b1_t2*time2 + b1_t3*time3 + b1_sect8*section8 + b1_sect8t1*sect8t1 + b1_sect8t2*sect8t2 + b1_sect8t3*sect8t3 + u1; z2 = b2_0 + b2_t1*time1 + b2_t2*time2 + b2_t3*time3 + b2_sect8*section8 + b2_sect8t1*sect8t1 + b2_sect8t2*sect8t2 + b2_sect8t3*sect8t3 + u2; IF (Housing=0) THEN p = 1 / (1 + EXP(z1) + EXP(z2)); ELSE IF (Housing=1) THEN p = EXP(z1) / (1 + EXP(z1) + EXP(z2)); ELSE IF (Housing=2) THEN p = EXP(z2) / (1 + EXP(z1) + EXP(z2)); LogLike = LOG(p); MODEL Housing ~ GENERAL(LogLike); RANDOM u1 u2 ~ NORMAL([0,0],[var1,c12,var2]) SUBJECT=Id; ESTIMATE ICC1 var1 /((((ATAN(1)*4)**2)/3) + var1); ESTIMATE ICC2 var2 /((((ATAN(1)*4)**2)/3) + var2); ESTIMATE re corr c12/sqrt(var1*var2); RUN; 88

89 Summary Models for longitudinal ordinal and nominal data as developed as models for continuous and dichotomous data Proportional odds models Non and partial proportional odds models Nominal models (with reference-cell contrasts) Grouped-time survival analysis models clust.pdf SuperMix can do it all, including 3-level models, also for counts full likelihood solution using adaptive quadrature 89