Multilevel and longitudinal modeling using Stata.

Volume II Categorical responses, counts, and survival

Sophia Rabe-Hesketh, Anders Skrondal.

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Contents List of tables xvii List of figures xix List of displays xxv V Models for categorical responses 555 10 Dichotomous or binary responses 557 10.1 Introduction..............................................................557 10.2 Single-level logit and probit regression models for dichotomous responses.................................................................557 10.2.1 Generalized linear model formulation...............................558 Labor-participation data......................................... 561 Estimation using logit........................................... 561 Estimation using glm..............................................565 10.2.2 Latent-response formulation........................................566 Logistic regression...............................................568 Probit regression.................................................568 Estimation using probit...........................................569 10.3 Which treatment is best for toenail infection?............................571 10.4 Longitudinal data structure ..............................................571 10.5 Proportions and fitted population-averaged or marginal probabilities.............................................................573 Estimation using logit............................................575 10.6 Random-intercept logistic regression......................................577 10.6.1 Model specification...............................................577 Reduced-form specification....................................... 577 Two-stage formulation.............................................578 vi Contents 10.6.2 Model assumptions....................................................578 10.6.3 Estimation ......................................................... 579 Using xtlogit.......................................................580 Using melogit.......................................................584 Using gllamm........................................................585 10.7 Subject-specific or conditional versus population-averaged or marginal relationships......................................................586 10.8 Measures of dependence and heterogeneity.....................................590 10.8.1 Conditional or residual intraclass correlation of the latent responses.................................................................. 590 10.8.2 Median odds ratio................................................... 591 10.8.3 ♦♦♦ Measures of association for observed responses at median fixed part of the model................................................592 10.9 Inference for random-intercept logistic models...............................594 10.9.1 Tests and confidence intervals for odds ratios.....................594 10.9.2 Tests of variance components.....................................595 10.10 Maximum likelihood estimation..............................................596 10.10.1 ♦♦♦ Adaptive quadrature.............................................596 10.10.2 Some speed and accuracy considerations..............................599 Integration methods and number of quadrature points . . . 599 Starting values.....................................................601 Using melogit and gllamm for collapsible data.......................602 Spherical quadrature in gllamm......................................602 10.11 Assigning values to random effects........................................ 603 10.11.1 Maximum “likelihood” estimation ................................... 603 10.11.2 Empirical Bayes prediction..........................................604 10.11.3 Empirical Bayes modal prediction ...................................606 10.12 Different kinds of predicted probabilities.................................608 10.12.1 Predicted population-averaged or marginal probabilities . . 608 10.12.2 Predicted subject-specific probabilities............................609 Contents ѴІІ Predictions for hypothetical subjects: Conditional probabilities ........................................................609 Predictions for the subjects in the sample: Posterior mean probabilities.......................................... 611 10.13 Other approaches to clustered dichotomous data............................ 617 10.13.1 Conditional logistic regression................................. 617 Estimation using clogit...........................................618 10.13.2 Generalized estimating equations (GEE).......................... 619 Estimation using xtgee............................................620 10.14 Summary and further reading............................................... 622 10.15 Exercises................................................................. 624 11 Ordinal responses 635 11.1 Introduction................................................................635 11.2 Single-level cumulative models for ordinal responses........................635 11.2.1 Generalized linear model formulation............................ 636 11.2.2 Latent-response formulation..................................... 637 11.2.3 Proportional odds .............................................. 641 11.2.4 ♦♦♦ Identification...............................................642 11.3 Are antipsychotic drugs effective for patients with schizophrenia? . 645 11.4 Longitudinal data structure and graphs..................................... 645 11.4.1 Longitudinal data structure..................................... 646 11.4.2 Plotting cumulative proportions................................. 647 11.4.3 Plotting cumulative sample logits and transforming the time scale....................................................... 648 11.5 Single-level proportional-odds model .......................................650 11.5.1 Model specification............................................. 650 Estimation using ologit ......................................... 651 11.6 Random-intercept proportional-odds model................................... 654 11.6.1 Model specification............................................. 654 Estimation using meologit.........................................654 Estimation using gllamm ..........................................655 viii Contents 11.6.2 Measures of dependence and heterogeneity..........................657 Residual intraclass correlation of latent responses...............657 Median odds ratio.................................................657 11.7 Random-coefficient proportional-odds model................................658 11.7.1 Model specification................................................658 Estimation using meologit.........................................658 Estimation using gllamm ......................................... 660 11.8 Different kinds of predicted probabilities................................662 11.8.1 Predicted population-averaged or marginal probabilities . . 662 11.8.2 Predicted subject-specific probabilities: Posterior mean . . 665 11.9 Do experts differ in their grading of student essays?.....................669 11.10 A random-intercept probit model with grader bias..........................670 11.10.1 Model specification...............................................670 Estimation using gllamm ......................................... 670 11.11 Including grader-specific measurement-error variances.....................672 11.11.1 Model specification...............................................672 Estimation using gllamm ..........................................673 11.12 ♦♦♦ Including grader-specific thresholds..................................675 11.12.1 Model specification...............................................675 Estimation using gllamm ..........................................676 11.13 Other link functions..................................................681 Cumulative complementary log-log model............................681 Continuation-ratio logit model....................................681 Adjacent-category logit model.....................................683 Baseline-category logit and stereotype models.....................683 11.14 Summary and further reading...............................................684 11.15 Exercises................................................................ 685 12 Nominal responses and discrete choice 695 12.1 Introduction..............................................................695 12.2 Single-level models for nominal responses..............................696 ix 696 700 701 705 707 710 711 712 713 714 715 716 718 721 721 722 723 723 724 725 731 735 737 739 741 742 744 745 12.2.1 Multinomial logit models ........................................ Transport data version 1....................................... Estimation using mlogit........................................ 12.2.2 Conditional logit models with alternative-specific covariates Transport data version 2: Expanded form........................ Estimation using clogit........................................ Estimation using cmclogit...................................... 12.2.3 Conditional logit models with alternative- and unit-specific covariates.............................................................. Estimation using clogit........................................ Estimation using cmclogit...................................... Independence from irrelevant alternatives............................... Utility-maximization formulation ....................................... Does marketing affect choice of yogurt?................................. Single-level conditional logit models................................... 12.6.1 Conditional logit models with alternative-specific intercepts Estimation using clogit........................................ Estimation using cmclogit...................................... Multilevel conditional logit models..................................... 12.7.1 Preference heterogeneity: Brand-specific random intercepts Estimation using cmxtmixlogit.................................. Estimation using gllamm ....................................... 12.7.2 Response heterogeneity: Marketing variables with random coefficients........................................................ Estimation using cmxtmixlogit.................................. Estimation using gllamm ....................................... 12.7.3 ♦♦♦ Preference and response heterogeneity ....................... Estimation using cmxtmixlogit.................................. Estimation using gllamm ....................................... Prediction of marginal choice probabilities............................. x Contents 12.9 Prediction of random effects and household-specific choice probabilities .......................................................................747 12.10 Summary and further reading...............................................751 12.11 Exercises................................................................ 753 VI Models for counts 761 13 Counts 763 13.1 Introduction............................................................. 763 13.2 What are counts?......................................................... 763 13.2.1 Counts versus proportions.......................................... 763 13.2.2 Counts as aggregated event-history data............................ 764 13.3 Single-level Poisson models for counts................................... 765 13.4 Did the German healthcare reform reduce the number of doctor visits?................................................................... 767 13.5 Longitudinal data structure ............................................. 767 13.6 Single-level Poisson regression.......................................... 768 13.6.1 Model specification................................................ 768 Estimation using poisson ........................................ 769 Estimation using glm............................................. 771 13.7 Random-intercept Poisson regression...................................... 772 13.7.1 Model specification................................................ 772 13.7.2 Measures of dependence and heterogeneity............................773 13.7.3 Estimation .........................................................773 Using xtpoisson.................................................. 773 Using mepoisson.................................................. 775 Using gllamm..................................................... 776 13.8 Random-coefficient Poisson regression.................................... 778 13.8.1 Model specification.............................................. 778 Estimation using mepoisson....................................... 779 Estimation using gllamm ......................................... 782 13.9 Overdispersion in single-level models.................................... 784 Contents xi 13.9.1 Normally distributed random intercept............................784 Estimation using xtpoisson........................................ 785 13.9.2 Negative binomial models........................................... 786 Mean dispersion or NB2............................................ 786 Constant dispersion or NB1........................................ 788 13.9.3 Quasilikelihood.................................................... 788 Estimation using glm.............................................. 789 13.10 Level-1 overdispersion in two-level models ................................ 790 13.10.1 Random-intercept Poisson model with robust standard errors............................................................ 791 Estimation using mepoisson........................................ 791 13.10.2 Three-level random-intercept model................................ 792 13.10.3 Negative binomial models with random intercepts ...................792 Estimation using menbreg.......................................... 793 13.10.4 The HHG model..................................................... 794 13.11 Other approaches to two-level count data................................... 794 13.11.1 Conditional Poisson regression.................................... 794 Estimation using xtpoisson, fe.................................... 796 Estimation using Poisson regression with dummy variables for clusters................................................ 796 13.11.2 Conditional negative binomial regression.......................... 797 13.11.3 Generalized estimating equations.................................. 797 Estimation using xtgee............................................ 798 13.12 Marginal and conditional effects when responses are MAR................799 ♦♦♦ Simulation.................................................... 799 13.13 Which Scottish counties have a high risk of lip cancer?...................803 13.14 Standardized mortality ratios ............................................. 804 13.15 Random-intercept Poisson regression.........................................806 13.15.1 Model specification................................................806 Estimation using gllamm .......................................... 807 13.15.2 Prediction of standardized mortality ratios .......................808 xii Contents 13.16 ♦♦♦ Nonparametric maximum likelihood estimation.........................811 13.16.1 Specification ..................................................811 Estimation using gllamm ........................................811 13.16.2 Prediction......................................................816 13.17 Summary and further reading.............................................816 13.18 Exercises...............................................................818 VII Models for survival or duration data 827 Introduction to models for survival or duration data (part VII) 829 14 Discrete-time survival 835 14.1 Introduction.............................................................835 14.2 Single-level models for discrete-time survival data......................835 14.2.1 Discrete-time hazard and discrete-time survival..................835 Promotions data.................................................835 14.2.2 Data expansion for discrete-time survival analysis...............838 14.2.3 Estimation via regression models for dichotomous responses.......................................................840 Estimation using logit..........................................842 14.2.4 Including time-constant covariates...............................845 Estimation using logit..........................................846 14.2.5 Including time-varying covariates................................849 Estimation using logit..........................................853 14.2.6 Multiple absorbing events and competing risks....................855 Estimation using mlogit.........................................858 14.2.7 Handling left-truncated data.....................................860 14.3 How does mother’s birth history affect child mortality?..................861 14.4 Data expansion.......................................................... 862 14.5 ♦♦♦ Proportional hazards and interval-censoring......................... 864 14.6 Complementary log-log models.............................................865 14.6.1 Marginal baseline hazard .......................................866 Estimation using cloglog....................................... 867 Contents xiii 14.6.2 Including covariates........................................................................................................868 Estimation using cloglog....................................................................................................869 14.7 Random-intercept complementary log-log model........................................................................................872 14.7.1 Model specification ........................................................................................................872 Estimation using mecloglog..................................................................................................872 14.8 ♦♦♦ Population-averaged or marginal vs. cluster-specific or conditional survival probabilities 875 14.9 Summary and further reading.........................................................................................................879 14.10 Exercises • • • 880 15 Continuous-time survival 887 15.1 Introduction........................................................................................................................887 15.2 What makes marriages fail?........................................................................................................ 888 15.3 Hazards and survival............................................................................................................... 889 15.4 Proportional hazards models.........................................................................................................895 15.4.1 Piecewise exponential model..................................................................................................897 Estimation using streg......................................................................................................900 Estimation using poisson ...................................................................................................905 15.4.2 Cox regression model..............................................................................................906 Estimation using stcox......................................................................................................907 15.4.3 Cox regression via Poisson regression for expanded data . . 910 Estimation using xtpoisson, fe............................................................................................911 15.4.4 Approximate Cox regression: Poisson regression, smooth baseline hazard......................................................................................................911 Estimation using poisson ...................................................................................................912 15.5 Accelerated failure-time models....................................................................................................914 15.5.1 Log-normal model ..........................................................................................................916 Estimation using streg......................................................................................................917 Estimation using stintreg...................................................................................................919 15.6 Time-varying covariates.............................................................................................................920 Estimation using streg . ..................................................................................................923 15.7 Does nitrate reduce the risk of angina pectoris?....................................................................................924 xiv Contents 15.8 Marginal modeling ........................................................926 15.8.1 Cox regression with occasion-specific dummy variables . . . 927 Estimation using stcox....................................................928 15.8.2 Cox regression with occasion-specific baseline hazards . . . 929 Estimation using stcox, strata............................................930 15.8.3 Approximate Cox regression.........................................930 Estimation using poisson .........................................932 15.9 Multilevel proportional hazards models....................................933 15.9.1 Cox regression with gamma shared frailty...........................934 Estimation using stcox, shared....................................935 15.9.2 Approximate Cox regression with log-normal shared frailty 938 Estimation using mepoisson................................................939 15.9.3 Approximate Cox regression with normal random intercept and coefficient......................................................940 Estimation using mepoisson........................................942 15.10 Multilevel accelerated failure-time models................................943 15.10.1 Log-normal model with gamma shared frailty........................943 Estimation using streg............................................944 15.10.2 Log-normal model with log-normal shared frailty...................945 Estimation using mestreg..........................................946 15.10.3 Log-normal model with normal random intercept and random coefficient........................................................947 Estimation using mestreg..........................................948 15.11 Fixed-effects approach ...................................................949 15.11.1 Stratified Cox regression with subject-specific baseline hazards...................................................................949 Estimation using stcox, strata....................................950 15.12 ♦♦♦ Different approaches to recurrent-event data .........................951 15.12.1 Total-time risk interval..........................................952 15.12.2 Counting-process risk interval....................................956 15.12.3 Gap-time risk interval............................................958 Contents XV 15.13 Summary and further reading.............................................. 959 15.14 Exercises................................................................ 960 VIII Models with nested and crossed random effects 969 16 Models with nested and crossed random effects 971 16.1 Introduction..............................................................971 16.2 Did the Guatemalan-immunization campaign work?............................971 16.3 A three-level random-intercept logistic regression model..................973 16.3.1 Model specification................................................ 974 16.3.2 Measures of dependence and heterogeneity............................974 Types of residual intraclass correlations of the latent responses............................................... 974 Types of median odds ratios...................................... 975 16.3.3 Three-stage formulation............................................ 975 16.3.4 Estimation ........................................................ 976 Using melogit.................................................... 976 Using gllamm..................................................... 980 16.4 A three-level random-coefficient logistic regression model................984 16.4.1 Estimation ........................................................ 985 Using melogit.................................................... 985 Using gllamm..................................................... 988 16.5 Prediction of random effects............................................. 991 16.5.1 Empirical Bayes prediction......................................... 991 16.5.2 Empirical Bayes modal prediction .................................. 993 16.6 Different kinds of predicted probabilities............................... 994 16.6.1 Predicted population-averaged or marginal probabilities: New clusters .................................................... 994 16.6.2 Predicted median or conditional probabilities.......................995 16.6.3 Predicted posterior mean probabilities: Existing clusters . . 996 16.7 Do salamanders from different populations mate successfully? . . . 998 16.8 Crossed random-effects logistic regression...............................1000 16.8Л Setup for estimating crossed random-effects model using melogit .....................................................1001 16.8.2 Approximate maximum likelihood estimation....................1003 Estimation using melogit ....................................1003 16.8.3 Bayesian estimation..........................................1007 Brief introduction to Bayesian inference.....................1007 Priors for the salamander data...............................1010 Estimation using bayes: melogit .............................1011 16.8.4 Estimates compared...........................................1019 16.8.5 Fully Bayesian versus empirical Bayesian inference for random effects.......................................................1020 16.9 Summary and further reading............................................1026 16.10 Exercises.............................................................1027 A Syntax for gllamm, eq, and gllapred: The bare essentials 1035 В Syntax for gllamm 1041 С Syntax for gllapred 1053 D Syntax for gllasim 1057 References 1061 Author index 1077 Subject index 1085
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