Mixed Models for Longitudinal Binary Outcomes. Don Hedeker Department of Public Health Sciences University of Chicago.
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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
31 31
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
37 37
38 Note Optimization Method should be adaptive quadrature 38
39 39
40 40
41 41
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 <
55 56
56 Select Model-based Graphs > Equations 57
57 58
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
Mixed Models for Longitudinal Ordinal and Nominal Outcomes. Don Hedeker Department of Public Health Sciences University of Chicago
Mixed Models for Longitudinal Ordinal and Nominal Outcomes Don Hedeker Department of Public Health Sciences University of Chicago hedeker@uchicago.edu https://hedeker-sites.uchicago.edu/ Hedeker, D. (2008).
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