Logistic Regression_Kleinbaum_2010

Bibliography 693 *McLaws, N., Irwig, L.M., Mock, P., Berry, G., and Gold, J., Predictors of surgical wound infection in Australia: A national study, Med. J. Aust. 149: 591–595, 1988. Pregibon, D., Logistic regression diagnostics, Ann. Stat. 9(4): 705–724, 1981. Prentice, R.L., and Pyke, R., Logistic disease incidence models and case-control studies, Biometrika 66(3): 403–411, 1979. Rezende, N.A., Blumberg, H.M., Metzger, B.S., Larsen, N.M., Ray, S.M., and McGowan, J.E., Risk factors for methicillin-resistance among patients with staph- ylococcus aureus bacteremia at the time of hospital admission, Am. J. Med. Sci., 323 (3): 117–123, 2002. Rothman, K.J., No adjustments are Needed for multiple comparisons, Epidemiology, 1(1): 43–46, 1990. Rothman, K.J., Greenland, S., and Lash, T.L. Modern Epidemiology, 3rd ed., Lippin- cott Williams & Wilkins, Philadelphia, PA, 2008. SAS Institute, SAS/STAT User’s Guide, Version 8.0, SAS Institute, Inc., Cary, NC, 2000. Sidack, Z., Rectangular confidence region for the means of multivariate normal distributions, J. Am. Stat. Assoc., 62: 626–633, 1967. Tigges, S., Pitts, S., Mukundan, S., Morrison, D., Olson, M., and Shahriara, A., External validation of the Ottawa Knee Rules in an Urban Trauma Center in the United States, AJR, 172: 1069–1071, 1999. Wikipedia contributors. Receiver operating characteristic Wikipedia, The Free Ency- clopedia. February 23, 2009, 20: 13 UTC. Available at: http://en.wikipedia.org/w/ index.php?title=Receiver_operating_characteristic&oldid=272792033. Accessed March 16, 2009. Wolfinger, R., and O’Connell, M., Generalized linear models: A pseudo-likelihood approach, J. Stat. Comput. Simul. 48: 233–243, 1993. Zack, M., Singleton, J., and Satterwhite, C., Collinearity macro (SAS), Unpublished, Department of Epidemiology RSPH at Emory University, (contact dkleinb@sph. emory.edu), 2009. *Sources for practice exercises or test questions presented at the end of several chapters.

Index A C Additive interaction, 53, 77 C measures, 274 Adjusted odds ratio, 26, 27, 82 Case-control study, 11–15, 392, 400–403 ALR. See Alternating logistic intercept in, 116 regressions pair-matched, 109–111, 395, 400–401 Alternating logistic regressions (ALR), Category matching, 393 Causal diagrams, 175–179 570–575 for common effect, 177–178 Area under the ROC curve (AUC), for confounding, 177 example with several exposures, 263 355–365 Change-of-estimate rule, 251, 255 c statistic formula, 359 Chi-square tests, 134, 138 geometrical approach, 362–365 Mantel–Haenszel, 134, 396 Aspirin-Heart Bypass Study, 551–555, Chunk test, 207–208 several exposure variables, 248 572–575 Classification table, 348, 350–352 Asymptotic properties, 516 CLR. See Conditional logistic regression Autoregressive correlation structure, Coding for dichotomous variables, 77–82 512 tests dependent on, 183 AR1, 498, 513, 545 tests independent of, 183–184 Collinearity, 270–275 B condition indices, 271–273 Backdoor path, 177 EVW model, 173–174 Background odds, 20 diagnosing, 271 Backward elimination approach, qualitative, 180 variance decomposition proportions 184–185 criticized, 264 (VDP’s), 271–273 an option for screening variables, 263 when screening variables, 263, 267 Baseline odds, 20 Common effect from two or more Berkson’s bias, 178 “Best” model, guidelines for, 168 causes, 177–178 Biased estimates, 117, 123 conditioning on, 178 in causal diagrams, 175–179 Complete separation of points, 358 Binomial-based model, 112 Computer data sets, 599–602 Black/White Cancer Survival Study, Conditional estimate, 111, 403 434, 470 Block diagonal matrix, 509 Bonferroni correction, 281–282 695

696 Index Conditional likelihood, 107–111, Covariance 115–116 correlation and, 500–502 defined, 500 Conditional logistic regression, 398, 403, matrix forms, 117–118, 132–133, 671 575–579 Covariate pattern, 307–310 SAS and, 612–614 number of covariate patterns (G), SPSS and, 642–643 307–308, 314–317 Stata and, 656–657 Condition indices (CNI’s), 271 Cox proportional hazards regression, Confidence intervals, 121 169 estimation of, 120, 140–141 interactions and, 142–146 Cross-sectional studies, 11–15 large-sample formula, 121, 447, 473 Cut-point (for classification), 348–352 matching, 393 narrower, 214 D odds ratios and, 141, 472–475 Data layout for correlated analysis, 499 one coefficient, 140–142 Data sets, computer. See Computer data Confounding assessment, 171–172, 204–230 sets causal diagrams for, 177 Deciles of risk, 318–319 change-of-estimate rule, 251, 255 Degrees of freedom, 134, 449 general procedure, 216 Delta-beta, 275–276 interactions and, 215–223 Dependence, and coding, 182–183 modeling strategy for, 204–220 Descriptive analyses, 181 odds ratio and, 87–91, 221 Deviance, 312–317 potential, 56, 57, 60, 62, 399 precision and, 211 events trials deviance (DevET), 314–315 screening, 264 subject specific deviance (DevSS), several exposure variables, 250–251, 254 Consistency, 516, 565 315–317 Controls, 392 Diagnostic table, 348 Control variables, 55 Diagonal matrix, 508 Cook’s distance, 276 Dichotomous data, 5, 440, 502–503 Cook’s distance-type measures, 274 Correlation. See also specific procedures, coding schemes for, 77–82 GEE and, 646–648, 659–660 parameters Discriminant function analysis, 107 autoregressive, 498, 512–513, 545 Discrimination measures block diagonal matrix and, 509 c statistic, 359, 367–368 covariance and, 500–502 gamma, 367–368 defined, 500 Somer’s D, 367–368 effects of ignoring, 497 Tau-a, 367–368 exchangeable, 511 Discriminatory performance (DP), fixed, 515 independent, 511 304–305, 356 matrix forms, 507–509 Dispersion, 523 stationary m-dependent, 514 Dummy variables, 82, 398, 399 structures, 507–510 types of, 511–516 interaction and, 405 unstructured, 510, 514–515 matching and, 109 E Effect modifiers, 57, 405 Eligible subsets, 214 Elimination, candidates for, 216 Epidemiologic study framework, 8

Index 697 Estimated variance-covariance matrix, Heartburn Relief Study, 555–557 117–118, 132–133 Infant Care Study, 542–550 matrix notation for, 524–525 Evans County Heart Disease Study, 61, parameters in, 526 62, 118, 146, 158, 188, 223–230, score-like equations and, 524–528 599 statistical properties of, 516 General odds ratio formula, 84 Events-trials (ET) format, 307 GLM. See Generalized linear models E, V, W model, 55–64 GLMM. See Generalized linear mixed general, 56–57 model logit form of, 76 Gold standard model, 212–215, 219 several exposure variables, 85–92, 245–246 Goodness of fit (GOF), 304–325 Exchangeable correlation structure, 511 Group prediction, 307, 310 Exchangeable sets, 407 Group saturated, 307, 310 Exposure variables, 4, 46 EVW model, 245–247 H odds ratio for, 87–91 Hat symbol, 9 single, 45 Heartburn Relief Study, 555–557 Extra variation, 523 Hierarchical backward-elimination F approach (HBWE), 184–185 False negatives (FN), 350 Hierarchically well-formulated (HWF) False positive rate (FPR), 355 model, 181–184, 406 1–specificity (synonym for FPR), 349, Hierarchy principle, 185 353, 355 applying, 216 False positives, (FP), 350 product terms and, 191 Family-wise error rate, 280 rationale for, 186 Fixed correlation structure, 515 retained variables, 185–187 Fixed effects, 576, 579 Hosmer–Lemeshow statistic, 318–325 Follow-up studies, 11–15 examples, 320–325 Forward selection approach formula, 320 table of observed and expected cases, an option for screening variables, 264 criticized, 264 319 Full model, 135 HWF. See Hierarchically well- Fully parameterized model, 308–311 formulated model G Hypothesis Generating Model, 260 GEE. See Generalized estimating Hypothesis testing, 9, 117, 120, 132–153 equations I Generalized linear mixed model Identity matrix, 511 Independent correlation structure, 511 (GLMM), 580–587 Independent of coding tests, 182–183 Generalized linear models (GLM), Independent variables, 4–5 Indicator variables, 398–399 503–506, 526 Individual prediction, 310 Generalizing estimating equations Infant care study, 493–498, 542–550, 559 Inference, statistical, 117–121, 130–153, (GEE) model, 492–538, 540–565 ALR model and, 569, 574 441–444. See also specific Aspirin-Heart Bypass Study, 551–555 methods asymptotic properties of, 516 Influential observations, 173, 275–278 defined, 506–507 dichotomous data and, 646–648, 658–660 GLM and, 526

698 Index Information matrix (IÀ1), 271 Likelihood statistic, log, 134 Interactions Linear regression, 169 Link function, 505, 506 additive, 51 Logistic function, shape of, 6–7 assessment of, 170–171, 190, 207–210 Logistic model, 5–8. See also Logistic coefficients of, 398 confidence interval estimation with, regression application of, 9–11 142–146 defined, 8 confounding and, 215–223 follow-up study and, 14–15 dummy variables and, 405 interaction, 49–55, 84 likelihood ratio test, 150 matching and, 397–400 matching, 404–406 multiplicative interaction, 49–55 modeling strategy for, 204–230 simple analysis, 43–46 multiplicative, 49–55 special cases of, 42–66 no interaction, 51–52, 63, 211–215 Logistic regression. See also Logistic odds ratio for several exposure variables model with, 87–91 ALR, 570–575 precision and, 211–215 basic features of, 4–7 product terms, 405 computing odds ratio in, 74–91 screening, 265 conditional, 398, 403, 575–579, 612–614, several exposure variables, 246 variables for, 56 642–643, 656–657 Wald tests, 448 defined, 5 Intercept term, 82, 116 introduction to, 2–32 Interval estimation. See Confidence matching and, 392–406 multiple standard, 479–481 interval estimation ordinal, 466, 481, 620–621, 644–646, Interval variables, 79 Invariance, 467–468 658 Iterative methods, 113 polytomous, 434–457, 617–619, 643, 657 statistical inferences for, 117–121, J Joint probability, 112, 114, 451 130–153, 441–444 stratified analysis, 398 L unconditional, 602–612, 635–640, L. See Likelihood function Large-sample formula, 121, 447, 473 649–654 Large versus small number of Logit form, of EVW model, 76 Logit transformation, 16–22 parameters debate, 108–110 Least squares (LS) estimation, 106 logistic model and, 17 Likelihood function (L), 111–117 log odds and, 19–21 Log likelihood statistic, 134 for conditional method, 114, 115 Log odds, logit and, 19–21 for ordinal model, 478–479 LR statistic. See Likelihood ratio for polytomous model, 450–452 LS. See Least squares estimation for saturated model, 311 for unconditional method, 114 M Likelihood ratio (LR) statistic, 134–138, Main effect variables, 27, 53 Mantel–Haenszel odds ratio (MOR), 449, 519 carrying out, 148 23, 396 defined, 120 Marginal model, 19, 20 interaction terms, 150 Matching, 116 application of, 400–403

Index 699 basic features of, 392–394 Multilevel outcomes, 432 case-control studies, 392–409 Multiple linear regression, 169 category matching, 393–394 Multiple standard logistic regressions, cohort studies, 409–413 confidence intervals, 393 453, 479–481 exchangeability and, 407–408 Multiple testing, 172 follow-up data, 409–413 Multiplicative interaction, 49–53 interaction and, 404–406 Multivariable problem, 4–5 logistic model and, 397–400 major disadvantage, 394 N matching factors, 392 No interaction model, 63, 78–79, 83–85, ML estimation and, 401 pooling, 407–409 149–150, 152–153, 211–215 precision and, 393 Nominal exposure variable, 82–84 stratification, 394–397 Normal distribution, 141 validity and, 394 Nuisance parameters, 116, 526, 575 Mathematical models, 5 Null hypotheses, 54, 280 Matrix algebra, 117–118, 132–133, O 507, 524 Odds, 18–19 Maximum likelihood (ML) methods, Odds ratio (OR), 11–13 106–107 adjusted, 26, 27, 77 numerical example, 147–153 computation of, 25–26, 64, 74–91 overview, 106 confidence limits, 141, 472–475 statistical inferences using, 130–153 confounders and, 87–91, 221 subject-specific saturated model, 312 correlation measure and, 571 unconditional versus conditional, examples of, 22–23 exchangeable, 573 107–109 as fixed, 60, 79 McNemar test, 396–397, 412 formula for, 22–25, 84 Meaningful change in OR, invariance of, 468 logistic regression and, 74–91 concept of, 217 MOR and, 396 Methicillin-resistance infection (MRSA) risk ratio and, 15–16 three categories, 437–441 example, 244 One-to-one matching, 393 Method 0, 265–270 OR. See Odds ratio MI dataset, 600 Ordinal logistic regression, 466–472 Mixed logistic model (MLM), SAS and, 620–621 SPSS and, 644–646 580, 584 Stata and, 658 ML. See Maximum likelihood methods Ordinal models, 466–472 MLM. See Mixed logistic model Ordinal variables, 79 Modeling strategy P confounding and, 203–230 Pair matching, 110–111, 393–398, example of, 188–192 guidelines for, 165–192 400–403 interaction and, 203–230 Parameterizing, of model, 470 overview of, 169–173 Parameters, number of, 108 rationale for, 168–169 Pearson statistic, 322 several exposure variables, 244–262 Perfect fit, 305 Moderate samples, 121 MOR. See Mantel–Haenszel odds ratio Multicollinearity, 172, 280

700 Index Perfect prediction, 310 Risk odds ratio (ROR), 23–24 Polytomous logistic regression, 434–437 estimated, 26 general formula for, 47–48 adding variables, 444–448 product formula for, 25 extending, 444–449 likelihood function for, 450–452 Risk ratio (RR), 11–13, 15–16 odds ratios from, 440 Robust conditions, 12 ordinal model and, 472 ROR. See Risk odds ratio proportional odds model, 468–469 RR. See Risk ratio SAS and, 617–619 R-to-1 matching, 393 SPSS and, 643–644 Stata and, 657 S Pooling, 407–409 Sample size, 121 Potential confounders, 56, 57, 60, 65, SAS software, 107, 553, 602–634 Saturated model, 305–307, 311 399 Scale factor, 522, 526–527 Precision Score equations, 521–523 Score-like equations, 521–528 confounding and, 211 Score statistic, 140 consideration of, 171, 222–223 Score test, 472–473, 480, 519–523 gaining, 214 Screening variables, 263–270 interaction and, 215–223 matching and, 393 assessing confounding, 264 validity and, 211–212 assessing interaction, 264 Predicted risk, 10 collinearity, 264, 267 Prediction, 167 Sensitivity, 349 Probability, 6, 18–19, 45, 112 Simple analysis, 46–48 Product terms, 28, 62, 119, 174 Single exposure variables, 45 hierarchy principle, 190–192, 196 Small samples, 121 interaction and, 210, 399, 405 Small versus large number of Proportional odds model, 466–472 alternate formulation of, 469 parameters debate, 108–110 polytomous model, 468–469 Software packages. See Computer data Q sets; specific programs Quasi-likelihood Specificity, 349 SPSS software, 107, 553, 635–648 estimating equations, 522 Standard error, estimated, 140–141 methods, 506 Standard logistic regression, 441 Stata software, 107, 553, 649–665 R Stationary m-dependent correlation Random effects, 579–585 Rare disease assumption, 16 structure, 514 Receiver operating characteristic (ROC) Statistical inference, 117–121, curve, 349, 355–358 130–153 SAS and, 614–617 Statistical tests for GEE, 519–520 SAS and, 640–642 Stratification, 116, 398 SAS and, 654–656 Reduced model, 54, 135 logistic regression, 398 Referent group, 50, 392 matching and, 394–397 Retaining variables, 185–186 Study variables, 4 Risk, 10, 43 Subject-specific effects, 580–585 Risk estimates, 13 Subsets, eligible, 214 Symmetric matrix, 508

Index 701 T V Test-wise error rate, 280 Validity, 169 Threshold idea, 7 Time-dependent variables, 494 matching and, 394 Time-independent variables, 494 precision and, 171–172 Trapezoid method, 360, 363–364 Variable specification, 169, 173–175, True negatives (TN), 349 True positive rate (TPR), 355 180–181, 245 True positives (TP), 349 Variables, retaining, 184–185 Variance-covariance matrix, 117–118, U Unbiased estimates, 117 121, 123, 132–133 Unconditional estimate, 403 collinearity assessment, 272 Unconditional logistic regression Variance decomposition proportions, 271 Variance estimators, 516–519 SAS and, 602–612 SPSS and, 635–640 W Stata and, 649–654 Wald tests, 138–140 Unconditional ML approach, carrying out, 148 107–111 defined, 120 Unknown parameters, 8 GEE models and, 519 Unstructured correlation structure, 510, ordinal model and, 475 polytomous model and, 443, 448 512–513 Z Z statistic, 139–140