| 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 |
Relative LR \(\chi^2\) explained. Interaction effects are added to main effects. |
| Figure 5.2 |
Relative explained variation due to each predictor. Interaction effects are added to main effects. Intervals are 0.95 bootstrap percentile confidence intervals. |
| Figure 5.3 |
Empirical and population cumulative distribution function |
| Figure 5.4 |
Estimating properties of sample median using the bootstrap |
| Figure 5.5 |
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 and 0.9 quantile of 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 \(\log_{10}(\text{months}+1\)) |
| Figure 10.9 |
Log odds of significant coronary artery disease modeling age with two dummy variables |
| Figure 10.10 |
Local regression fit for log odds of significant coronary disease vs. age and cholesterol |
| Figure 10.11 |
Linear spline surface for logit(significant disease) for males |
| Figure 10.12 |
Restricted cubic spline surface in two variables, each with \(k=4\) knots |
| Figure 10.13 |
Restricted cubic spline fit with age \(\times\) spline(cholesterol) and cholesterol \(\times\) spline(age) |
| Figure 10.14 |
Spline fit with nonlinear effects of cholesterol and age and a simple product interaction |
| Figure 10.15 |
Predictions from linear interaction model with mean age in tertiles indicated. |
| Figure 10.16 |
Partial residuals for binary logistic model |
| Figure 10.17 |
Effects of predictors on odds of coronary disease |
| Figure 10.18 |
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 10.19 |
Fitted logistic models in two variables, with and without interaction |
| Figure 10.20 |
Nomogram for predicting \(\Pr(\text{CAD})\) |
| Figure 10.21 |
Nomogram for predicting \(\Pr(\), Bacterial meningitis\()\) |
| Figure 11.1 |
Ranking of apparent importance of predictors of cause of death using LR statistics |
| 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 |
Patterns of missing Titanic data |
| Figure 12.8 |
Univariable descriptions of proportion of passengers with missing age |
| ?fig-titanic-nasingle |
Predicted log odds of survival in Titanic using casewise deletion |
| Figure 12.11 |
Distribution of imputed and actual ages |
| ?fig-titanic-calibrate |
Estimated calibration curves for the Titanic risk model, accounting for multiple imputation |
| Figure 12.14 |
Predicted Titanic survival using multiple imputation |
| Figure 12.15 |
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 |
| Figure 24.1 |
Plot of the degree of symmetry of the distribution of a variable (value of 1.0 is most symmetric) vs. the number of distinct values of the variable. Hover over a point to see the variable name and detailed characteristics. |
| Figure 24.2 |
Spearman rank correlation matrix. Positive correlations are blue and negative are red. |