Subject Index

Active predictors, 235, 238–245 Bootstrap analysis, 149, 174–179 Added-variable plots, 51–55, 67–68, 70, 71, bias, 177 78 bias corrected and accelerated (BCa ), properties, 69 176–178 as a regression diagnostic, 224–225 case resampling, 176–178 relation to t -test, 68 for a median, 175 Additive models, 199 nonlinear regression, 225, 226 Akaike Information Criterion (aic ), residual resampling, 179 238–245, 248 tests, 179 Allometric model, 187–188 Box–Cox transformation method, 190–191, Analysis of covariance, 107–108 312–314 Analysis of deviance, 284 automatic choice of predictor Poisson regression, 278–279, 281–283 transformations, 195–196 Analysis of variance (anova ), 133, 138–150 linearly related regressors, 194–195 types, 140–141 nonpositive variables, 198–199 unbalanced, 140 response transformations, 196–198 Autoregressive errors, 163, 168 Boxplots one-way factor model, 99–102 Backsolving, 308 transformations, 186 Backward elimination, 240–243 Basis functions, 113–116 C a u s a t i o n , 8 7 Bayes Information Criterion (bic ), 239, Censoring, 121–122 244–245 Central composite design, 128 Bernoulli distribution, counted data, Central limit theorem, nonlinear regression, 270–271 256–257 Bertalanffy function, 267–268 Chi-squared random variable, 27 Best linear unbiasedCOPYRIGHTED estimates, 29 noncentral, MATERIAL 143 Binomial distribution, 271 C l a u s i u s – C l a p e y r o n f o r m u l a , 4 2 Binomial regression. See Logistic regression Cluster sampling, 161 Block diagonal, generalized least squares C o e f fi cient of determination (R 2 ) model, 168–169 adjusted, 36 Bonferroni inequality, regression diagnostics, multiple linear regression, 66–68, 92–93 outliers, 217–218 regression through origin, 93

Applied Linear Regression, Fourth Edition. Sanford Weisberg. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc. 331 332 subject index

Coeffi cient of determination (R2) (continued) BlowBS , 274 sampling conditions, 91–93 Blowdown , 275, 276, 278, 285, 289 simple linear regression, 35–36, 91–92 brains , 186 variable selection, 235–237 cakes , 111, 126, 152, 183 C o e f fi cient estimates cathedral , 45, 129 logistic regression, 311–312 caution , 210 multiple regression, 55–65 Challeng , 288 simple regression, 22–28 cloud , 231 spline basis, 116 domedata , 131 Collinearity, 79–81 domedata1 , 132 Complex regressors Donner , 287 factors, 56, 98–109 Downer , 286 interactions, 56, 104–108 drugcost , 232 polynomials, 56, 109–113 dwaste , 250 principal components, 57, 116–119 fl orida , 230 splines, 57, 113–116 Forbes , 5, 24, 42, 129 Complexity, predictor variable discovery, ftcollinssnow , 8, 41 238–239 ftcollinstemp , 41 C o n fi dence intervals, 30 fuel2001 , 15, 57, 151, 183, 203, 229 bootstrap analysis, 175–179 galapagos , 250 Box–Cox transformations, 191 galtonpeas , 181 Poisson and binomial regression, 284 Heights , 2, 46 simple linear regression, 30–34 Highway , 191, 249, 269 Constant variance, test for, 164–167 Hooker , 42, 129 Construction set, variable selection, 247 Htwt , 38 Cook ’ s distance, 220–225 jevons , 182 Correlation, 23, 284 lakes , 239 matrix, 58, 119, landrent , 231 partial, 54, 68 lathe , 230 relation to t -test, 46 lathe1 , 128 sample, 174 mantel , 249 Counted response models, 270 mile , 183 distributions, 270–272 MinnLand , 119, 123, 127, 128, 153, regression models, 272–279 154 simple and multiple regression, 283–284 MinnWater , 79, 97, 249 C o v a r i a n c e Mitchell , 18 matrix, 295, 302 MWwords , 48 multiple linear regression, 66 oldfaith , 18, 49 sample, 59 physics , 157 simple linear regression, 29 physics1 , 180 Credit scoring, 246 pipeline , 237 Cross-sectional data, 8 prodscore , 155 Cross-validation, variable selection, 247 rat , 221 Curvature testing, 212–213 Rateprof , 19, 117, 152, 159, 286 Rpdata , 226 Data mining, 235 salary , 130, 136 D a t a s e t s salarygov , 126, 153, 180, 201 AMSsurvey , 280, 289 segreg , 263 anscombe , 12 sleep1 , 265 baeskel , 199 snake , 48 BGSall , 70, 129 sniffer , 164 BGSboys , 70, 250 Stevens , 169 BGSgirls , 70, 93 stopping , 181, 200 BigMac2003 , 41, 202 swan96 , 268 subject index 333

Transact , 94, 184 Challenger, 288 turk0 , 257 cloud seeding, 281 turkey , 9, 261 credit scoring, 246 twins , 153 Donner party, 287 UBSprices , 39, 40, 41 downer, 286–287 ufcwc , 189 drug costs, 232 UN11 , 17, 47, 51, 69, 95, 123, 150, 151, 183, electrical energy consumption, 263–265 249 Epworth sleepiness scale, 245–246 walleye , 268 feedlots, 87–89 water , 19, 71, 200, 228 Florida election in 2000, 230 wblake , 7, 18, 47 Forbes ’ s data, 5-7, 11-12, 24-27, 30-38, Whitestar , 288 135-138 wm1 , 49, 184 Ft. Collins snowfall, 8, 11, 31, 41, 136 wm2 , 154 Ft. Collins temperature, 41 Wool , 108, 131, 153, 203 fuel consumption, 15–17, 57–59, 63–66, Degrees of freedom, 26 73–79, 211–212 Delta method, 172–174 Galápagos Island data, 250–251 Dependent variable. See Response variable Galton ’ s peas, 181 Deviance, logistic, and Poisson regression, Government salary, 126–127, 153, 180, 201 277–279 height inheritance, 2–5, 10–12, 14–15, 36–37, Discovery, variable selection, 237–245 91–93 information criteria, 238–239 Hooker, 44, 129 regularized methods, 244–245 Jevon ’ s gold coins, 182 stepwise regression, 239–244 lake diversity, 229 subset selection, 245 Lake Mary, 267 Discovery probability, hypothesis testing, land rent, 231 147–148 land valuation, 155 Dummy variables, 56–57, 100–102 lathe, 128, 230 nonlinear regression, 260–262 mammal species, 186 outlier testing, 215–216 Mantel, 249 mathematical sciences PhDs, 280–283 Ecological regression, 160–162 metrodome, 131 Effects plots, 74–75, 105–108, 111–113, mile run, 183 141–142, 278–279 Minnesota agricultural land sales, 119–122, Eigenvectors and eigenvalues, 118–119, 309 144–145 Elastic net, 244–245 Minnesota highway accidents, 191–196, Epworth sleepiness scale, 245–246 240–245 E r r o r s e Minnesota water use, 79–81 assumptions, 21 Old Faithful Geyser, 18–19, 49, 113–116 multiple linear regression, 61 Oxygen uptake, 250 regression diagnostic residuals, 205–206 professor ratings, 117–119, 159–162, Estimated variances, simple linear 247–248 regression, 29–30 rat data, 221–225 E x a m p l e s segmented regression, 263 Alaska pipeline, 227 sex discrimination, 130 Anscombe, 12–13 sleep in mammals, 267 Berkeley Guidance study, 75–78, 119, 149, smallmouth bass, 7–8, 10–12, 14–15 250 Snake River, 48 blowdown, 274–279 sniffer data, 164–167 brain weight, 187–188 Stefanski, 226–227 cakes, 111–113, 138 stopping distances, 181 California water, 19, 200, 228 strong interaction, 157–159 cathedrals, 129–130 surface tension, 199 caution, 210 Swan Lake black crappies, 268 334 subject index

Examples (continued) General correlation structure, 168–169 Titanic, 288 General likelihood ratio tests, 138 transactions, 94, 175–178, 184, 226, 227 General linear hypothesis, Wald tests, 146 turkey growth, 9–10, 252–259 Generalized least squares (gls ), twin study, 153 autoregressive, 168 UBS prices, 39 block diagonal, 168-168 United Nations, 51–55, 98–108, 137–138, compound symmetry, 168 140, 149, 207–208, 212–214, Generalized linear models, 285 219–225 Geometric mean, 190–191 Upper Flat creek, 189–191 Goodness of fi t tests, Poisson regression, walleye growth, 268 282–283 weather prediction, 8–9, 246–247 weight gain, 260–262 Hat matrix, 205–208 Whitestar, 288 Hawthorne effect, 150 windmills, 49, 154–155, 184 HC3 estimates, misspecifi ed variances, wool, 108–109, 136–138, 141–142 163–167 Zipf ’ s law, 48 Hessian matrix, nonlinear regression, Expected information matrix, 314 254–256 Experimentation vs. observation, 86–89 Hierarchical regression, mixed models, 171 Exponential family distribution, 285 Hot deck, missing data, 122 Hyperplane, multiple regression model, 55 Factors, 56, 98–109 H y p o t h e s i s t e s t i n g nonlinear regression, 260–262 analysis of variance and, 133–150 False discovery, 147–148, 150 counted response, 284 Family-wise error rate, 150 false results, 147–148 FICO score, 246 fi le drawer effects, 150 File drawer effects, 150 general linear hypothesis, 146 Finite population approach to sample goodness of fi t, 282 surveys, 162 Hawthorne effect, 150 Fisher scoring, 311 interpreting p -values, 146–150 F i t t e d m e a n f u n c t i o n likelihood ratio tests, 138, 146, 195 multiple regression model, 55 logistic regression, 277–279 nonlinear regression, 259–262 marginality principle, 139 F i t t e d v a l u e s multiple testing, 150 inverse fi tted value plot, 196–198 nonadditivity, 212 multiple linear regression, 68–69 one coeffi cient, 67–68 simple linear regression Poisson regression, 279–281 confi dence intervals, 33–34 population vs . sampling, 149 estimation of, 32–34 power, 143–145 ordinary least squares, 22–24 reported signifi cance levels, 149 F i x e d - s i g n i fi cance level, p -value t -tests, 30–34, 67–68 interpretation, 147 types, analysis of variance, 135–136 Focal predictors, variable selection, 235–237 unbalanced, 135 Forward selection, predictor variable discovery, 240–242 Imputation, missing data, 122 F-tests, 133–138 Inclusion probability, sample surveys, analysis of variance, 139–142 161–162 interpretation, 146–150 Independent variable. See Predictor overall test, 135–138 variables power and non-null distributions, 144–145 I n fl uence, 204, 218–225 Wald tests, 145–146 Cook ’ s distance, 220–224 Information criteria, 238–239 Gauss–Markov theorem, 28–29 Interactions, 56, 104–106, 139–142, Gauss–Newton algorithm, 255–256 211–213 subject index 335

Intercept, 21–22, 56, 100–102 response in, 82–83 confi dence interval, 30–34 variance stabilization, 172 Interquartile range (IQR), 99–102 Logistic regression, 272–277 I n v a r i a n c e , 4 3 deviance, 277–279 Inverse fi tted value plot, 196–198 goodness of fi t tests, 282–283 Inverse regression, 183 Logit function, 273–277 Inverse response plot, 198, 202, 203 Log-likelihood profi le, Box–Cox method, 196–198 Jittering scatterplots, 3–5 Log-odds, 273–277 Longitudinal studies, 8 Kernel mean function, 253, 272–277 Lsmeans, 103, 108, 153 Lurking variables, 88–89 L a c k - o f - fi t testing, 211–212 Lasso, 244–245 Machine learning, 235, 247–248 Leaps and bounds algorithm, Main effects interpretation, 73–93 239–245 analysis of variance, 139–142 Least squares estimates. See Ordinary least experimentation vs . observation, 86–89 squares factor models Level means comparison, factor models, continuous predictors, 104–106 102–103 one-factor model, 106–108 Leverage values, 204 multiple factors, 109 residuals, 207–209 normal population sampling, 89–91 scatterplots, 4–5 parameter estimates, 73–83 Li–Duan theorem, 194–195 regressor omission, 84–86 Likelihood ratio tests, 134 Marginal plot, 52–55 transformations, automatic predictor Marginality principle, analysis of variance, selection, 195–196 139–142 Wald tests comparison, 146 Matrices, 290–309 Linear dependence, 78–79 inverse, 301 Linear independence, 78–79 multiple linear regression, 60–61 L i n e a r p r e d i c t o r partitioned matrix, 71–72 binomial regression, 272–277 QR factorization, 307–308 Poisson regression, 280–283 rank, 76–81, 301 L i n e a r r e g r e s s i o n scatterplot matrices, 15–17 basic properties, 1–2 simple regression, 63–66 coeffi cients, 133 spectral decomposition, 309 F -tests, 134–138 Maximum likelihood estimates, 309–313 mean functions, 10–12 Poisson regression, 280–283 multiple linear regression, 51–69 regression parameters, 90–91 scatterplots, 2–10 M e a n f u n c t i o n s simple linear regression, 21–38 additive model transformation, 199 summary graph, 12–13 Box–Cox transformation, 190–191 variable selection, 235–237 F -tests, 135–138 Linearly related regressors, 194–195 main effects109 L i n k f u n c t i o n multiple linear regression, 58–59 binomial regression, 273–277 nonlinear regression, 252–256 Poisson regression, 279–283 one-factor models, 100–102 loess smoother, 14–15 outlier models, 214–218 Log rule, power transformations, parameter estimation, 75–78 188 parameter regressors, omission, 84–86 L o g a r i t h m s polynomial regression, 109–113 base, 24 quadratic regression, 109–113 power transformations, 187–188 rank defi cient and overparameterized mean regressors in, 81–82 functions, 78–79 336 subject index

Mean functions (continued) Nonlinear regression, 252–269 regression, 10–12 bootstrap inference, 262–265 scaled power transformations, 189–190 large sample inference, 256–257 simple linear regression, 21–22 literature sources, 265 least squares estimates, 29 mean function estimation, 253–256 regressor addition, 51–55 starting values, 257–262 smoothers, 14–15 Non-null distributions, analysis of variance, Mean shift outlier model, regression 143–145 diagnostics, 214–218 Nonparametric estimation, mean functions, Mean square, 26, 134–138 10–12 Means comparison Nonpositive variables, transformation, analysis of variance, 142 198–199 level means, 102–103 N o r m a l d i s t r i b u t i o n Measure, correlate, predict method, 154–155 multivariate, 89–91 Missing data, 119–122 sampling from, 89–91 missing at random (MAR), 121–122 Normal equations, 293 multiple imputation, 122 Normal probability plot, 225–226 M i s s p e c i fi ed variance, 162–167 N o r m a l i t y accommodation, 163–164 Box–Cox transformation to, 191 constant variance test, 164–167 power transformations to, 195–196 Mixed models, 169–171 Normality assumption, regression Model averaging, 247 diagnostics, 225–226 Multilevel and hierarchical models, 171 N o t a t i o n Multiple comparisons, 102, 108 aic , 238 Multiple correlation coeffi cient. See anova , 139 Coeffi cient of determination bic , 239 Multiple linear regression, 51–69 case, 2 coeffi cient of determination (R 2 ) , 66–67, correlation ρ , 292 92–93 covariance, Cov, 291 collinearity, 79–81 df , 26 delta method, 173–174 expectation E, 290 factors, 98–108 gls , 168 model, 55 hats, 22 ordinary least squares, 58–68 h ii , 207 overall F -test, 136 NID, 29 predictions, fi tted values, and linear ols , 22 combinations, 68–69 p ′ , 64 regressors, 51–58 predictor, 16 2 residual plots, 210 RYX, , 236 transformations, 193–196 regressor, 16 Multiple testing, 150 RSS , 24 Multiplicative error, 187–188 rxy , 23 Multistage sample surveys, 161–162 SD , 23 Multivariate normality, 89–91 se , 28 SSreg , 35 Natural logarithms. See Logarithms SXX , 23

Neural networks, 247 sxy , 23 Newton–Raphson algorithm, 311 SXY , 23 Noncentrality parameter, power SYY , 23 and non-null distributions, 143–145 typewriter font , 2 Nonconstant variance variance VAR, 291 regression diagnostics, 213–214 wls , 156 tests for, 164–167 x , 23 subject index 337

Null plot Per-test error rate, 150 characteristics, 14 Poisson distribution, 271–272 simple linear regression, 36–38 generalized linear models, 283–285 variance stabilizing transformations, Observational data, 75 171–172 Odds of success, binomial regression, Poisson regression, 270–289 273–277 deviance, 277–279 Offset, 249 goodness of fi t tests, 282–283 One-dimensional estimation, Polynomial regressors, 109–113 linearly related regressors, 194–195 multiple predictors, 111–112 One-factor model, one-way anova , multiple regression model, 56 99–102 numerical issues, 112–113 Ordinary least squares (ols ) estimation, 22, Power calculators, 144 24–26, 58–68 Power family computing formulas, 61 modifi ed power family, 190–191 matrix version, 304 scaled power transformations, misspecifi ed variances, 163–167 188–190 nonlinear regression, 258–259 transformations, 186–188 properties, 27–29, 305–307 Power of the test, analysis of variance, Orthogonal factors, 141–142 143–145 Orthogonal polynomials, 112–113 Predicted residual (PRESS residual), Orthogonal projection, 206–208 230 Outliers, 214–218 Prediction, 32–34 scatterplots, 4–5, 13 weighted least squares, 159 Overall F -test Predictor variables. See also Regressors multiple regression, 136 active vs . inactive, 235 simple regression, 135–136 complex regressors, 98–122 Overparameterized mean function principal components, 117–119 one-factor models, 100–102 discovery, 238–245 parameter estimates, 78–79 experimentation vs . observation, 86–89 Pairwise comparisons, 102–103 multiple linear regression, 55–58, 68–69 Parameters, 73–93, 95–114 one-factor models, 100–102 aliased, 78 polynomial regression, 109–113 collinearity, 79–81 scatterplots, 2–5 F -tests, 138 matrix, 16–17 intercept, 10, 21 selection methods, 234–251 multiple regression model, 55 single variable transformation, 188–190 not the same as estimates, 24 transformations, 193–196 partial slope, 73 automatic selection, 195–196 rank defi cient or overparameterized mean Principal component analysis functions, 78–79 complex regressors, 116–119 signs of estimates, 75 multiple regression model, predictors and simple linear regression, 21–22 regressors, 57 slope, 10, 21 Probability plot, 225–226 variable selection and assessment of, p -value 235–237 hypothesis testing, 133 Partial R 2 , 236 interpretation, 146–147 P a r t i a l s l o p e , 7 3 means comparison, 103 Pearson residuals, 208 outlier tests, 217–218 Poisson and binomial regression, power and non-null distributions, 284–285 144–145 Pearson ’ s χ2 , 283 Wald tests, 145–146 338 subject index

QR factorization, 228, 307–308 splines, 113–116 Quadratic regression, 109–113 transformed predictors, 56 curvature testing with, 212–213 Regularized methods, 244–245 delta method for a maximum or minimum, Reliability of hypothesis testing, 148 174 Repeated measures, 171 R e p o r t e d s i g n i fi cance levels, 149 R packages R e s e a r c h fi ndings, test interpretation, alr4 , ii, 290 147–148 car , 140 Residual mean square, 26–27 effects , 108, 153 Residual plots, 166, 209–226 lsmeans , 153 Residual sampling, bootstrap analysis, 179 nlme , 168 Residual variance, 90–91 R2 . See Coeffi cient of determination Residuals, 23, 25, 35–38, 204–218 R a n d o m c o e f fi cients model, 170–171 Pearson, 208 Random forests, 247 predicted, 230 Random vectors, 303 standardized, 216 Range rule, power transformations, 188 studentized, 216 R a n k d e fi cient mean function, 78–79 supernormality, 225–226 R e g r e s s i o n c o e f fi cients weighted, 156 complex models, 98-113 R e s p o n s e v a r i a b l e interpretation, 73-91 logarithmic scale, 82–83 Regression diagnostics, 204–233 scatterplots, 2–5 hat matrix, 205 transformations, 196–198 weighted hat matrix, 208 infl uential cases, 218–225 Sample surveys, 161–162 added-variable plots, 224–225 Sampling weight, 162 Cook ’ s distance, 220–221 Sandwich estimator, 163–167 nonconstant variance, 213–214 Scad, 244 normality assumption, 225–226 Scaled power transformations, 189–190 outliers, 214–218 Box–Cox method, 191 level signifi cance, 217–218 Scatterplot, 2 methodology, 218 Scatterplot matrix, 15–17 test, 215–216 Score test, nonconstant variance, 166–167 weighted least squares, 216 Score vector, 254–256 Poisson and binomial regression, 284–285 S e c o n d - o r d e r m e a n f u n c t i o n residuals, 204–212 analysis of variance, 141–142 curvature testing, 212–213 polynomial regressors, 111–113 error vectors, 205–206 Segmented regression, 263–265 hat matrix, 206–208 Separated points, scatterplots, 4–5 plots of, 209–210 Sequential analysis of variance (Type I), weighted hat matrix, 208 140–141 Regression through the origin, 93 Signs of parameter estimates, 75 Regressors, 16, 51, 55–58 S i n g l e c o e f fi cient hypotheses, 133 class variable, 101 multiple linear regression, 68–69 colinear, 79 Wald tests, 145–146 dropping, 84 Single linear combination, Wald tests, 146 dummy variables, 56, 100 Size, scatterplots, 14 effects coding, 125 Slices, scatterplots, 4–5 factors, 98–109 Slope parameter intercept, 56 estimates, 73–83 linearly dependent, 78 simple linear regression, 21–22 linearly related, 194–195 S m o o t h e r s polynomial, 56, 109–113 loess , 14, 296–298 principal component, 116–119 splines, 113–116 subject index 339

Sparcity principle, 244–245 response, 196–198 Spectral decomposition, 309 scaled power, 189, 252 Splines, 113-116 scatterplots, 14 Square-root transformation, variance single predictor variable, 188–190 stabilization, 172 variance stabilization, 171–172 Stacking the deck, hypothesis testing, Yeo–Johnson, 198–199 149–150 True discovery, hypothesis testing, 147–148 Standard deviation, simple linear regression, t-Tests 29–30 misspecifi ed variances, 163–167 Standard error of prediction, 33, 68, 159 multiple linear regression, 68 bootstrap analysis, 176–179 one-factor models, 102 delta method, 172–174 main effects model, 107–108 Standard error of regression, 29–30, 61 Poisson and binomial regression, 284 Starting values, nonlinear regression, regression diagnostics, outliers, 217–218 257–262 simple linear regression, 30–34 Statistical error, 21–22 two sample, 44 Stepwise regression, 238, 239–245 Tukey ’ s test for nonadditivity, 212–213 S t r a t i fi ed random sample, sample surveys, Type II analysis of variance, 140–141 161–162 Summary graph, 12–14 Uncorrected sums of squares, 61–62 S u m s o f s q u a r e s Uncorrelated data, scatterplots, 8–9 regression, 35, 63, 134 Unexplained variation residual, 22, 24, 63 multiple linear regression, coeffi cient of total, 35 determination (R 2 ), 67–68 Superpopulation, sample surveys, 162 simple linear regression, coeffi cient of S y m b o l s , d e fi nitions table, 23 determination (R 2 ), 35–36 Univariate summary statistics Taylor series approximation, 254–256 multiple regression, 57–58 Test interpretation, 146–150 simple linear regression, 23–24 bootstrap analysis, 179 Poisson and binomial regression, 284 Validation set, variables selection, 247 regression diagnostics, outliers, 215–218 Variable selection, 234–251 Term. See Regressors discovery, 237–245 Test statistics, power transformations, information criteria, 238–239 automatic predictor selection, regularized methods, 244–245 195–196 stepwise regression, 239–244 Third-order mean function, 109 subset selection, 245 Transformation family, 186–188 parameter assessment, 235–237 Transformations, 56, 185–203 Poisson and binomial regression, additive models, 199 285 automatic predictor selection, prediction, model selection for, 245–248 195–196 cross-validation, 247 basic power transformation, 186 professor ratings, 247–248 basic principles, 185–186 Variance estimation Box–Cox method, 190–191, 194–199, bootstrap method, 174–179 312–314 nonlinear parameter functions, 178 linearly related regressors, 194–195 regression inference, no normality, log rule, 188 175–178 modifi ed power, 190 residual bootstrap, 179 methodology and examples, 191–196 delta method, 174 multivariate, 195 multiple linear regression, 66 nonpositive variables, 198–199 nonlinear regression, 253–256 power transformations, 186–188 simple linear regression, 26–27 range rule, 188 tests, 179 340 subject index

V a r i a n c e i n fl ation factor, 249 Weighted least squares (wls ) V a r i a n c e s constant variance test, 166–167 general correlation structures, regression diagnostics 168–169 outliers, 216 misspecifi ed variance, 162–167 weighted hat matrix, residuals, 208 accommodation, 163–164 variances, 156–162 constant variance test, 164–167 group means weighting, 159–161 mixed models, 169–171 sample surveys, 161–162 multiple linear regression, 58–59 Wilkinson–Rogers notation, 101, 106–109, overview, 156–179 139, 151, 259 Poisson and binomial regression, 284 binomial regression, 276–277 scatterplots, 12–14 Working residual, nonlinear mean function simple linear regression, 21–22 estimation, 255 stabilizing transformations, 171–172 W statistic, regression diagnostics, 226 weighted least squares, 156–162 Yeo–Johnson transformation, nonpositive Wald tests, 133, 145–146 variables, 198–199 likelihood ratio test comparison, 146 single coeffi cient hypotheses, 145–146 Z i p f ’ s l a w , 4 8