INDEX

Activation function in a neural network, 527

Adjusted R², 87, 333

Akaike information criterion (AIC), 336

All possible regressions, 336, 342

Allocated codes and indicator variables, 273

Analysis of covariance, 272

Analysis of variance (ANOVA) in regression, 25, 84

Analysis of variance identity for regression, 26

Analysis of variance model as a regression model, 275

Artificial neural networks, 526

Assumptions in regression, 129

Asymptotic covariance matrix, 406

Asymptotic efficiency, 511

Asymptotic inference in nonlinear regression, 409, 411

Asymptotically unbiased estimators, 52

Autocorrelation function, 475

Autocorrelation in regression data, 474

Average prediction variance, 531

Backpropagation, 528

Backward elimination, 346

Bayesian estimation, 312

Bernoulli random variable, 432

Best linear unbiased estimators, 19, 587

Biasing parameter in ridge regression, 306

BIC, 336

Binary response variable, 422. See also Logistic regression

Binomial distribution, 422, 608

Bivariate normal distribution, 53

Bonferroni method for confidence intervals, 102

Bootstrap confidence intervals, 519

Bootstrapping, 411, 517

cases, 518

residuals, 518

Box–Behnken design, 532

Box–Cox method, 182

Breakdown point, 510

Calibration problem, 513, 516

Candidate regressors, 328

Canonical link, 451

Categorical variables in regression, 260

Central composite design, 243, 532

Central limit theorem, 575

Chi-square distribution, definition, 575

Classical calibration estimator, 514

Classification and regression trees (CART), 524

Cluster analysis to identify jointly influential points, 220

Cochrane–Orcutt method, 482

Coefficient of determination, 35. See also R²

Collinearity, 7. See also Multicollinearity

Comparing regression models, 272

Complimentary log-log link, 442, 452

Condition indices of X′X, 298

Condition number of X′X, 298

Confidence interval

in logistic regression, 437

on the mean response, 30, 46, 99

on model parameters, 29, 46, 98

Consistent estimators, 52

Cook's distance measure, 216, 600

Correlation coefficicent, 53

Correlation matrix

of the parameter estimates, 80

of the regressors, 114, 292

as a multicollinearity diagnostic, 292, 294

COVRATIO_i, 219

Cubic splines, 230, 231

Cuboidal design region, 533

Degrees of freedom in ANOVA, 26

Deleted residuals, 134

Deletion diagnostic, 217

Dependent variable, 2

Designed experiment, 5, 8, 71, 243, 275, 529, 530

Detecting multicollinearity, 292

Determinant of X′X, 302

Deviance, 431, 525

residuals, 440, 458

DFBETAS_ij, 217, 601

DFFITS_i, 217, 600

D-optimal design, 530

Double exponential distribution, 502

Durbin–Watson test for autocorrelation, 475, 477

Efficiency of an estimator, 511

Eigenvalues of X′X, 299

Empirical model, 3, 68

Empirical selection of a transformation, 172

Errors in the regressor variables, 511

Estimation of σ², 21

Exponential family, 421, 450, 605

Externally studentized residuals, 135

Extra sum of squares method, 89, 585

Extrapolation in regression, 33, 42, 107, 225

Extreme values, 130

Factorial design, 529

F-distribution (definition), 576

Finite-sample efficiency, 511

First-order autoregressive process, 475

Forward selection, 345

Fraction of design space plot, 536

Fractional increments, 408

Gauss–Markov theorem, 19, 587, 597

Generalized least squares, 188, 189

Generalized linear model, 421, 450

Generalized ridge regression, 313

Gompertz growth model, 412

Goodness of fit tests in logistic regression, 431, 432

G-optimal design, 531

Growth models, 412

Hat matrix in regression, 73, 110, 131, 212

Hessian matrix, 437

Hidden extrapolation in regression, 107

Hierarchy, 226, 395, 396

High-breakdown-point estimators, 510

Hosmer–Lemeshow goodness-of-fit test, 433

Idempotent matrix (definition), 577

Identity link, 445

Ill-conditioning, 226. See also Multicollinearity

Independent variable, 2

Indicator variables, 260, 268, 270, 173, 274, 275

Influence function, 504

Influential point, 132, 134, 211, 220

Integrated variance, 531

Interaction, 69

Intrinsically linear model, 176, 178, 398

Inverse estimation, 513, 516

I-optimal design, 532

Iteratively reweighted least squares (IRLS), 503, 605, 611

Joint confidence region on the model parameters, 101

Jointly influential points, 220

Lack of fit test in regression, 156, 159, 162

Lag one autocorrelation, 476

Least absolute deviation (L₁ norm) regression, 502

Least squares

estimators, 14, 72, 73

normal equations, 14, 71, 72

Leverage point, 132, 211, 212, 213

Likelihood function, 51

Likelihood ratio tests in logistic regression, 430

Linear dependence of regressors, 117

Linear predictor, 423, 451

Linear regression model, 2, 389

Linearization of a nonlinear model, 400

Linearly independent regressors, 73

Link function, 445

Locally weighted regression (Loess), 237, 239

Log link, 445

Logistic growth model, 412

Logistic regression, 421, 423, 428, 430, 432, 433, 437, 440, 442, 444

Logistic response function, 423

Mallow's C_p, 334

Marquardt's compromise, 408

Matrix of scatter plots, 82

Maximum likelihood estimation

in the generalized linear model, 452, 454, 608

in logistic regression, 424, 601

in time series regression models, 485

Maximum likelihood estimators, 51, 83

Maximum likelihood score equations, 602

Mean shift outlier model, 594

Mechanistic model, 3, 411

M-estimators, 503, 509

Method of least squares, 13, 70, 395

Method of maximum likelihood, 51

Method of steepest decent, 407

Minimum variance estimators, 52

Mixed model, 195, 202

Model adequacy checking, 4, 15, 129, 130, 136, 139, 141, 142, 143, 149, 475

Model building, 91, 225, 304, 327, 338, 345, 346, 348, 351

Model independent estimate of error, 157, 160

Model misspecification, 329

Model-dependent estimate of error, 21, 81

Multicollinearity, 7, 117, 285, 292, 396

effects, 288

Multiple linear regression model, 4, 67

Near neighbors, 160

Neural networks, 526

Newton–Raphson method, 603

No-intercept regression model, 45

Nominal logistic regression, 442

Noncentral chi-square distribution (definition), 576

Noncentral F distribution (definition), 576

Noncentral t distribution (definition), 575

Nonlinear least squares, 396

Nonlinear regression model, 389

Nonparametric regression, 236, 237

Nonsense relationships in regression, 44

Normal distribution, 22, 129, 136, 574, 606

Normal probability plot of residuals, 136

Observational study, 5, 8, 71

Odds ratio, 428, 429

Ordinal logistic regression, 444

Orthogonal design, 530

Orthogonal polynomials, 248

Orthogonal regressors, 93, 118

Orthonormal marix (definition), 578

Outliers, 7, 43, 139, 59

Overdispersion, 461

Overfitting, 88, 528

Partial deviance, 434

Partial F-test, 90

Partial regression

coefficients, 68, 115

plots, 143, 144

Partial residual plots, 143, 146

Pearson chi-square goodness-of-fit test, 432

Pearson residuals, 440

Piecewise polynomial models, 229, 234

Poisson distribution, 444, 607

Poisson regression, 444

Polynomial regression models, 223, 242

Polynomial and trigonometric terms, 235

Population regression model, 13

Positive definite and positive semidefinite matrix, definition, 578

Power family link, 452

Power family transformations, 182

Prediction, 9, 33, 45, 104, 488

in time series regression models, 488

Prediction interval on a new observation, 33, 46, 104, 488, 491

Predictor variable, 2. See also Regressor variable

PRESS residuals, 134

PRESS statistic, 151, 152, 337, 591

Principal component regression, 313, 315

Principal components, 314

Probit analysis, 442

Probit link, 452

Properties of ridge regression, 309

Pure error, 157

Quasi-likelihood, 462

R², 35, 36, 46, 87, 332

for prediction, 151

Random regressor variable, 52

Rank of a matrix, definition, 577

Regression analysis, 1

Regression coefficients, 13

Regression models

with concurrent lines, 27

with parallel lines, 272

with random effects, 194

Regression or model sum of squares, 26, 84, 86

Regressor variable, 2, 52, 68, 73, 93, 118, 184, 260, 273, 274, 275, 285, 292, 328, 511

REML, 196, 200, 201

Residual analysis, 22, 130, 328

Residual mean square, 21, 80, 333

Residual plotting, 130, 136, 139, 141, 142, 143, 149, 475

Residual sum of squares, 20, 84, 86

Residuals, 73, 130

Response surface, 242

methodology, 242, 532

Response variable, 2, 68, 422, 444

Retrospective study, 5, 6

Ridge regression, 304, 307, 309, 312, 319

Ridge trace, 307

Robust estimation of parameters, 221. See also Robust regression

Robust regression, 500, 502, 503, 504, 510

Rotatability, 248, 533

R-student, 135

Sample correlation coefficient, 53

Sample regression model, 13

Scaled residuals, 130

Scatter diagram, 1, 82

Second-order model, 69, 242

Sherman–Morrison–Woodbury theorem, 590

Shrinkage estimators, 310

Significance of regression, 24, 25, 28, 84

Simple linear regression model, 2, 12

Simultaneous inference, 33, 100, 102, 103

Singular value decomposition, 299

Sources of multicollinearity in regression, 286

Spherical design region, 533

Splines, 229

Standard error of parameter estimates, 23

Standard error of regression, 21

Standardized Pearson residuals, 437

Standardized regression coefficients, 111, 115

Standardized residuals, 130

Starting values in nonlinear regression, 408

Statistical tests on residuals, 150

Stepwise regression, 345, 348, 350

Strength of a transformation, 172

Studentized residuals, 131

Sufficient statistics, 52

t-distribution (definition), 576

Testing the equality of regression coefficients, 96

Testing the general linear hypothesis, 95

Tests based on deviance, 433

Time series data, 474

Trace of a matrix (definition), 578

Transformation of the regressor variables, 184, 187

Transformations, 139, 171, 176, 182, 184, 224, 397

to linearize a model, 397

t-tests on model parameters, 22, 28, 55, 88, 585

Unbiased estimators, 18, 52, 79, 81

Unit length scaling, 114, 117

Unit normal scaling, 113

Variable selection in regression, 88, 327, 338, 342, 346, 348, 351

Variance decomposition proportions, 300

Variance inflation factors, 118, 296

Variance stabilizing transformations, 172

Vector of increments, 401

V-optimal design, 531

Wald inference, 436

Weibull growth model, 412

Weighted least squares, 188, 190

estimator, 190, 191

Woodbury matrix identity, 590

Wrong signs for regression coefficients, 119