Index A B Absolute standardized mean difference Bayesian approach, 218 (ASMD), 121–122 BlackBoost, 214 ACEAIPW precision Boosting model, 119–120 known propensity score model arbitrary function, 77 disadvantage, 79 C Monte Carlo computations, 79 cART. See Combined antiretroviral therapies quadratic function minimization, 78 (cART) simulated 100 datasets, 79, 80 Causal inference variance, 78 counterfactual outcome weighted mean squared error, 79 mediation, treatment effect, 8, 9 known response regression model, 80–81 post-treatment confounders, RCT, 7–8 Acquired immunodeficiency syndrome potential outcome, 5–6 (AIDS), 203 randomization, 5 AdaBoost, 213 selection bias, observational studies, 7 AIDS Clinical Trials Group (ACTG) Study epidemiology and clinical trials, 4 A5095, 204 ITT, 4 ASMD. See Absolute standardized mean statistical models difference (ASMD) case-control designs, 10 ATE. See Average treatment effect (ATE) causal mediation, 19–20 ATT. See Average treatment effect among the MAR mechanism, 9 treated (ATT) matching and propensity score Augmented inverse probability weighted matching, 10–11 (AIPW) estimator missing data, 9 construction, 75 MSMs, 12–13 correct PM, 74 post-treatment confounders, RCT, correct RRM, 74 13–19 double robustness property, 76 sequential ignorability (SI) and model HT estimator, 74, 75 identification, 20, 21 Average causal mediation effect (ACME), Causal mediation models 20 ACME, 20, 23 Average treatment effect (ATE), 112, 121 causal diagram, 244, 254 Average treatment effect among the treated CDF, random variable, 21 (ATT), 112, 121 direct effect/natural direct effect, 19 © Springer International Publishing Switzerland 2016 315 H. He et al. (eds.), Statistical Causal Inferences and Their Applications in Public Health Research, ICSA Book Series in Statistics, DOI 10.1007/978-3-319-41259-7 316 Index Causal mediation models (cont.) CDE. See Controlled direct effect (CDE) equivalence of different choices, 260 CFI. See Comparative fit index (CFI) estimation of parameters Child Resilience Project (CRP), 219, 234–236 general treatment X; 252–253 Chronic thromboembolic pulmonary maximum likelihood estimation, 250 hypertension (CTEPH), 104 moment estimation, 250 Cognitive behavioral smoking cessation three-value treatment, 251–252 therapy (CBT), 187 GMM, 243 Combined antiretroviral therapies (cART) identifiability of parameters ACTG A5095, 204, 209–211 continuous mediator M; 248–249 AdaBoost, 213 discrete variable M; 248 BlackBoost, 214 general conditions, 246–248 HIV-1 infected patients, 203 linear model of M; 249–250 methods indirect and direct effects, 241 missing data, 205–207 logistic regression equation, 258 two-stage designs, 207–209 LSEM, 21, 22 variance estimate, 209 matrix Geff, 261 non-parametric estimator, 212 means of estimates, 255 simulation studies, 211–212 mediator–outcome relationship, 242 The Commit to Quit (CTQ) study moderated-mediation model, 255–257 compliance model, 196–197 necessity and sufficiency theorem, 258–259 compliance regions, 200 notation and definitions, 243–246 data, 195–196 OLS estimates, 254 estimated causal effects, 199 OLS regression, 243 maximum likelihood estimates, 198 pure indirect effect, 20 principal effects, 196–197 three linear models, 242 two-stage ML approach, 200 total effect of treatment, 20 The Commit to Quit (CTQ) trials, 187 treatment–outcome relationship, 242 Comparative fit index (CFI), 303 unobserved pre-treatment confounder, 242 Compliance behavior, 13 Causal models Complier average causal effect (CACE), 14 CBT, 187 Controlled direct effect (CDE), 19, 269 continuous measures, 187 Cox regression, 104 CTQ study CRP. See Child Resilience Project (CRP) compliance model, 196–197 CTQ trials. See The Commit to Quit (CTQ) compliance regions, 200 trials data, 195–196 Cumulative distribution function (CDF), 21 estimated causal effects, 199 maximum likelihood estimates, 198 principal effects, 196–197 D two-stage ML approach, 200 Data-adaptive matching score, 117–118 likelihood and inference methods DomEXT Baseline, 234–235 compliance regions, 193–195 Donsker classes, 158–161 contribution, 192 two-stage approach, 192–193 placebo-controlled trials, 188 E principal stratification approach, 188 Empirical processes structural principal effects model average treatment effect, 161–163 compliance distributions, 191–192 Donsker classes, 158–161 ITT effects, 190–191 estimating equation, 157 notation and assumptions, 189–190 motivation and setup, 157–158 Causal relative risk, 176, 177 Estimated propensity variable (EPV), 62, CBT. See Cognitive behavioral smoking 63 cessation therapy (CBT) Estimating equation (EE), 37 CD4 cell, 210–211 Exposure to agents, 7 Index 317 F J Face-value average causal effect (FACE), 52 Jackknife method, 182–183 Fisher’s linear discriminant (LD), 61 Functional response models (FRM), 221 L Latent growth modeling (LGM), 296 G Likelihood and inference methods Generalized boosted model (GBM), 116–117 compliance regions, 193–195 Generalized Linear Structural Equation Models contribution, 192 (GLSEM), 21 two-stage approach, 192–193 Generalized method of moments (GMM), 243 Linear predictor (LP), 60 Genetic Epidemiology Network of Salt Linear SEM (LSEM), 21, 22 Sensitivity (GenSalt) Study LISREL formulation, 299–300 covariate adjustment, 43–44 Logistic regression covariates, 40 model construction, 70–71 outcomes, 39 propensity analysis, custodial sanctions parameter estimations, 40, 41 study, 71–73 pre vs. post score matching, 41–43 propensity score weighting approach, 43 treatment conditions, 39–40 M GMM. See Generalized method of moments Mahalanobis distance, 33 (GMM) Mahalanobis metric matching, 33 Greedy algorithm, 33 Mann-Whitney-Wilcoxon rank sum test, 222 MAR. See Missing at random (MAR) H Marginal structural models (MSMs), 12–13, Heteroscedasticity 274 balancing property of PS/PV, 67 mboost package, 207, 212 covariance matrices, 66 MCAR. See Missing Complete at Random linear discriminant, 67 (MCAR) QD, 67 Mean squared error (MSE), 278 simulations, 68–70 Missing at random (MAR), 9, 31, 302 High-Risk Youth Demonstration Grant Missing Complete at Random (MCAR), 235, Programs, 95 296, 308 Homoscedasticity MMDP. See Monotone missing data patterns asymptotic variance analysis (MMDP) EPV, 62, 63 Moderated-mediation model, 255–257 propensity variable, 62, 63 Monotone missing data patterns (MMDP), 227 sample size, 63 Monte Carlo (MC) cross-validation criteria, variance multiplier of coefficient, 63–64 122–123 model construction, 60–61 Monte Carlo (MC) mean, 278 precision, propensity analysis, 62 Monte Carlo (MC) replications, 212 simulations, 65, 66 MSE. See Mean squared error (MSE) Horvitz-Thompson (HT) estimator, 74, 75 MSMs. See Marginal structural models (MSMs) Multinomial logistic regression (MLR), 115 I Important variables stratification (IVS), 128 Intention to treat (ITT) approach, 4, 218, 219 N Inverse probability of treatment weights Natural direct effects (NDEs), 268–269 (IPTW), 102 Natural indirect effects (NIEs), 268–269 Inverse probability weighiting (IPW), 36, 106, Newton’s method, 170 273, 309 Nonparametric black-box algorithms, 112 ITT approach. See Intention to treat (ITT) Nonparametric curve regression methods, 37 approach Nonparametric density estimation, 119 318 Index Nonparametric models, 147 controlled effects, 274–275 Non-randomized controlled trials (non-RCTs), natural effects, 274 91 principal strata effects, 273–274 Nucleoside reverse-transciptase inhibitor identification, 289 (NRTI), 210–211 controlled effects, 272–274 natural effects, 271–272 principal strata effects, 270–271 O limitations, 289 Observational data, 91 RPM G-estimator, 275, 287 OLS regression. See Ordinary least squares simulation study (OLS) regression conditions, 275 Optimal pair matching (OPM) method data generation, 276–278 advantages, 126 IPWCDE, 286 control-Philadelphia creation MC SD, 286 covariate balance before and after no confounders, 279–282 matching, 130, 132–133 population values, 278–279 PSS illustration, 130, 131 post-T confounder, M and Y, 284 standardized differences, 133 pre-T confounder, M and Y, 281–284 stratification tree, 130, 132 pre-T confounder, T, M, and Y, 285–286 stratification variables and stratification squared MC SD, 278, 280 intervals, 129–130 TSLS IV estimator, 275, 285, 287 tolerance number of subclasses, 129 Principal stratification (PST), 15–16, 218 tolerance size of distance matrix, 129 PROC SYSLIN, 175 covariate balance, 126 Propensity analysis massive obstetric unit closures in balancing score, 58 Philadelphia, 127, 134 logistic regression, 70–73 rank-based Mahalanobis distances, 126, normal linear model (see 127 Heteroscedasticity; R package, 134 Homoscedasticity) stratification tree construction propensity variable, 59 checking matching feasibility, 127–128 PS, 58, 59 checking statistical criteria, 127 Propensity score (PS) estimation checking the number of strata after assessment steps, 97–98 propensity, 128 definition, 92 flowchart, 128, 129 empirical example IVS, 128 approximated Type IV Pearson PSS, 128 distribution, 95, 96 structure of data, 127 composite score, 30-day substance use, Ordinary least squares (OLS) regression, 243 95 empirical distribution of robustness, 95, 97 P empirical distribution of sensitivity Pearl’s causal framework, 50 indices, 95, 96 PHQ-9, 304 High-Risk Youth Demonstration Grant Potential outcome approaches Programs, 95 causal inference, 265–266 in literature, 105 define mediation effects, 287–288 missing confounder data CDE, 269 balance assessment, 107–108 controlled effects, 266 IPTW, 107 identification