List of Figures

Figure Short Caption
Figure 2.1 A linear spline function with knots at \(a = 1, b = 3, c = 5\).
Figure 2.2 Cubic spline function and its derivatives
Figure 2.3 Restricted cubic spline component variables for 5 knots
Figure 2.4 Some typical restricted cubic spline functions
Figure 2.5 Regression assumptions for one binary and one continuous predictor
Figure 2.6 Probability of hemorrhagic stroke vs. blood pressures
Figure 4.1 Fitting errors to withstand or to avoid
Figure 4.2 Means from 20 \(U(0,1)\) samples
Figure 4.3 transcan transformations for two physiologic variables
Figure 4.4 HR vs. BP before and after transcan transformations
Figure 5.1 Empirical and population cumulative distribution function
Figure 5.2 Estimating properties of sample median using the bootstrap
Figure 5.3 Bootstrap confidence limits for ranks of predictors
Figure 7.1 Time profiles for individual subjects, stratified by study site and dose
Figure 7.2 Quartiles of TWSTRS stratified by dose
Figure 7.3 Mean responses and nonparametric bootstrap 0.95 confidence limits for population means, stratified by dose
Figure 7.4 Variogram, with assumed correlation pattern superimposed
Figure 7.5 Three residual plots to check for absence of trends in central tendency and in variability. Upper right panel shows the baseline score on the \(x\)-axis. Bottom left panel shows the mean \(\pm 2\times\) SD. Bottom right panel is the QQ plot for checking normality of residuals from the GLS fit.
Figure 7.6 Results of anova.rms from generalized least squares fit with continuous time AR1 correlation structure
Figure 7.7 Estimated effects of time, baseline TWSTRS, age, and sex
Figure 7.8 Contrasts and 0.95 confidence limits from GLS fit
Figure 7.9 Nomogram from GLS fit. Second axis is the baseline score.
Figure 8.1 Spearman \(\rho\) rank correlations of predictors
Figure 8.2 ’Hierarchical clustering
Figure 8.3 Simultaneous transformation and imputation using transcan
Figure 8.4 Variance of the system explained by principal components.
Figure 8.5 AIC vs. number of principal components
Figure 8.6 Sparse principal components
Figure 8.7 Performance of sparse principal components
Figure 8.8 Transformation of variables using ACE
Figure 9.1 Log-likelihood function for binomial distribution with 2 sample sizes
Figure 9.2 Tests arising from maximum liklihood estimation
Figure 9.3 Bootstrap confidence interval choices, from Carpenter & Bithell (2000)
Figure 9.4 Bootstrap confidence intervals
Figure 10.1 Logistic function
Figure 10.2 Absolute benefit as a function of risk in a control subject and the relative effect
Figure 10.3 Data, subgroup proportions, and fitted logistic model
Figure 10.4 Average maximum error with continuous predictor
Figure 10.5 Logistic regression assumptions for one binary and one continuous predictor
Figure 10.6 Logit proportions of significant CAD by sex and age
Figure 10.7 Duration of symptoms and severe CAD
Figure 10.8 Duration of symptoms and \(\log_{10}(\text{months}+1\))
?fig-lrm-cholxage Log odds of significant coronary artery disease modeling age with two dummy variables
Figure 10.11 Local regression fit for log odds of significant coronary disease vs. age and cholesterol
?fig-lrm-iacholxage-lsp Linear spline surface for logit(significant disease) for males
?fig-lrm-iacholxage-a Restricted cubic spline surface in two variables, each with \(k=4\) knots
?fig-lrm-iacholxage-b Restricted cubic spline fit with age \(\times\) spline(cholesterol) and cholesterol \(\times\) spline(age)
?fig-lrm-iacholxage-c Spline fit with nonlinear effects of cholesterol and age and a simple product interaction
Figure 10.20 Predictions from linear interaction model with mean age in tertiles indicated.
Figure 10.21 Partial residuals for binary logistic model
Figure 10.22 Effects of predictors on odds of coronary disease
Figure 10.23 Linear spline fit for probability of bacterial vs. viral meningitis as a function of age at onset (Spanos et al., 1989). Points are simple proportions by age quantile groups.
Figure 10.24 Fitted logistic models in two variables, with and without interaction
Figure 10.25 Nomogram for predicting \(\Pr(\text{CAD})\)
Figure 10.26 Nomogram for predicting \(\Pr(\), Bacterial meningitis\()\)
Figure 11.1 Ranking of apparent importance of predictors of cause of death
Figure 11.2 Partial effects in cause of death model
Figure 11.3 Interquartile-range odds ratios and confidence limits
Figure 11.4 Nomogram for obtaining \(X\hat{\beta}\) and \(\hat{P}\) from step-down model
Figure 11.5 Bootstrap nonparametric calibration curve for reduced cause of death model
Figure 11.6 Model approximation vs. LR \(\chi^2\) preserved
Figure 11.7 Approximate nomogram for predicting cause of death
Figure 12.1 Univariable summaries of Titanic survival
Figure 12.2 Multi-way summary of Titanic survival
Figure 12.3 Nonparametric regression for age, sex, class, and passenger survival
Figure 12.4 Relationship between age and survival stratified by family size variables
Figure 12.5 Effects of predictors on probability of surviving the Titanic
Figure 12.6 Effect of number of siblings/spouses on survival
Figure 12.7 Bootstrap overfitting-corrected loess nonparametric calibration curve for casewise deletion model
Figure 12.8 Univariable descriptions of proportion of passengers with missing age
Figure 12.9 Predicted log odds of survival in Titanic using casewise deletion
Figure 12.10 Distribution of imputed and actual ages
Figure 12.11 Predicted Titanic survival using multiple imputation
Figure 12.12 Odds ratios for some predictor settings
Figure 13.1 Simple method for checking PO assumption using stratification
Figure 13.2 Checking impact of the PO assumption
Figure 13.3 Checking assumptions of PO and parametric model
Figure 15.1 Examining normality and ordinal model assumptions
Figure 15.2 Assumptions of linear vs. semiparametric models
Figure 15.3 Six methods for estimating quantiles or means.
Figure 15.4 Observed and predicted distributions
Figure 15.5 Estimated intercepts from probit model
Figure 15.6 Variable clustering for all potential predictors
Figure 15.7 Median height vs. age
Figure 15.8 Median leg length vs. age
Figure 15.9 Generalized squared rank correlations
Figure 15.10 Estimated mean and quantiles from casewise deletion model.
Figure 15.11 ANOVA for reduced model after multiple imputation
Figure 15.12 Partial effects after multiple imputation
Figure 15.13 Partial effects (means) after multiple imputation
Figure 15.14 Partial effect for age with bootstrap and Wald confidence bands
Figure 15.15 Predicted mean, median, and 0.9 quantile of r hba
Figure 15.16 Nomogram of log-log ordinal model for \(\text{HbA}_{1c}\)
Figure 16.1 Transformations estimated by avas
Figure 16.2 Checking estimated against optimal transformation
Figure 16.3 Predicted y as a function of x2 and x3
Figure 17.1 Survival function
Figure 17.2 Cumulative hazard function
Figure 17.3 Hazard function
Figure 17.4 Some censored data. Circles denote events.
Figure 17.5 Some Weibull hazard functions with \(\alpha=1\) and various values of \(\gamma\)
Figure 17.6 Kaplan-Meier and Nelson–Aalen estimates
Figure 18.1 Absolute clinical benefit as a function of survival in a control subject and the relative benefit
Figure 18.2 PH model with one binary predictor
Figure 18.3 PH model with one continuous predictor
Figure 18.4 PH model with one continuous predictor
Figure 18.5 Regression assumptions, linear additive PH or AFT model with two predictors
Figure 18.6 AFT model with one predictor
Figure 18.7 AFT model with one continuous predictor
Figure 18.8 Examples of checking parametric survival model assumptions
Figure 18.9 Fitted log-logistic model
Figure 18.10 Checking AFT distributional assumption using residuals
Figure 18.11 Estimated log-logistic hazard functions
Figure 19.1 Cluster analysis of missingness in SUPPORT
Figure 19.2 Clustering of predictors in SUPPORT using Hoeffding \(D\)
Figure 19.3 \(\Phi^{-1}(S_{ ext{KM}}(t))\) stratified by dzgroup
Figure 19.4 Distributions of residuals from log-normal model
Figure 19.5 Generalized Spearman \(\rho^2\) rank correlation between predictors and truncated survival time
Figure 19.6 Somers’ \(D_{xy}\) rank correlation between predictors and original survival time
Figure 19.7 Partial \(\chi^{2}\) statistics from saturated main effects model
Figure 19.8 Effect of predictors on log survival time in SUPPORT
Figure 19.9 Contribution of variables in predicting survival time in log-normal model
Figure 19.10 Survival time ratios from fitted log-normal model
Figure 19.11 Bootstrap validation of calibration curve for log-normal model
Figure 19.12 Nomogram for simplified log-normal model
Figure 20.1 Nonparametric and Cox–Breslow survival estimates
Figure 20.2 Unadjusted (Kaplan–Meier) and adjusted survival estimates
Figure 20.3 Kaplan–Meier log \(\Lambda\) estimates by sex and deciles of age
Figure 20.4 Cox PH model stratified on sex, using spline function for age
Figure 20.5 Cox PH model stratified on sex,with interaction between age spline and sex
Figure 20.6 Spline estimate of relationship between LVEF and relative log hazard
Figure 20.7 Smoothed martingale residuals vs. LVEF
Figure 20.8 \(\Lambda\) ratio plot
Figure 20.9 Stratified hazard ratios for pain/ischemia index over time
Figure 20.10 Smoothed Schoenfeld residuals
Figure 20.11 Bootstrap calibration of random survival predictions
Figure 20.12 Display of an interactions among treatment, extent of disease, and year
Figure 20.13 Cox–Kalbfleisch–Prentice survival estimates stratifying on treatment and adjusting for several predictors
Figure 20.14 Cox model predictions with respect to a continuous variable
Figure 20.15 Survival estimates for model stratified on sex, with interaction.
Figure 20.16 Nomogram for stratified Cox model
Figure 21.1 Schoenfeld residuals for dose of estrogen in Cox model
Figure 21.2 Shapes of predictors for log hazard in prostate cancer
Figure 21.3 Bootstrap estimates of calibration accuracy in prostate cancer model
Figure 21.4 Hazard ratios for prostate survival model
Figure 21.5 Nomogram for predicting death in prostate cancer trial
Figure 22.1 Transition proportions from data simulated from VIOLET
Figure 22.2 State occupancy proportions from simulated VIOLET data with death carried forward
Figure 22.3 Estimated time trends in relative log transition odds
Figure 22.4 Variogram-like graph
Figure 22.5 State occupancy probabilities for each treatment
Figure 22.6 Relationship between bootstrap log ORs and differences in mean days unwell