Format:
1 Online-Ressource (xx, 476 Seiten)
,
Illustrationen
Edition:
Second edition
ISBN:
9781498728348
Series Statement:
Texts in statistical science
Content:
Cover -- Half Title -- Title Page -- Copyright Page -- Table of Contents -- Preface -- 1: Linear Models -- 1.1 A simple linear model -- Simple least squares estimation -- 1.1.1 Sampling properties of β -- 1.1.2 So how old is the universe? -- 1.1.3 Adding a distributional assumption -- Testing hypotheses about β -- Confidence intervals -- 1.2 Linear models in general -- 1.3 The theory of linear models -- 1.3.1 Least squares estimation of β -- 1.3.2 The distribution of β -- 1.3.3 ( βi - βi)/σβi ~ tn-p -- 1.3.4 F-ratio results I -- 1.3.5 F-ratio results II -- 1.3.6 The influence matrix -- 1.3.7 The residuals, ϵ, and fitted values, µ -- 1.3.8 Results in terms of X -- 1.3.9 The Gauss Markov Theorem: What's special about least squares? -- 1.4 The geometry of linear modelling -- 1.4.1 Least squares -- 1.4.2 Fitting by orthogonal decompositions -- 1.4.3 Comparison of nested models -- 1.5 Practical linear modelling -- 1.5.1 Model fitting and model checking -- 1.5.2 Model summary -- 1.5.3 Model selection -- 1.5.4 Another model selection example -- A follow-up -- 1.5.5 Confidence intervals -- 1.5.6 Prediction -- 1.5.7 Co-linearity, confounding and causation -- 1.6 Practical modelling with factors -- 1.6.1 Identifiability -- 1.6.2 Multiple factors -- 1.6.3 'Interactions' of factors -- 1.6.4 Using factor variables in R -- 1.7 General linear model specification in R -- 1.8 Further linear modelling theory -- 1.8.1 Constraints I: General linear constraints -- 1.8.2 Constraints II: 'Contrasts' and factor variables -- 1.8.3 Likelihood -- 1.8.4 Non-independent data with variable variance -- 1.8.5 Simple AR correlation models -- 1.8.6 AIC and Mallows' statistic -- 1.8.7 The wrong model -- 1.8.8 Non-linear least squares -- 1.8.9 Further reading -- 1.9 Exercises -- 2: Linear Mixed Models -- 2.1 Mixed models for balanced data -- 2.1.1 A motivating example
Content:
The wrong approach: A fixed effects linear model -- The right approach: A mixed effects model -- 2.1.2 General principles -- 2.1.3 A single random factor -- 2.1.4 A model with two factors -- 2.1.5 Discussion -- 2.2 Maximum likelihood estimation -- 2.2.1 Numerical likelihood maximization -- 2.3 Linear mixed models in general -- 2.4 Linear mixed model maximum likelihood estimation -- 2.4.1 The distribution of b|y, β given θ -- 2.4.2 The distribution of β given θ -- 2.4.3 The distribution of θ -- 2.4.4 Maximizing the profile likelihood -- 2.4.5 REML -- 2.4.6 Effective degrees of freedom -- 2.4.7 The EM algorithm -- 2.4.8 Model selection -- 2.5 Linear mixed models in R -- 2.5.1 Package nlme -- 2.5.2 Tree growth: An example using lme -- 2.5.3 Several levels of nesting -- 2.5.4 Package lme4 -- 2.5.5 Package mgcv -- 2.6 Exercises -- 3: Generalized Linear Models -- 3.1 GLM theory -- 3.1.1 The exponential family of distributions -- 3.1.2 Fitting generalized linear models -- 3.1.3 Large sample distribution of β -- 3.1.4 Comparing models -- Deviance -- Model comparison with unknown ø -- AIC -- 3.1.5 Estimating ø, Pearson's statistic and Fletcher's estimator -- 3.1.6 Canonical link functions -- 3.1.7 Residuals -- Pearson residuals -- Deviance residuals -- 3.1.8 Quasi-likelihood -- 3.1.9 Tweedie and negative binomial distributions -- 3.1.10 The Cox proportional hazards model for survival data -- Cumulative hazard and survival functions -- 3.2 Geometry of GLMs -- 3.2.1 The geometry of IRLS -- 3.2.2 Geometry and IRLS convergence -- 3.3 GLMs with R -- 3.3.1 Binomial models and heart disease -- 3.3.2 A Poisson regression epidemic model -- 3.3.3 Cox proportional hazards modelling of survival data -- 3.3.4 Log-linear models for categorical data -- 3.3.5 Sole eggs in the Bristol channel -- 3.4 Generalized linear mixed models -- 3.4.1 Penalized IRLS
Content:
3.4.2 The PQL method -- 3.4.3 Distributional results -- 3.5 GLMMs with R -- 3.5.1 glmmPQL -- 3.5.2 gam -- 3.5.3 glmer -- 3.6 Exercises -- 4: Introducing GAMs -- 4.1 Introduction -- 4.2 Univariate smoothing -- 4.2.1 Representing a function with basis expansions -- A very simple basis: Polynomials -- The problem with polynomials -- The piecewise linear basis -- Using the piecewise linear basis -- 4.2.2 Controlling smoothness by penalizing wiggliness -- 4.2.3 Choosing the smoothing parameter, λ, by cross validation -- 4.2.4 The Bayesian/mixed model alternative -- 4.3 Additive models -- 4.3.1 Penalized piecewise regression representation of an additive model -- 4.3.2 Fitting additive models by penalized least squares -- 4.4 Generalized additive models -- 4.5 Summary -- 4.6 Introducing package mgcv -- 4.6.1 Finer control of gam -- 4.6.2 Smooths of several variables -- 4.6.3 Parametric model terms -- 4.6.4 The mgcv help pages -- 4.7 Exercises -- 5: Smoothers -- 5.1 Smoothing splines -- 5.1.1 Natural cubic splines are smoothest interpolators -- 5.1.2 Cubic smoothing splines -- 5.2 Penalized regression splines -- 5.3 Some one-dimensional smoothers -- 5.3.1 Cubic regression splines -- 5.3.2 A cyclic cubic regression spline -- 5.3.3 P-splines -- 5.3.4 P-splines with derivative based penalties -- 5.3.5 Adaptive smoothing -- 5.3.6 SCOP-splines -- 5.4 Some useful smoother theory -- 5.4.1 Identifiability constraints -- 5.4.2 'Natural' parameterization, effective degrees of freedom and smoothing bias -- 5.4.3 Null space penalties -- 5.5 Isotropic smoothing -- 5.5.1 Thin plate regression splines -- Thin plate splines -- Thin plate regression splines -- Properties of thin plate regression splines -- Knot-based approximation -- 5.5.2 Duchon splines -- 5.5.3 Splines on the sphere -- 5.5.4 Soap film smoothing over finite domains -- 5.6 Tensor product smooth interactions
Content:
5.6.1 Tensor product bases -- 5.6.2 Tensor product penalties -- 5.6.3 ANOVA decompositions of smooths -- Numerical identifiability constraints for nested terms -- 5.6.4 Tensor product smooths under shape constraints -- 5.6.5 An alternative tensor product construction -- What is being penalized? -- 5.7 Isotropy versus scale invariance -- 5.8 Smooths, random fields and random effects -- 5.8.1 Gaussian Markov random fields -- 5.8.2 Gaussian process regression smoothers -- 5.9 Choosing the basis dimension -- 5.10 Generalized smoothing splines -- 5.11 Exercises -- 6: GAM theory -- 6.1 Setting up the model -- 6.1.1 Estimating β given λ -- 6.1.2 Degrees of freedom and scale parameter estimation -- 6.1.3 Stable least squares with negative weights -- 6.2 Smoothness selection criteria -- 6.2.1 Known scale parameter: UBRE -- 6.2.2 Unknown scale parameter: Cross validation -- Leave-several-out cross validation -- Problems with ordinary cross validation -- 6.2.3 Generalized cross validation -- 6.2.4 Double cross validation -- 6.2.5 Prediction error criteria for the generalized case -- 6.2.6 Marginal likelihood and REML -- 6.2.7 The problem with log |Sλ|+ -- 6.2.8 Prediction error criteria versus marginal likelihood -- Unpenalized coefficient bias -- 6.2.9 The 'one standard error rule' and smoother models -- 6.3 Computing the smoothing parameter estimates -- 6.4 The generalized Fellner-Schall method -- 6.4.1 General regular likelihoods -- 6.5 Direct Gaussian case and performance iteration (PQL) -- 6.5.1 Newton optimization of the GCV score -- 6.5.2 REML -- log |Sλ|+ and its derivatives -- The remaining derivative components -- 6.5.3 Some Newton method details -- 6.6 Direct nested iteration methods -- 6.6.1 Prediction error criteria -- 6.6.2 Example: Cox proportional hazards model -- Derivatives with respect to smoothing parameters
Content:
Prediction and the baseline hazard -- 6.7 Initial smoothing parameter guesses -- 6.8 GAMM methods -- 6.8.1 GAMM inference with mixed model estimation -- 6.9 Bigger data methods -- 6.9.1 Bigger still -- 6.10 Posterior distribution and confidence intervals -- 6.10.1 Nychka's coverage probability argument -- Interval limitations and simulations -- 6.10.2 Whole function intervals -- 6.10.3 Posterior simulation in general -- 6.11 AIC and smoothing parameter uncertainty -- 6.11.1 Smoothing parameter uncertainty -- 6.11.2 A corrected AIC -- 6.12 Hypothesis testing and p-values -- 6.12.1 Approximate p-values for smooth terms -- Computing Tr -- Simulation performance -- 6.12.2 Approximate p-values for random effect terms -- 6.12.3 Testing a parametric term against a smooth alternative -- 6.12.4 Approximate generalized likelihood ratio tests -- 6.13 Other model selection approaches -- 6.14 Further GAM theory -- 6.14.1 The geometry of penalized regression -- 6.14.2 Backfitting GAMs -- 6.15 Exercises -- 7: GAMs in Practice: mgcv -- 7.1 Specifying smooths -- 7.1.1 How smooth specification works -- 7.2 Brain imaging example -- 7.2.1 Preliminary modelling -- 7.2.2 Would an additive structure be better? -- 7.2.3 Isotropic or tensor product smooths? -- 7.2.4 Detecting symmetry (with by variables) -- 7.2.5 Comparing two surfaces -- 7.2.6 Prediction with predict.gam -- Prediction with lpmatrix -- 7.2.7 Variances of non-linear functions of the fitted model -- 7.3 A smooth ANOVA model for diabetic retinopathy -- 7.4 Air pollution in Chicago -- 7.4.1 A single index model for pollution related deaths -- 7.4.2 A distributed lag model for pollution related deaths -- 7.5 Mackerel egg survey example -- 7.5.1 Model development -- 7.5.2 Model predictions -- 7.5.3 Alternative spatial smooths and geographic regression -- 7.6 Spatial smoothing of Portuguese larks data
Content:
7.7 Generalized additive mixed models with R
Additional Edition:
ISBN 9781498728331
Additional Edition:
Erscheint auch als Druck-Ausgabe Wood, Simon N. Generalized additive models Boca Raton : CRC Press/Taylor & Francis Group, 2017 ISBN 9781498728331
Language:
English
Subjects:
Economics
,
Psychology
,
Mathematics
Keywords:
Stochastischer Prozess
;
Verallgemeinertes lineares Modell
;
Regressionsanalyse
;
Irrfahrtsproblem
;
R
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