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tb: Biostatistical Methods Lachin
Lachin, John M. 2010. Biostatistical Methods: The Assessment of Relative Risks. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc. http://doi.wiley.com/10.1002/9780470907412.
Table of Content below are copied from book web sites
website, online material (not checked)
http://ca.wiley.com/WileyCDA/WileyTitle/productCd-0470508221.html
http://www.bsc.gwu.edu/bsc/webpage.php?no=30
I didn't read anything of it. I read related articles.
Interesting part is emphasis on risk and proportions, i.e. stats.proportion and related, logit and loglinear poisson
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- Biostatistics and Biomedical Science
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- Relative Risk Estimates and Tests for Independent Groups
- stats proportion, contingency table
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- Sample Size, Power, and Efficiency
- stats proportion, power, weightstatws, ...
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- Stratified-Adjusted Analysis for Independent Groups
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- Case-Control and Matched Studies
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- Applications of Maximum Likelihood and Efficient Scores
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- Logistic Regression Models
- GLM, Logit, Related: Probit
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- Analysis of Count Data
- GLM, Poisson, NegativeBinomial
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- Analysis of Event-Time Data
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- Appendix - Statistical Theory
1 Biostatistics and Biomedical Science.
1.1 Statistics and the Scientific Method.
1.2 Biostatistics.
1.3 Natural History of Disease Progression.
1.4 Types of Biomedical Studies.
1.5 Studies of Diabetic Nephropathy.
2 Relative Risk Estimates and Tests for Independent Groups.
2.1 Probability As a Measure of Risk.
2.2 Measures of Relative Risk.
2.3 Large Sample Distribution.
2.4 Sampling Models Likelihoods.
2.5 Exact Inference.
2.6 Large Sample Inferences.
2.7 SAS PROC FREQ.
2.8 Other Measures of Differential Risk.
2.9 Polychotomous and Ordinal Data.
2.10 Two Independent Groups With Polychotomous Response.
2.11 Multiple Independent Groups.
2.12 Problems.
3 Sample Size, Power, and Efficiency.
3.1 Estimation Precision.
3.2 Power of Z-Tests.
3.3 Test for Two Proportions.
3.4 Power of Chi-Square Tests.
3.5 SAS PROC POWER.
3.6 Efficiency.
3.7 Problems.
4 Stratified-Adjusted Analysis for Independent Groups.
4.1 Introduction.
4.2 Mantel-Haenszel Test and Cochran’s Test.
4.3 Stratified-Adjusted Estimators.
4.4 Nature of Covariate Adjustment.
4.5 Multivariate Tests of Hypotheses.
4.6 Tests of Homogeneity.
4.7 Efficient Tests of No Partial Association.
4.8 Asymptotic Relative Efficiency of Competing Tests.
4.9 Maximin-Efficient Robust Tests.
4.10 Random Effects Model.
4.11 Power and Sample Size for Tests of Association.
4.12 Polychotomous and Ordinal Data.
4.13 Problems.
5 Case-Control and Matched Studies.
5.1 Unmatched Case-Control (Retrospective) Sampling.
5.2 Matching.
5.3 Tests of Association for Matched Pairs.
5.4 Measures of Association for Matched Pairs.
5.5 Pair-Matched Retrospective Study.
5.6 Power Function of McNemar’s Test.
5.7 Stratified Analysis of Pair-Matched Tables.
5.8 Multiple Matching-Mantel-Haenszel Analysis.
5.9 Matched Polychotomous Data.
5.10 Kappa Index of Agreement.
5.11 Problems.
6 Applications of Maximum Likelihood and Efficient Scores.
6.1 Binomial.
6.2 2x2 Table: Product Binomial (Unconditionally).
6.3 2x2 Table, Conditionally.
6.4 Score-Based Estimate.
6.5 Stratified Score Analysis of Independent 2x2 Tables.
6.6 Matched Pairs.
6.7 Iterative Maximum Likelihood.
6.8 Problems.
7 Logistic Regression Models.
7.1 Unconditional Logistic Regression Model.
7.2 Interpretation of the Logistic Regression Model.
7.3 Tests of Significance.
7.4 Interactions.
7.5 Measures of the Strength of Association.
7.6 Conditional Logistic Regression Model for Matched Sets.
7.7 Models for Polychotomous or Ordinal Data.
7.8 Random Effects and Mixed Models.
7.9 Models for Multivariate or Repeated Measures.
7.10 Problems.
8 Analysis of Count Data.
8.1 Event Rates and the Homogeneous Poisson Model.
8.2 Over Dispersed Poisson Model.
8.3 Poisson Regression Model.
8.4 Over Dispersed and Robust Poisson Regression.
8.5 Conditional Poisson Regression for Matched Sets.
8.6 Negative Binomial Models.
8.7 Power and Sample Size.
8.8 Multiple Outcomes.
8.9 Problems.
9 Analysis of Event-Time Data.
9.1 Introduction to Survival Analysis.
9.2 Lifetable Construction.
9.3 Family of Weighted Mantel-Haenszel Tests.
9.4 Proportional Hazards Models.
9.5 Evaluation of Sample Size and Power.
9.6 Additional Models.
9.7 Analysis of Recurrent Events.
9.8 Problems.
Appendix Statistical Theory.
A.1 Introduction.
A.2 Central Limit Theorem and the Law of Large Numbers.
A.3 Delta Method.
A.4 Slutsky’s Convergence Theorem.
A.5 Least Squares Estimation.
A.6 Maximum Likelihood Estimation and Efficient Scores.
A.7 Tests of Significance.
A.8 Explained Variation.
A.9 Robust Inference.
A.10 Generalized Linear Models and Quasi-Likelihood.
A.11 Generalized Estimating Equations (GEE).
1 Biostatistics and Biomedical Science, 1
1.1 Statistics and the Scientific Method, 1 1.2 Biostatistics, 2 1.3 Natural History of Disease Progression, 3 1.4 Types of Biomedical Studies, 5 1.5 Studies of Diabetic Nephropathy, 7
2 Relative Risk Estimates and Tests for Independent Groups, 13
2.1 Probability as a Measure of Risk, 14 2.1.1 Prevalence and Incidence, 14 2.1.2 Binomial Distribution and Large Sample Approximations, 14 2.1.3 Asymmetric Confidence Limits, 16 2.1.3.1 Exact Confidence Limits, 16 2.1.3.2 Logit Confidence Limits, 16 2.1.3.3 Complementary log-log Confidence Limits, 17 2.1.3.4 Test-Inverted Confidence Limits, 18 2.1.4 Case of Zero Events, 19 2.2 Measures of Differential or Relative Risk, 19 2.3 Large Sample Distribution, 23 2.3.1 Risk Difference, 23 2.3.2 Relative Risk, 24 2.3.3 Odds Ratio, 26 2.4 Sampling Models: Likelihoods, 28 2.4.1 Unconditional Product Binomial Likelihood, 28 2.4.2 Conditional Hypergeometric Likelihood, 28 2.4.3 Maximum Likelihood Estimates, 30 2.4.4 Asymptotically Unbiased Estimates, 30 2.5 Exact Inference, 32 2.5.1 Confidence Limits, 32 2.5.2 Fisher-Irwin Exact Test, 33 2.6 Large Sample Inferences, 36 2.6.1 General Considerations, 36 2.6.2 Unconditional Test, 38 2.6.3 Conditional Mantel-Haenszel Test, 39 2.6.4 Cochran's Test, 40 2.6.5 Likelihood Ratio Test, 42 2.6.6 Test-Based Confidence Limits, 42 2.6.7 Continuity Correction, 43 2.6.8* Establishing Equivalence or Noninferiority, 45 2.6.8.1* Equivalence, 45 2.6.8.2* Noninferiority, 46 2.7 SAS PROC FREQ, 48 2.8 Other Measures of Differential Risk, 52 2.8.1 Attributable Risk Fraction, 52 2.8.2 Population Attributable Risk, 53 2.8.3 Number Needed to Treat, 56 2.9* Polychotomous and Ordinal Data, 56 2.9.1* Multinomial Distribution and Large Sample Approximation, 56 2.9.2* Pearson Chi-Square Test, 57 2.9.3* Pearson Goodness-of-Fit Test, 59 2.9.4* Logits, 60 2.10* Two Independent Groups with Polychotomous Response, 61 2.10.1* Large Sample Test of Proportions, 61 2.10.2* The Pearson Contingency Chi-Square Test, 62 2.10.3* Odds Ratios, 63 2.10.4* Rank Tests: Cochran-Mantel-Haenszel Mean Scores Test, 63 2.11* Multiple Independent Groups, 67 2.11.1* The Pearson Test, 67 2.11.2* Measures of Association, 69 2.11.3* Logits, 69 2.11.4* Multiple Tests, 69 2.11.5* Rank and Correlation Tests, 73 2.11.6* The Cochran-Armitage Test for Trend, 74 2.11.7* Exact Tests, 76 2.12 Problems, 76
3 Sample Size, Power, and Efficiency, 85
3.1 Estimation Precision, 86 3.2 Power of Z-Tests, 87 3.2.1 Type I and II Errors and Power, 87 3.2.2 Power and Sample Size, 90 3.3 Test for Two Proportions, 92 3.3.1 Power of the Z-Test, 93 3.3.2 Relative Risk and Odds Ratio, 95 3.3.3* Equivalence, 96 3.3.4* Noninferiority, 98 3.4 Power of Chi-Square Tests, 99 3.4.1* Noncentral Chi-Square Distribution, 99 3.4.2* Pearson Chi-Square Tests, 100 3.4.2.1* A Multinomial Distribution, 100 3.4.2.2* RxC Contingency Table, 101 3.4.3* The Mean Score (Rank) Test, 102* 3.4.4* The Cochran-Armitage Test of Trend, 104 3.5* SAS PROC POWER, 106 3.5.1* Test for Two Proportions, 106 3.5.2* Wilcoxon Mann-Whitney Test, 107 3.6 Efficiency, 108 3.6.1 Pitman Efficiency, 108 3.6.2 Asymptotic Relative Efficiency, 110 3.6.3 Estimation Efficiency, 111 3.6.4 Stratified Versus Unstratified Analysis of Risk Differences, 112 3.7 Problems, 115
4 Stratified-Adjusted Analysis for Independent Groups, 119
4.1 Introduction, 119 4.2 Mantel-Haenszel Test and Cochran's Test, 121 4.2.1 Conditional Within-Strata Analysis, 121 4.2.2 Marginal Unadjusted Analysis, 121 4.2.3 Mantel-Haenszel Test, 123 4.2.4 Cochran's Test, 125 4.3 Stratified-Adjusted Estimators, 126 4.3.1 Mantel-Haenszel Estimates, 126 4.3.2 Test-Based Confidence Limits, 127 4.3.3 Large Sample Variance of the Log Odds Ratio, 128 4.3.4 Maximum Likelihood Estimate of the Common Odds Ratio, 130 4.3.5 Minimum Variance Linear Estimators, 131 4.3.6 MVLE Versus Mantel-Haenszel Estimates, 134 4.3.7 SAS PROC FREQ, 135 4.4 Nature of Covariate Adjustment, 136 4.4.1 Confounding and Effect Modification, 137 4.4.2 Stratification Adjustment and Regression Adjustment, 138 4.4.3 When Does Adjustment Matter?, 140 4.5 Multivariate Tests of Hypotheses, 145 4.5.1 Multivariate Null Hypothesis, 145 4.5.2 Omnibus Test, 146 4.5.3* Multiple Tests, 148 4.5.4 Partitioning of the Omnibus Alternative Hypothesis, 149 4.6 Tests of Homogeneity, 150 4.6.1 Contrast Test of Homogeneity, 151 4.6.2 Cochran's Test of Homogeneity, 153 4.6.3 Zelen's Test, 154 4.6.4 Breslow-Day Test for Odds Ratios, 155 4.6.5* Tarone Test for Odds Ratios, 156 4.7 Efficient Tests of No Partial Association, 156 4.7.1 Restricted Alternative Hypothesis of Association, 156 4.7.2 Radhakrishna Family of Efficient Tests of Association, 158 4.8 Asymptotic Relative Efficiency of Competing Tests, 163 4.8.1 Family of Tests, 163 4.8.2 Asymptotic Relative Efficiency, 165 4.9 Maximin-Efficient Robust Tests, 169 4.9.1 Maximin Efficiency, 169 4.9.2 Gastwirth Scale Robust Test, 170 4.9.3 Wei-Lachin Test of Stochastic Ordering, 171 4.9.4 Comparison of Weighted Tests, 174 4.10 Random Effects Model, 175 4.10.1 Measurement Error Model, 175 4.10.2 Stratified-Adjusted Estimates from Multiple 2x2 Tables, 177 4.11 Power and Sample Size for Tests of Association, 183 4.11.1 Power Function of the Radhakrishna Family, 184 4.11.2 Power and Sample Size for Cochran's Test, 186 4.12* Polychotomous and Ordinal Data, 188 4.12.1* Cochran-Mantel-Haenszel Tests, 188 4.12.2* Stratified-Adjusted Estimates, 189 4.12.3* Vector Test of Homogeneity, 191 4.12.4* Stratified Mean Scores Estimate and Test, 191 4.12.5* Stratified Cochran-Armitage Test of Trend, 192 4.13 Problems, 193
5 Case-Control and Matched Studies, 201
5.1 Unmatched Case-Control (Retrospective) Sampling, 201 5.1.1 Odds Ratio, 202 5.1.2 Relative Risk, 204 5.1.3 Attributable Risk, 205 5.2 Matching, 206 5.2.1 Frequency Matching, 207 5.2.2 Matched Pairs Design: Cross-Sectional or Prospective, 208 5.3 Tests of Association for Matched Pairs, 211 5.3.1 Exact Test, 211 5.3.2 McNemar's Large Sample Test, 212 5.3.3 SAS PROC FREQ, 213 5.4 Measures of Association for Matched Pairs, 214 5.4.1 Conditional Odds Ratio, 214 5.4.2 Confidence Limits for the Odds Ratio, 215 5.4.2.1 Exact Limits, 215 5.4.2.2 Large Sample Limits, 216 5.4.3 Conditional Large Sample Test and Confidence Limits, 217 5.4.4 Mantel-Haenszel Analysis, 217 5.4.5 Relative Risk for Matched Pairs, 218 5.4.6 Attributable Risk for Matched Pairs, 219 5.5 Pair-Matched Retrospective Study, 220 5.5.1 Conditional Odds Ratio, 221 5.5.2 Relative Risks from Matched Retrospective Studies, 222 5.6 Power Function of McNemar's Test, 223 5.6.1 Unconditional Power Function, 223 5.6.2 Conditional Power Function, 224 5.6.3 Other Approaches, 225 5.6.4 Matching Efficiency, 226 5.7 Stratified Analysis of Pair-Matched Tables, 227 5.7.1 Pair and Member Stratification, 227 5.7.2 Stratified Mantel-Haenszel Analysis, 228 5.7.3 MVLE, 229 5.7.4 Tests of Homogeneity and Association, 229 5.7.4.1 Conditional Odds Ratio, 229 5.7.4.2 Marginal Relative Risk, 230 5.7.4.3* Robust Tests, 231 5.7.5 Random Effects Model Analysis, 232 5.8* Multiple Matching: Mantel-Haenszel Analysis, 232 5.9* Matched Polychotomous Data, 234 5.9.1* McNemar's Test, 234 5.9.2* Bowker's Test of Symmetry, 234 5.9.3* Marginal Homogeneity and Quasi-symmetry, 235 5.10* Kappa Index of Agreement, 235 5.10.1* Duplicate Gradings, Binary Characteristic, 235 5.10.2* Duplicate Gradings, Polychotomous or Ordinal Characteristic, 237 5.10.3* Multiple Gradings, Intraclass Correlation, 239 5.11 Problems, 239
6 Applications of Maximum Likelihood and Efficient Scores, 247
6.1 Binomial, 247 6.2 2x2 Table: Product Binomial (Unconditionally), 249 6.2.1 MLEs And Their Asymptotic Distribution, 249 6.2.2 Logit Model, 250 6.2.3 Tests of Significance, 254 6.2.3.1 Wald Test, 255 6.2.3.2 Likelihood Ratio Test, 255 6.2.3.3 Efficient Score Test, 256 6.3 2x2 Table, Conditionally, 257 6.4 Score-Based Estimate, 258 6.5 Stratified Score Analysis of Independent 2x2 Tables, 260 6.5.1 Conditional Mantel-Haenszel Test and the Score Estimate, 260 6.5.2 Unconditional Cochran Test as a C(alpha) Test, 261 6.6 Matched Pairs, 263 6.6.1 Unconditional Logit Model, 263 6.6.2 Conditional Logit Model, 265 6.6.3 Conditional Likelihood Ratio Test, 268 6.6.4 Conditional Score Test, 268 6.6.5 Matched Case-Control Study, 268 6.7 Iterative Maximum Likelihood, 269 6.7.1 Newton-Raphson (or Newton's Method), 269 6.7.2 Fisher Scoring (Method of Scoring), 270 6.8 Problems, 275
7 Logistic Regression Models, 283
7.1 Unconditional Logistic Regression Model, 283 7.1.1 General Logistic Regression Model, 283 7.1.2 Logistic Regression and Binomial Logit Regression, 286 7.1.3 SAS Procedures, 288 7.1.4 Stratified 2x2 Tables, 291 7.1.5 Family of Binomial Regression Models, 293 7.2 Interpretation of the Logistic Regression Model, 294 7.2.1 Model Coefficients and Odds Ratios, 294 7.2.2* Class Effects in PROC LOGISTIC, 300 7.2.3 Partial Regression Coefficients, 302 7.2.4 Model Building: Stepwise Procedures, 304 7.2.5 Disproportionate Sampling, 307 7.2.6 Unmatched Case-Control Study, 308 7.3 Tests of Significance, 309 7.3.1 Likelihood Ratio Tests, 309 7.3.1.1 Model Test, 309 7.3.1.2 Test of Model Components, 309 7.3.2 Efficient Scores Test, 310 7.3.2.1 Model Test, 310 7.3.2.2 Test of Model Components, 312 7.3.3 Wald Tests, 312 7.3.4 SAS PROC GENMOD, 314 7.3.5 Robust Inferences, 317 7.3.6 Power and Sample Size, 321 7.3.6.1* Univariate Effects, 321 7.3.6.2* Multiple Categorical Effects, 322 7.3.6.3* Models Adjusted for Multiple Factors, 324 7.3.6.4* SAS PROC POWER, 325 7.4 Interactions, 325 7.4.1 Qualitative--Qualitative Covariate Interaction, 326 7.4.2 Interactions with a Quantitative Covariate, 330 7.5 Measures of the Strength of Association, 333 7.5.1 Squared Error Loss, 333 7.5.2 Entropy Loss, 334 7.6 Conditional Logistic Regression Model for Matched Sets, 337 7.6.1 Conditional Logistic Model, 337 7.6.2 Matched Retrospective Study, 340 7.6.3 Fitting the General Conditional Logistic Regression Model, 341 7.6.4* Allowing for Clinic Effects in a Randomized Trial, 341 7.6.5 Robust Inference, 345 7.6.6 Explained Variation, 348 7.6.7* Power and Sample Size, 348 7.6.7.1* Quantitative Covariate, 350 7.6.7.2* Binary Covariate, 351 7.6.7.3* Multivariate Model, 351* 7.7* Models for Polychotomous or Ordinal Data, 352 7.7.1* Multinomial Logistic Model, 352 7.7.2* Proportional Odds Model, 357 7.7.3* Conditional Models for Matched Sets, 359 7.8* Random Effects and Mixed Models, 359 7.8.1* Random Intercept Model, 359 7.8.2* Random Treatment Effect, 361 7.9* Models for Multivariate or Repeated Measures, 363 7.9.1* GEE Repeated Measures Models, 364 7.9.2* GEE Multivariate Models, 368 7.9.3* Random Coefficient Models, 369 7.10 Problems, 370
8 Analysis of Count Data, 381
8.1 Event Rates and the Homogeneous Poisson Model, 382 8.1.1 Poisson Process, 382 8.1.2 Doubly Homogeneous Poisson Model, 382 8.1.3 Relative Risks, 384 8.1.4 Violations of the Homogeneous Poisson Assumptions, 388 8.2 Overdispersed Poisson Model, 389 8.2.1 Two-Stage Random Effects Model, 389 8.2.2 Relative Risks, 392 8.2.3 Stratified-Adjusted Analyses, 393 8.3 Poisson Regression Model, 393 8.3.1 Homogeneous Poisson Regression Model, 393 8.3.2 Explained Variation, 401 8.3.3 Applications of Poisson Regression, 401 8.4 Overdispersed and Robust Poisson Regression, 402 8.4.1 Quasi-likelihood Overdispersed Poisson Regression, 402 8.4.2 Robust Inference Using the Information Sandwich, 404 8.4.3* Zeros-inflated Poisson Regression Model, 407 8.5 Conditional Poisson Regression for Matched Sets, 410 8.6* Negative Binomial Models, 412 8.6.1* The Negative Binomial Distribution, 412 8.6.2* Negative Binomial Regression Model, 414 8.7* Power and Sample Size, 416 8.7.1* Poisson Models, 416 8.7.2* Negative Binomial Models, 418 8.8* Multiple Outcomes, 419 8.9 Problems, 420
9 Analysis of Event-Time Data, 429
9.1 Introduction to Survival Analysis, 430 9.1.1 Hazard and Survival Function, 430 9.1.2 Censoring at Random, 431 9.1.3 Kaplan-Meier Estimator, 432 9.1.4 Estimation of the Hazard Function, 435 9.1.5 Comparison of Survival Probabilities for Two Groups, 436 9.2 Lifetable Construction, 441 9.2.1 Discrete Distributions: Actuarial Lifetable, 443 9.2.2 Modified Kaplan-Meier Estimator, 444 9.2.3 SAS PROC LIFETEST: Survival Estimation, 446 9.3 Family of Weighted Mantel-Haenszel Tests, 449 9.3.1 Weighted Mantel-Haenszel Test, 449 9.3.2 Mantel-Logrank Test, 450 9.3.3 Modified Wilcoxon Test, 451 9.3.4 G(rho) Family of Tests, 452 9.3.5 Measures of Association, 453 9.3.6 SAS PROC LIFETEST: Tests of Significance, 455 9.4 Proportional Hazards Models, 456 9.4.1 Cox's Proportional Hazards Model, 456 9.4.2 Stratified Models, 460 9.4.3 Time-Dependent Covariates, 461 9.4.4 Fitting the Model, 461 9.4.5 Robust Inference, 463 9.4.6 Adjustments for Tied Observations, 464 9.4.6.1 Discrete and Grouped Failure Time Data, 465 9.4.6.2 Cox's Adjustment for Ties, 466 9.4.6.3 Kalbfleisch-Prentice Marginal Model, 467 9.4.6.4 Peto-Breslow Adjustment for Ties, 467 9.4.7* Survival Function Estimation, 468 9.4.8 Model Assumptions, 469 9.4.9 Explained Variation, 471 9.4.10 SAS PROC PHREG, 473 9.5 Evaluation of Sample Size and Power, 483 9.5.1 Exponential Survival, 483 9.5.2 Cox's Proportional Hazards Model, 486 9.5.2.1* Qualitative Covariate, 487 9.5.2.2* Quantitative Covariate, 488 9.5.2.3* Adjusted Analysis, 489 9.5.2.4* Maximum Information Design, 489 9.6* Additional Models, 491 9.6.1* Competing Risks, 492 9.6.2* Interval Censoring, 495 9.6.3* Parametric Models, 496 9.6.4* Multiple Event Times, 497 9.7 Analysis of Recurrent Events, 499 9.7.1 Counting Process Formulation, 500 9.7.2 Nelson-Aalen Estimator, Kernel Smoothed Estimator, 502 9.7.3 Aalen-Gill Test Statistics, 504 9.7.4 Multiplicative Intensity Model, 507 9.7.5* Robust Estimation: Proportional Rate Models, 511 9.7.6* Stratified Recurrence Models, 512 9.8 Problems, 513
Appendix Statistical Theory, 535
A.1 Introduction, 535 A.1.1 Notation, 535 A.1.2 Matrices, 536 A.1.3 Partition of Variation, 537 A.2 Central Limit Theorem and the Law of Large Numbers, 537 A.2.1 Univariate Case, 537 A.2.2 Multivariate Case, 540 A.3 Delta Method, 541 A.3.1 Univariate Case, 541 A.3.2 Multivariate Case, 542 A.4 Slutsky's Convergence Theorem, 543 A.4.1 Convergence in Distribution, 543 A.4.2 Convergence in Probability, 544 A.4.3 Convergence in Distribution of Transformations, 544 A.5 Least Squares Estimation, 546 A.5.1 Ordinary Least Squares, 546 A.5.2 Gauss-Markov Theorem, 548 A.5.3 Weighted Least Squares, 548 A.5.4 Iteratively Reweighted Least Squares, 550 A.6 Maximum Likelihood Estimation and Efficient Scores, 551 A.6.1 Estimating Equation, 551 A.6.2 Efficient Score, 552 A.6.3 Fisher's Information Function, 553 A.6.4 Cramer-Rao Inequality: Efficient Estimators, 555 A.6.5 Asymptotic Distribution of the Efficient Score and the MLE, 556 A.6.6 Consistency and Asymptotic Efficiency of the MLE, 557 A.6.7 Estimated Information, 558 A.6.8 Invariance Under Transformations, 558 A.6.9 Independent But Not Identically Distributed Observations, 560 A.7 Tests of Significance, 561 A.7.1 Wald Tests, 561 A.7.1.1 Elementwise Tests, 562 A.7.1.2 Composite Test, 562 A.7.1.3 Test of a Linear Hypothesis, 562 A.7.2 Likelihood Ratio Tests, 563 A.7.2.1 Composite Test, 563 A.7.2.2 Test of a Subhypothesis, 564 A.7.3 Efficient Scores Test, 565 A.7.3.1 Composite Test, 565 A.7.3.2 Test of a Subhypothesis: C(alpha) Tests, 565 A.7.3.3 Relative Efficiency Versus the Likelihood Ratio Test, 567 A.8 Explained Variation, 569 A.8.1 Squared Error Loss, 570 A.8.2 Residual Variation, 572 A.8.3 Negative Log-Likelihood Loss, 573 A.8.4 Madalla's R(LR) Squared, 573 A.9 Robust Inference, 574 A.9.1 Information Sandwich, 574 A.9.1.1 Correct Model Specification, 575 A.9.1.2 Incorrect Model Specification, 576 A.9.2 Robust Confidence Limits and Tests, 579 A.10 Generalized Linear Models and Quasi-likelihood, 579 A.10.1 Generalized Linear Models, 580 A.10.2 Exponential Family of Models, 581 A.10.3 Deviance and the Chi-Square Goodness of Fit, 584 A.10.4 Quasi-likelihood, 585 A.10.5 Conditional GLMs, 588 A.11 Generalized Estimating Equations (GEE), 588