Mixed Models for Longitudinal Binary Outcomes. Don Hedeker Department of Public Health Sciences University of Chicago.

Transcription

1 Mixed Models for Longitudinal Binary Outcomes Don Hedeker Department of Public Health Sciences University of Chicago Hedeker, D. (2005). Generalized linear mixed models. In B. Everitt & D. Howell (Eds.), Encyclopedia of Statistics in Behavioral Science. Wiley. Hedeker, D. & Gibbons, R.D. (2006). Longitudinal Data Analysis, chapter 9. Wiley. 1

2 Random-intercept Logistic Regression Model Consider the model with p covariates for the response Y ij for subject i (i = 1, 2,..., N) at time j (j = 1, 2,..., n i ): log P (Y ij = 1) = x 1 P (Y ij = 1) ijβ + υ 0i where Y ij = dichotomous response for subject i at time j x ij = (p + 1) 1 covariate vector (includes 1 for intercept) β = (p + 1) 1 vector of unknown parameters υ 0i = subject effects distributed N ID(0, σ 2 υ) and assumed independent of x variables 2

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

4 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) 4

5 Scaling of regression coefficients β estimates from mixed-effects 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 model V (y) = σ 2 in fixed-effects model difference depends on size of random-effects variance σ 2 υ In clustered case, σ 2 υ is usually relatively small and the difference in size of β estimates is also small. However, for longitudinal data, σ 2 υ is usually relatively large and the difference in size of β estimates is therefore much larger. 5

6 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 2 = borderline mentally ill 3 = mildly ill 4 = moderately ill 5 = markedly ill 6 = severely ill 7 = 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? 6

7 Observed proportions moderately ill week 0 week 1 week 3 week 6 placebo drug Observed odds moderately ill week 0 week 1 week 3 week 6 placebo drug ratio Observed log odds moderately ill week 0 week 1 week 3 week 6 placebo drug difference exp (odds ratio)

8 Stata code: schizb plots.do log using U:\Data\StatHorizons\schizb_plots.log, replace infile id imps79 imps79b imps79o inter tx week sweek txswk /// using 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 summarize collapse (mean) p1=imps79b, by (tx week) gen odds1 = p1 / (1-p1) gen logit1 = log(odds1) list tx week p1 odds1 logit1 * graph probabilities & logits graph twoway (connected p1 week if tx==0) /// (connected p1 week if tx==1), /// legend(label(1 "placebo") label(2 "drug")) /// yscale(range(0 1)) xlabel(0(1)6) graph twoway (connected logit1 week if tx==0) /// (connected logit1 week if tx==1), /// legend(label(1 "placebo") label(2 "drug")) /// yscale(range( )) xlabel(0(1)6) 8

9 gen srweek = sqrt(week) graph twoway (connected logit1 srweek if tx==0) /// (connected logit1 srweek if tx==1), /// legend(label(1 "placebo") label(2 "drug")) /// yscale(range( )) xlabel(0(.5)2.5) 9

10 . collapse (mean) p1=imps79b, by (tx week). gen odds1 = p1 / (1-p1). gen logit1 = log(odds1). list tx week p1 odds1 logit tx week p1 odds1 logit

11 Observed Proportions across Time by Condition. graph twoway (connected p1 week if tx==0) /// > (connected p1 week if tx==1), /// > legend(label(1 "placebo") label(2 "drug")) /// > yscale(range(0 1)) xlabel(0(1)6) (mean) imps79b week placebo drug model is not linear in terms of probabilities 1 P r(y ij = 1) = 1 + exp x ij β + υ 0i 11

12 Observed Logits across Time by Condition. graph twoway (connected logit1 week if tx==0) /// > (connected logit1 week if tx==1), /// > legend(label(1 "placebo") label(2 "drug")) /// > yscale(range( )) xlabel(0(1)6) logit week placebo drug model is linear in terms of logits: log P (Y ij = 1) = x 1 P (Y ij = 1) ijβ+υ 0i 12

13 . gen srweek = sqrt(week). graph twoway (connected logit1 srweek if tx==0) /// > (connected logit1 srweek if tx==1), /// > legend(label(1 "placebo") label(2 "drug")) /// > yscale(range( )) xlabel(0(.5)2.5) logit srweek placebo drug time effect is approximately linear when expressed as week 13

14 Within-Subjects / Between-Subjects components Within-subjects model - level 1 (j = 1,..., n i obs) logit ij = 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 0i N ID(0, σ 2 υ) Put together, logit ij = (β 0 + β 2 Grp i + υ 0i ) + (β 1 + β 3 Grp i ) W eek j 14

15 Stata code: schizb.do log using c:\stata_examples\schizb.log, replace infile id imps79 imps79b imps79o inter tx week sweek txswk /// using clear recode imps79 imps79b imps79o (-9=.) summarize * random intercept model melogit imps79b sweek tx txswk, nolog id: scalar m1 = e(ll) estat icc * random intercept and trend model melogit imps79b sweek tx txswk, nolog id:sweek,covariance(unstructured) scalar m2 = e(ll) * obtain the random effects for each subject predict u1 u0, reffects * list the random effects for the first few subjects by id, sort: generate tolist = (_n==1) list id u0 u1 in 1/30 if tolist 15

16 * add random and fixed effects together, list out a few generate intercept = _b[_cons] + _b[tx]*tx + u0 generate slope = _b[sweek] + _b[txswk]*tx + u1 list id intercept slope in 1/30 if tolist * plot subjects intercepts and slopes twoway scatter intercept slope * generate an estimated spaghetti plot predict fitimps79b, fitted sort id sweek twoway connected fitimps79b sweek, connect(l) * get LR test for comparing models display "chibar(12) = " 2*(m2-m1) display "Prob > chibar2(12) = ".5*chi2tail(1, 2*(m2-m1)) + ///.5*chi2tail(2, 2*(m2-m1)) log close 16

17 . summarize Variable Obs Mean Std. Dev. Min Max id 3, imps79 1, imps79b 1, imps79o 1, inter 3, tx 3, week 3, sweek 3, txswk 3,

18 Mixed-effects logistic regression Number of obs = 1,603 Group variable: id Number of groups = 437 Obs per group: min = 2 avg = 3.7 max = 5 Integration method: mvaghermite Integration pts. = 7 Wald chi2(3) = Log likelihood = Prob > chi2 = imps79b Coef. Std. Err. z P> z [95% Conf. Interval] sweek tx txswk _cons id var(_cons) LR test vs. logistic model: chibar2(01) = Prob >= chibar2 =

19 Intraclass Correlation (ICC). estat icc Residual intraclass correlation Level ICC Std. Err. [95% Conf. Interval] id ICC = π 2 /3 = average pairwise correlation of repeated outcomes =.58 proportion of (unexplained) variation at the subject level is 58% 19

20 Within-Subjects / Between-Subjects components Within-subjects model - level 1 (j = 1,..., n i obs) logit ij = 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 ) Put together, logit ij = (β 0 + β 2 Grp i + υ 0i ) + (β 1 + β 3 Grp i + υ 1i ) W eek j 20

21 . melogit imps79b sweek tx txswk, nolog id: sweek, covariance(unstructured) Mixed-effects logistic regression Number of obs = 1,603 Group variable: id Number of groups = 437 Obs per group: min = 2 avg = 3.7 max = 5 Integration method: mvaghermite Integration pts. = 7 Wald chi2(3) = Log likelihood = Prob > chi2 = imps79b Coef. Std. Err. z P> z [95% Conf. Interval] sweek tx txswk _cons id 21

22 var(sweek) var(_cons) id cov(_cons,sweek) LR test vs. logistic model: chi2(3) = Prob > chi2 = Note: LR test is conservative and provided only for reference. 22

23 Postestimation: schizb.do * obtain the random effects for each subject predict u1 u0, reffects * list the random effects for the first few subjects by id, sort: generate tolist = (_n==1) list id u0 u1 in 1/30 if tolist id u0 u

24 * add random and fixed effects together, list out a few generate intercept = _b[_cons] + _b[tx]*tx + u0 generate slope = _b[sweek] + _b[txswk]*tx + u1 list id intercept slope in 1/30 if tolist id interc~t slope

25 * plot subjects intercepts and slopes twoway scatter intercept slope intercept slope 25

26 * generate an estimated spaghetti plot predict fitimps79b, fitted sort id sweek twoway connected fitimps79b sweek, connect(l) Linear prediction sweek 26

27 Model comparisons - Likelihood Ratio (LR) tests comparing random intercept to ordinary logistic regression LR test vs. logistic model: chibar2(01) = Prob >= chibar2 = H 0 : σ 2 υ = 0, H A : σ 2 υ > 0 one-sided test chibar2(01) refers to a 50:50 mixture of a χ 2 0 and a χ 2 1 distribution; chi-bar square distribution; p-value is obtained from χ 2 1, but is halved comparing random trend to ordinary logistic regression LR test vs. logistic model: chi2(3) = Prob > chi2 = Note: LR test is conservative and provided only for reference. H 0 : σ 2 υ 0 = σ 2 υ 1 = σ υ01 = 0 27

28 comparing random trend to random intercept model H 0 : σ 2 υ 1 = σ υ01 = 0 need chibar2(12), 50:50 mixture of a χ 2 1 and a χ 2 distribution; p-value is obtained from the average of χ 2 1 and χ 2 (i.e., q and q 1, where q is the number of (co)variance parameters in the null). * get LR test for comparing models. display "chibar(12) = " 2*(m2-m1) chibar(12) = display "Prob > chibar2(12) = ".5*chi2tail(1, 2*(m2-m1)) + /// >.5*chi2tail(2, 2*(m2-m1)) Prob > chibar2(12) = 8.600e-06 28

29 SAS code: schizb.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 intercept logistic regression via GLIMMIX */ PROC GLIMMIX DATA=one METHOD=QUAD NOCLPRINT; CLASS id; MODEL imps79b(desc) = tx sweek tx*sweek / SOLUTION DIST=BINARY LINK=LOGIT; RANDOM INTERCEPT / SUBJECT=id; /* random trend logistic regression via GLIMMIX */ PROC GLIMMIX DATA=one METHOD=QUAD NOCLPRINT; CLASS id; MODEL imps79b(desc) = tx sweek tx*sweek / SOLUTION DIST=BINARY LINK=LOGIT; RANDOM INTERCEPT sweek / SUBJECT=id TYPE=UN GCORR SOLUTION; ODS OUTPUT SOLUTIONR=ebest2; COVTEST test of random effects GLM; COVTEST test of random slope. 0 0; /* print out the estimated random effects */ PROC PRINT DATA=ebest2; RUN; 29

30 COVTEST test of random effects GLM compares model with random effects to ordinary (fixed) logistic regression H 0 : σ 2 υ 0 = σ 2 υ 1 = σ υ01 = 0 COVTEST test of random slope. trend to random intercept model H 0 : σ 2 υ 1 = σ υ01 = compares random using chibar2(12), 50:50 mixture of a χ 2 1 and a χ 2 2 distribution; p-value is obtained from the average of χ 2 1 and χ

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32 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) 32

33 C:\SuperMixEn Examples\Workshop\Binary\SCHIZX1.ss3 33

34 Select Imps79D column, then Edit > Set Missing Value 34

35 Under File click on Open Existing Model Setup Open C:\SuperMixEn Examples\Workshop\Binary\schizb1.mum (or C:\SuperMixEn Student Examples\Workshop\Binary\schizb1.mum) 35

36 Note Dependent Variable Type should be binary 36

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38 Note Optimization Method should be adaptive quadrature 38

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42 SuperMix is FAST for full-likelihood estimation up to 3-level models 42

43 Empirical Bayes Estimates of Random Effects Select Analysis > View Level-2 Bayes Results ID, random effect number, estimate, variance, name 43

44 Select Model-based Graphs > Equations 44

45 Estimated (subject-specific) Logits across Time by Condition: random-intercepts model log P r(y ij = 1) 1 P r(y ij = 1) = D i 1.50 T j 1.01 (D i T j ) + υ 0i υ 0i N ID(0, ˆσ 2 υ = 4.48) ˆβ change in (conditional) logit due to x for subjects with the same value of υ 0i (the above plot is for υ 0i = 0) 46

46 Random-intercepts Logistic Regression logit ij = x ijβ + υ 0i every subject has their own propensity for response (υ 0i ) the influence of covariates x is determined controlling (or adjusting) for the subject effect the covariance structure, or dependency, of the repeated observations is explicitly modeled 47

47 β 0 = log odds of response for a typical subject with x = 0 and υ 0i = 0 β = log odds ratio for response associated with unit changes in x for the same subject value υ 0i referred to as subject-specific how a subject s response probability depends on x συ 2 = degree of heterogeneity across subjects in the probability of response not attributable to x most useful when the objective is to make inference about subjects rather than the population average interest is in the heterogeneity of subjects 48

48 Estimated Subject-Specific Probabilites P r(y ij = 1) = exp [ ( D i 1.50 T j 1.01 D i T j + υ 0i )] where υ 0i = 1σ υ 1σ υ and ˆσ υ =

49 P (Y ij = 1) = exp [ ( D i 1.05 T j.60 D i T j )] obtained using the Population Average estimates from Supermix 50

50 Under File click on Open Existing Model Setup Open C:\SuperMixEn Examples\Workshop\Binary\schizb2.mum (or C:\SuperMixEn Student Examples\Workshop\Binary\schizb2.mum) 51

51 Note Dependent Variable Type should be binary 52

52 SqrtWeek is a level-2 (subject) random effect 53

53 Note Optimization Method should be adaptive quadrature 54

54 Comparing models: H0 : σ 2 υ 1 = σ υ01 = 0; χ 2 2 = = 21.97, p <

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56 Select Model-based Graphs > Equations 57

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58 Supermix - mixed models for binary outcomes link functions: logistic, probit, log-log, complementary log-log multiple random effects (correlated or independent) for up to 3-level models fast full-likelihood estimation using adaptive Gauss-Hermite quadrature discrete/grouped time survival analysis via person-period dataset Advanced > Level-2 (Co)variance Patterns > Unidimensional varying ICC model for MZ/DZ twin pair data: create indicator variables MZ and DZ, specify both as random effects, select Unidimensional Item-response theory (IRT) models: create indicator variables for items, specify all as random effects, select Unidimensional 59