R allows interaction spline functions, wide variety of predictor parameterizations, wide variety of models, unifying model formula language, model validation by resampling.

R is comprehensive:

Easy to write R functions for new models \(\rightarrow\) wide variety of modern regression models implemented (trees, nonparametric, ACE, AVAS, survival models for multiple events)

Designs can be generated for any model \(\rightarrow\) all handle “class” var, interactions, nonlinear expansions

Single R objects (e.g., fit object) can be self-documenting \(\rightarrow\) automatic hypothesis tests, predictions for new data

Superior graphics

Classes and generic functions

6.1 The R Modeling Language

R statistical modeling language:

Code

response ~ termsy ~ age + sex # age + sex main effectsy ~ age + sex + age:sex # add second-order interactiony ~ age*sex # second-order interaction +# all main effectsy ~ (age + sex + pressure)^2# age+sex+pressure+age:sex+age:pressure...y ~ (age + sex + pressure)^2- sex:pressure# all main effects and all 2nd order# interactions except sex:pressurey ~ (age + race)*sex # age+race+sex+age:sex+race:sexy ~ treatment*(age*race + age*sex) # no interact. with race,sexsqrt(y) ~ sex*sqrt(age) + race# functions, with dummy variables generated if# race is an R factor (classification) variabley ~ sex +poly(age,2) # poly generates orthogonal polynomialsrace.sex <-interaction(race,sex)y ~ age + race.sex # for when you want dummy variables for# all combinations of the factors

The formula for a regression model is given to a modeling function, e.g.

Code

lrm(y ~rcs(x,4))

is read “use a logistic regression model to model y as a function of x, representing x by a restricted cubic spline with 4 default knots”^{1}.

^{1}lrm andrcs are in the rms package.

update function: re-fit model with changes in terms or data:

Code

f <-lrm(y ~rcs(x,4) + x2 + x3)f2 <-update(f, subset=sex=="male")f3 <-update(f, .~.-x2) # remove x2 from modelf4 <-update(f, .~. +rcs(x5,5))# add rcs(x5,5) to modelf5 <-update(f, y2 ~ .) # same terms, new response var.

6.2 User-Contributed Functions

R is high-level object-oriented language.

R runs on all platforms, and can be run on chromebooks using RStudio Server

Multitude of user-contributed functions freely available

International community of users

Some R functions:

See Venables and Ripley

Hierarchical clustering: hclust

Principal components: princomp, prcomp

Canonical correlation: cancor

Nonparametric transform-both-sides additive models: ace, avas

Parametric transform-both-sides additive models: areg, areg.boot (Hmisc package in R))

datadist function to compute predictor distribution summaries

Code

y ~ sex +lsp(age,c(20,30,40,50,60)) + sex %ia%lsp(age,c(20,30,40,50,60))

E.g. restrict age \(\times\) cholesterol interaction to be of form \(AF(B) + BG(A)\):

Code

y ~lsp(age,30) +rcs(cholesterol,4) +lsp(age,30) %ia%rcs(cholesterol,4)

Special fitting functions by Harrell to simplify procedures described in these notes:

rms Fitting Functions

Function

Purpose

Related R Functions

ols

Ordinary least squares linear model

lm

lrm

Binary and ordinal logistic regression model with options for penalize MLE

glm

orm

Ordinal semi-parametric regression model for continuous \(Y\) and several link functions

polr,lrm

psm

Accelerated failure time parametric survival models

survreg

cph

Cox proportional hazards regression

coxph

bj

Buckley-James censored least squares model

survreg,lm

Glm

rms version of glm

glm

Gls

rms version of gls

gls (nlme package)

Rq

rms version of rq

rq (quantreg package)

rms Transformation Functions

Function

Purpose

asis

No post-transformation (seldom used explicitly)

rcs

Restricted cubic splines

pol

Polynomial using standard notation

lsp

Linear spline

catg

Categorical predictor (seldom)

scored

Ordinal categorical variables

matrx

Keep variables as group for anova and fastbw

strat

Non-modeled stratification factors (used for cph only)

The transformation functions work also with regular R functions, e.g. when predict() is called the predicted values are computed by looking up the knot locations for rcs.

Below notice that there are three graphic models implemented for depicting the effects of predictors in the fitted model: lattice graphics, a ggplot method using the ggplot2 package (which has an option to convert the result to plotly), and a direct plotly method. plotly is used to create somewhat interactive graphics with drill-down capability, and the rms package takes advantage of this capability. plotly graphics are best used with RStudio Rmarkdown html output.

rms After-fit Functions

Function

Purpose

Related Functions

print

Print parameters and statistics of fit

coef

Fitted regression coefficients

formula

Formula used in the fit

specs

Detailed specifications of fit

vcov

Fetch covariance matrix

logLik

Fetch maximized log-likelihood

AIC

Fetch AIC with option to put on chi-square basis

lrtest

Likelihood ratio test for two nested models

univarLR

Compute all univariable LR \(\chi^{2}\)

robcov

Robust covariance matrix estimates

bootcov

Bootstrap covariance matrix estimates and bootstrap distribution of estimates

pentrace

Find optimum penalty factors by tracing effective AIC for a grid of penalties

effective.df

Print effective d.f. for each type of variable in model, for penalized fit or pentrace results

summary

Summary of effects of predictors

plot.summary

Plot continuously shaded confidence bars for results of summary

anova

Wald tests of most meaningful hypotheses

plot.anova

Graphical depiction of anova

contrast

General contrasts, C.L., tests

gendata

Easily generate predictor combinations

predict

Obtain predicted values or design matrix

Predict

Obtain predicted values and confidence limits easily varying a subset of predictors and others set at default values

plot.Predict

Plot the result of Predict using lattice

ggplot.Predict

Plot the result of Predict using ggplot2

plotp.Predict

Plot the result of Predict using plotly

fastbw

Fast backward step-down variable selection

step

residuals

(or resid) Residuals, influence stats from fit

sensuc

Sensitivity analysis for unmeasured confounder

which.influence

Which observations are overly influential

residuals

latex

\(\LaTeX\) representation of fitted model

Function

Function

R function analytic representation of \(X\hat{\beta}\) from a fitted regression model

latex

Hazard

R function analytic representation of a fitted hazard function (for psm)

Survival

R function analytic representation of fitted survival function (for psm, cph)

Quantile

R function analytic representation of fitted function for quantiles of survival time (for psm, cph)

Mean

R function analytic representation of fitted function for mean survival time or for ordinal logistic

nomogram

Draws a nomogram for the fitted model

latex, plot

survest

Estimate survival probabilities (psm, cph)

survfit

survplot

Plot survival curves (psm, cph)

plot.survfit

survplotp

Plot survival curves with plotly features

survplot

validate

Validate indexes of model fit using resampling

val.prob

External validation of a probability model

lrm

val.surv

External validation of a survival model

calibrate

calibrate

Estimate calibration curve using resampling

val.prob

vif

Variance inflation factors for fitted model

naresid

Bring elements corresponding to missing data back into predictions and residuals

naprint

Print summary of missing values

impute

Impute missing values

aregImpute

rmsb: Bayesian Regression Modeling Strategies Package, Focusing on Semiparametric Univariate and Longitudinal Models

Function

Purpose

blrm

Bayesian binary and ordinal logistic model

stackMI

Bayesian posterior stacking for multiple imputation

stanDx

Stan diagnostics on fit

stanDxplot

Trace plots to check posterior sampling convergence

PostF

Creates R function for computing posterior probabilities

plot.rmsb

Plot posterior densities, intervals, point summaries

compareBmods

Compare two models using LOO-cv

HPDint

Compute highest posterior density interval

distSym

Compute meaure of symmetry of posterior distribution

Global options prType and grType control printed and some graphical output, respectively as shown in example code below. The default is plain output and static graphics. If using plotly interactive graphics through ggplot or plotp or with anova or summary functions it is best to do so with RStudio html output or html notebooks. If using html output you must be producing an html document or notebook. When setting grType to use \(\LaTeX\) or html it is highly recommended that you use the knitr package.

Example:

treat: categorical variable with levels "a","b","c"

num.diseases: ordinal variable, 0-4

age: continuous Restricted cubic spline

cholesterol: continuous (3 missings; use median) log(cholesterol+10)

Allow treat\(\times\)cholesterol interaction

Program to fit logistic model, test all effects in design, estimate effects (e.g. inter-quartile range odds ratios), plot estimated transformations

Code

require(rms) # make new functions availableoptions(prType='html') # print, summary, anova LaTeX output# others: 'html', 'plain'options(grType='plotly') # plotly graphics for ggplot, anova, summary# default is 'base' for static graphicsddist <-datadist(cholesterol, treat, num.diseases, age)# Could have used ddist <- datadist(data.frame.name)options(datadist="ddist") # defines data dist. to rmscholesterol <-impute(cholesterol)fit <-lrm(y ~ treat +scored(num.diseases) +rcs(age) +log(cholesterol+10) + treat:log(cholesterol+10))fit # outputs plain, LaTeX, or html markupdescribe(y ~ treat +scored(num.diseases) +rcs(age))# or use describe(formula(fit)) for all variables used in fit# describe function (in Hmisc) gets simple statistics on variables# fit <- robcov(fit) # Would make all statistics that follow# use a robust covariance matrix# would need x=T, y=T in lrm()specs(fit) # Describe the design characteristicsanova(fit) # plain, LaTex, or htmlanova(fit, treat, cholesterol) # Test these 2 by themselvesplot(anova(fit)) # Summarize anova graphicallysummary(fit) # Estimate effects using default ranges# prints plain, LaTeX, or htmlplot(summary(fit)) # Graphical display of effects with C.I.summary(fit, treat="b", age=60) # Specify reference cell and adjustment valsummary(fit, age=c(50,70)) # Estimate effect of increasing age from# 50 to 70summary(fit, age=c(50,60,70)) # Increase age from 50 to 70, adjust to# 60 when estimating effects of other# factors# If had not defined datadist, would have to define ranges for all var.# Estimate and test treatment (b-a) effect averaged over 3 cholesterolscontrast(fit, list(treat='b', cholesterol=c(150,200,250)),list(treat='a', cholesterol=c(150,200,250)),type='average')# See the help file for contrast.rms for several examples of# how to obtain joint tests of multiple contrasts and how to get# double differences (interaction contrasts)p <-Predict(fit, age=seq(20,80,length=100), treat, conf.int=FALSE)plot(p) # Plot relationship between age and log# or ggplot(p), plotp(p) # odds, separate curve for each treat,# no C.I.plot(p, ~ age | treat) # Same but 2 panelsggplot(p, groups=FALSE)bplot(Predict(fit, age, cholesterol, np=50))# 3-dimensional perspective plot for age,# cholesterol, and log odds using default# ranges for both variablesplot(Predict(fit, num.diseases, fun=function(x) 1/(1+exp(-x)), conf.int=.9),ylab="Prob") # Plot estimated probabilities instead of# log odds (or use ggplot())# can also use plotp() for plotly# Again, if no datadist were defined, would have to tell plot all limitslogit <-predict(fit, expand.grid(treat="b",num.dis=1:3,age=c(20,40,60),cholesterol=seq(100,300,length=10)))# Could also obtain list of predictor settings interactively}logit <-predict(fit, gendata(fit, nobs=12))# Since age doesn't interact with anything, we can quickly and# interactively try various transformations of age, taking the spline# function of age as the gold standard. We are seeking a linearizing# transformation.ag <-10:80logit <-predict(fit, expand.grid(treat="a", num.dis=0, age=ag,cholesterol=median(cholesterol)), type="terms")[,"age"]# Note: if age interacted with anything, this would be the age# "main effect" ignoring interaction terms# Could also use# logit <- Predict(f, age=ag, ...)$yhat,# which allows evaluation of the shape for any level of interacting# factors. When age does not interact with anything, the result from# predict(f, ..., type="terms") would equal the result from# Predict if all other terms were ignored# Could also specify# logit <- predict(fit, gendata(fit, age=ag, cholesterol=...))# Un-mentioned variables set to reference valuesplot(ag^.5, logit) # try square root vs. spline transform.plot(ag^1.5, logit) # try 1.5 powerlatex(fit) # invokes latex.lrm, creates fit.tex# Draw a nomogram for the model fitplot(nomogram(fit))# Compose R function to evaluate linear predictors analyticallyg <-Function(fit)g(treat='b', cholesterol=260, age=50)# Letting num.diseases default to reference value

To examine interactions in a simpler way, you may want to group age into tertiles:

Code

age.tertile <-cut2(age, g=3)# For automatic ranges later, add age.tertile to datadist inputfit <-lrm(y ~ age.tertile *rcs(cholesterol))

6.4 Other Functions

supsmu: Friedman’s “super smoother”

lowess: Cleveland’s scatterplot smoother

glm: generalized linear models (see Glm)

gam: Generalized additive models

rpart: Like original CART with surrogate splits for missings, censored data extension (Atkinson & Therneau)

validate.rpart: in rms; validates recursive partitioning with respect to certain accuracy indexes

loess: multi-dimensional scatterplot smoother

Code

f <-loess(y ~ age * pressure)plot(f) # cross-sectional plotsages <-seq(20,70,length=40)pressures <-seq(80,200,length=40)pred <-predict(f, expand.grid(age=ages, pressure=pressures))persp(ages, pressures, pred) # 3-d plot

Source Code

```{r setup, include=FALSE}require(Hmisc)getRs('qbookfun.r')```# R Software {#sec-r}`R` allows interaction spline functions, wide variety ofpredictor parameterizations, wide variety of models, unifying modelformula language, model validation by resampling. `R` is comprehensive:* Easy to write `R` functions for new models $\rightarrow$ wide variety of modern regression models implemented (trees, nonparametric, ACE, AVAS, survival models for multiple events)* Designs can be generated for any model $\rightarrow$ all handle "class" var, interactions, nonlinear expansions* Single `R` objects (e.g., fit object) can be self-documenting $\rightarrow$ automatic hypothesis tests, predictions for new data* Superior graphics* Classes and generic functions## The `R` Modeling Language`R` statistical modeling language:```{r eval=FALSE}response ~ termsy ~ age + sex # age + sex main effectsy ~ age + sex + age:sex # add second-order interactiony ~ age*sex # second-order interaction +# all main effectsy ~ (age + sex + pressure)^2# age+sex+pressure+age:sex+age:pressure...y ~ (age + sex + pressure)^2- sex:pressure# all main effects and all 2nd order# interactions except sex:pressurey ~ (age + race)*sex # age+race+sex+age:sex+race:sexy ~ treatment*(age*race + age*sex) # no interact. with race,sexsqrt(y) ~ sex*sqrt(age) + race# functions, with dummy variables generated if# race is an R factor (classification) variabley ~ sex +poly(age,2) # poly generates orthogonal polynomialsrace.sex <-interaction(race,sex)y ~ age + race.sex # for when you want dummy variables for# all combinations of the factors```The formula for a regression model is given to a modelingfunction, e.g.```{r eval=FALSE}lrm(y ~rcs(x,4))```is read "use a logistic regression model to model y as a function of x, representing x bya restricted cubic spline with 4 default knots"^[`lrm` and`rcs` are in the `rms` package.].`update` function: re-fit model with changes in terms or data:```{r eval=FALSE}f <-lrm(y ~rcs(x,4) + x2 + x3)f2 <-update(f, subset=sex=="male")f3 <-update(f, .~.-x2) # remove x2 from modelf4 <-update(f, .~. +rcs(x5,5))# add rcs(x5,5) to modelf5 <-update(f, y2 ~ .) # same terms, new response var.```## User-Contributed Functions* `R` is high-level object-oriented language. * `R` runs on all platforms, and can be run on chromebooks using [RStudio Server](https://www.rstudio.com/products/rstudio/#rstudio-server)* Multitude of user-contributed functions freely available* International community of usersSome `R` functions:* See Venables and Ripley * Hierarchical clustering: `hclust`* Principal components: `princomp, prcomp`* Canonical correlation: `cancor`* Nonparametric transform-both-sides additive models: <br>`ace, avas`* Parametric transform-both-sides additive models: <br>`areg`,`areg.boot` (`Hmisc` package in `R`))* Rank correlation methods: <br>`rcorr`, `hoeffd`,`spearman2` (`Hmisc`)* Variable clustering: `varclus` (`Hmisc`)* Single imputation: `transcan` (`Hmisc`)* Multiple imputation: `aregImpute` (`Hmisc`)* Restricted cubic splines: <br>`rcspline.eval` (`Hmisc`)* Re-state restricted spline in simpler form:<br>`rcspline.restate` (`Hmisc`)## The `rms` Package`r mrg(rmsdisc())`[[FAQ](http://datamethods.org/rms)]{.aside}* `datadist` function to compute predictor distribution summaries ```{r eval=FALSE}y ~ sex +lsp(age,c(20,30,40,50,60)) + sex %ia%lsp(age,c(20,30,40,50,60))```E.g. restrict age $\times$cholesterol interaction to be of form $AF(B) + BG(A)$:```{r eval=FALSE}y ~lsp(age,30) +rcs(cholesterol,4) +lsp(age,30) %ia%rcs(cholesterol,4)```Special fitting functions by Harrell to simplifyprocedures described in these notes:| Function | Purpose | Related `R` Functions||-----|-------------------------------|---|| `ols` | Ordinary least squares linear model | `lm` || `lrm` | Binary and ordinal logistic regression model with options for penalize MLE| `glm` || `orm` | Ordinal semi-parametric regression model for continuous $Y$ and several link functions |`polr`,`lrm` || `psm` | Accelerated failure time parametric survival models | `survreg` || `cph` | Cox proportional hazards regression | `coxph` || `bj` | Buckley-James censored least squares model | `survreg`,`lm` || `Glm` | `rms` version of `glm` | `glm` || `Gls` | `rms` version of `gls` | `gls` (`nlme` package) || `Rq` | `rms` version of `rq` | `rq` (`quantreg` package) |: `rms` Fitting Functions| Function | Purpose | Related `R` Functions ||-----|-----|| `asis` | No post-transformation (seldom used explicitly) | `I` || `rcs` | Restricted cubic splines | `ns` || `pol` | Polynomial using standard notation | `poly` || `lsp` | Linear spline | || `catg` | Categorical predictor (seldom) | `factor` || `scored` | Ordinal categorical variables | `ordered` || `matrx` | Keep variables as group for `anova` and `fastbw` | `matrix` || `strat` | Non-modeled stratification factors (used for `cph` only) | `strata` |: `rms` Transformation FunctionsThe transformation functions work also with regular R functions, e.g. when `predict()` is called the predicted values are computed by looking up the knot locations for `rcs`.Below notice that there are three graphic models implemented fordepicting the effects of predictors in the fitted model: `lattice`graphics, a `ggplot` method using the `ggplot2` package (whichhas an option to convert the result to `plotly`), and a direct`plotly` method. `plotly` is used to create somewhatinteractive graphics with drill-down capability, and the `rms`package takes advantage of this capability. `plotly` graphics arebest used with RStudio Rmarkdown html output.| Function | Purpose | Related Functions ||----------|--------------------------------|-----|| `print` | Print parameters and statistics of fit | || `coef` | Fitted regression coefficients | || `formula` | Formula used in the fit | || `specs` | Detailed specifications of fit | || `vcov` | Fetch covariance matrix | || `logLik` | Fetch maximized log-likelihood | || `AIC` | Fetch AIC with option to put on chi-square basis | || `lrtest` | Likelihood ratio test for two nested models | || univarLR | Compute all univariable LR $\chi^{2}$ | || `robcov` | Robust covariance matrix estimates | || `bootcov` | Bootstrap covariance matrix estimates and bootstrap distribution of estimates | || `pentrace` | Find optimum penalty factors by tracing effective AIC for a grid of penalties | || `effective.df`| Print effective d.f. for each type of variable in model, for penalized fit or `pentrace` results | || `summary` | Summary of effects of predictors | || `plot.summary` | Plot continuously shaded confidence bars for results of `summary`| || `anova` | Wald tests of most meaningful hypotheses | || `plot.anova` | Graphical depiction of anova | || `contrast` | General contrasts, C.L., tests | || `gendata` | Easily generate predictor combinations | || `predict` | Obtain predicted values or design matrix | || `Predict` | Obtain predicted values and confidence limits easily varying a subset of predictors and others set at default values| || `plot.Predict` | Plot the result of `Predict` using `lattice`| || `ggplot.Predict` | Plot the result of `Predict` using `ggplot2` | || `plotp.Predict` | Plot the result of `Predict` using `plotly` | || `fastbw` | Fast backward step-down variable selection | `step` || `residuals` | (or `resid`) Residuals, influence stats from fit | || `sensuc` | Sensitivity analysis for unmeasured confounder | || `which.influence` | Which observations are overly influential |`residuals` || `latex` | $\LaTeX$ representation of fitted model | `Function` || `Function`| `R` function analytic representation of $X\hat{\beta}$ from a fitted regression model| `latex` || `Hazard` | `R` function analytic representation of a fitted hazard function (for `psm`) | || `Survival` | `R` function analytic representation of fitted survival function (for `psm`, `cph`) | || `Quantile` | `R` function analytic representation of fitted function for quantiles of survival time (for `psm`, `cph`) | || `Mean` | `R` function analytic representation of fitted function for mean survival time or for ordinal logistic | || `nomogram` | Draws a nomogram for the fitted model | `latex`, `plot` || `survest` | Estimate survival probabilities (`psm`, `cph`) |`survfit` || `survplot` | Plot survival curves (`psm`, `cph`) | `plot.survfit` || `survplotp` | Plot survival curves with `plotly` features | `survplot` || `validate` | Validate indexes of model fit using resampling | || `val.prob` | External validation of a probability model | `lrm` || `val.surv` | External validation of a survival model | `calibrate` || `calibrate` | Estimate calibration curve using resampling | `val.prob` || `vif` | Variance inflation factors for fitted model | || `naresid` | Bring elements corresponding to missing data back into predictions and residuals | || `naprint` | Print summary of missing values | || `impute` | Impute missing values | `aregImpute` |: `rms` After-fit Functions| Function | Purpose ||-----|-----|| `blrm` | Bayesian binary and ordinal logistic model || `stackMI` | Bayesian posterior stacking for multiple imputation || `stanDx` | Stan diagnostics on fit || `stanDxplot` | Trace plots to check posterior sampling convergence || `PostF` | Creates R function for computing posterior probabilities || `plot.rmsb` | Plot posterior densities, intervals, point summaries || `compareBmods` | Compare two models using LOO-cv || `HPDint` | Compute highest posterior density interval || `distSym` | Compute meaure of symmetry of posterior distribution |: `rmsb`: Bayesian Regression Modeling Strategies Package, Focusing on Semiparametric Univariate and Longitudinal ModelsAn extensive overview of Bayesian capabilities of the `rmsb` packagemay be found at [hbiostat.org/R/rmsb/blrm.html](https://hbiostat.org/R/rmsb/blrm.html).Global options `prType` and `grType` control printed and somegraphical output, respectively as shown in example code below. Thedefault is plain output and static graphics. If using `plotly`interactive graphics through `ggplot` or `plotp` or with`anova` or `summary` functions it is best todo so with RStudio `html` outputor `html` notebooks. If using `html` output you must beproducing an `html` document or notebook. When setting `grType`to use $\LaTeX$ or `html` it is highly recommended that you use the`knitr` package.Example:* `treat`: categorical variable with levels `"a","b","c"`* `num.diseases`: ordinal variable, 0-4* `age`: continuous <br> Restricted cubic spline* `cholesterol`: continuous <br> (3 missings; use median) <br>`log(cholesterol+10)`* Allow `treat` $\times$ `cholesterol` interaction* Program to fit logistic model, test all effects in design, estimate effects (e.g. inter-quartile range odds ratios), plot estimated transformations```{r eval=FALSE}require(rms) # make new functions availableoptions(prType='html') # print, summary, anova LaTeX output# others: 'html', 'plain'options(grType='plotly') # plotly graphics for ggplot, anova, summary# default is 'base' for static graphicsddist <-datadist(cholesterol, treat, num.diseases, age)# Could have used ddist <- datadist(data.frame.name)options(datadist="ddist") # defines data dist. to rmscholesterol <-impute(cholesterol)fit <-lrm(y ~ treat +scored(num.diseases) +rcs(age) +log(cholesterol+10) + treat:log(cholesterol+10))fit # outputs plain, LaTeX, or html markupdescribe(y ~ treat +scored(num.diseases) +rcs(age))# or use describe(formula(fit)) for all variables used in fit# describe function (in Hmisc) gets simple statistics on variables# fit <- robcov(fit) # Would make all statistics that follow# use a robust covariance matrix# would need x=T, y=T in lrm()specs(fit) # Describe the design characteristicsanova(fit) # plain, LaTex, or htmlanova(fit, treat, cholesterol) # Test these 2 by themselvesplot(anova(fit)) # Summarize anova graphicallysummary(fit) # Estimate effects using default ranges# prints plain, LaTeX, or htmlplot(summary(fit)) # Graphical display of effects with C.I.summary(fit, treat="b", age=60) # Specify reference cell and adjustment valsummary(fit, age=c(50,70)) # Estimate effect of increasing age from# 50 to 70summary(fit, age=c(50,60,70)) # Increase age from 50 to 70, adjust to# 60 when estimating effects of other# factors# If had not defined datadist, would have to define ranges for all var.# Estimate and test treatment (b-a) effect averaged over 3 cholesterolscontrast(fit, list(treat='b', cholesterol=c(150,200,250)),list(treat='a', cholesterol=c(150,200,250)),type='average')# See the help file for contrast.rms for several examples of# how to obtain joint tests of multiple contrasts and how to get# double differences (interaction contrasts)p <-Predict(fit, age=seq(20,80,length=100), treat, conf.int=FALSE)plot(p) # Plot relationship between age and log# or ggplot(p), plotp(p) # odds, separate curve for each treat,# no C.I.plot(p, ~ age | treat) # Same but 2 panelsggplot(p, groups=FALSE)bplot(Predict(fit, age, cholesterol, np=50))# 3-dimensional perspective plot for age,# cholesterol, and log odds using default# ranges for both variablesplot(Predict(fit, num.diseases, fun=function(x) 1/(1+exp(-x)), conf.int=.9),ylab="Prob") # Plot estimated probabilities instead of# log odds (or use ggplot())# can also use plotp() for plotly# Again, if no datadist were defined, would have to tell plot all limitslogit <-predict(fit, expand.grid(treat="b",num.dis=1:3,age=c(20,40,60),cholesterol=seq(100,300,length=10)))# Could also obtain list of predictor settings interactively}logit <-predict(fit, gendata(fit, nobs=12))# Since age doesn't interact with anything, we can quickly and# interactively try various transformations of age, taking the spline# function of age as the gold standard. We are seeking a linearizing# transformation.ag <-10:80logit <-predict(fit, expand.grid(treat="a", num.dis=0, age=ag,cholesterol=median(cholesterol)), type="terms")[,"age"]# Note: if age interacted with anything, this would be the age# "main effect" ignoring interaction terms# Could also use# logit <- Predict(f, age=ag, ...)$yhat,# which allows evaluation of the shape for any level of interacting# factors. When age does not interact with anything, the result from# predict(f, ..., type="terms") would equal the result from# Predict if all other terms were ignored# Could also specify# logit <- predict(fit, gendata(fit, age=ag, cholesterol=...))# Un-mentioned variables set to reference valuesplot(ag^.5, logit) # try square root vs. spline transform.plot(ag^1.5, logit) # try 1.5 powerlatex(fit) # invokes latex.lrm, creates fit.tex# Draw a nomogram for the model fitplot(nomogram(fit))# Compose R function to evaluate linear predictors analyticallyg <-Function(fit)g(treat='b', cholesterol=260, age=50)# Letting num.diseases default to reference value```To examine interactions in a simpler way, you may want to group ageinto tertiles:```{r eval=FALSE}age.tertile <-cut2(age, g=3)# For automatic ranges later, add age.tertile to datadist inputfit <-lrm(y ~ age.tertile *rcs(cholesterol))```## Other Functions* `supsmu`: Friedman's "super smoother" * `lowess`: Cleveland's scatterplot smoother* `glm`: generalized linear models (see `Glm`)* `gam`: Generalized additive models* `rpart`: Like original CART with surrogate splits for missings, censored data extension (Atkinson \& Therneau)* `validate.rpart`: in `rms`; validates recursive partitioning with respect to certain accuracy indexes* `loess`: multi-dimensional scatterplot smoother```{r eval=FALSE}f <-loess(y ~ age * pressure)plot(f) # cross-sectional plotsages <-seq(20,70,length=40)pressures <-seq(80,200,length=40)pred <-predict(f, expand.grid(age=ages, pressure=pressures))persp(ages, pressures, pred) # 3-d plot```