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:

6.1 The R Modeling Language

R statistical modeling language:

Code 
response ~ terms

y ~ age + sex            # age + sex main effects
y ~ age + sex + age:sex  # add second-order interaction
y ~ age*sex              # second-order interaction +
                         # all main effects
y ~ (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:pressure
y ~ (age + race)*sex     # age+race+sex+age:sex+race:sex
y ~ treatment*(age*race + age*sex) # no interact. with race,sex
sqrt(y) ~ sex*sqrt(age) + race
# functions, with dummy variables generated if
# race is an R factor (classification) variable
y ~ sex + poly(age,2)    # poly generates orthogonal polynomials
race.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 model
f4 <- update(f, .~. + rcs(x5,5))# add rcs(x5,5) to model
f5 <- 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))
  • Rank correlation methods:
    rcorr, hoeffd, spearman2 (Hmisc)
  • Variable clustering: varclus (Hmisc)
  • Single imputation: transcan (Hmisc)
  • Multiple imputation: aregImpute (Hmisc)
  • Restricted cubic splines:
    rcspline.eval (Hmisc)
  • Re-state restricted spline in simpler form:
    rcspline.restate (Hmisc)

6.3 The rms Package

Big Picture

  • rms package handles many popular standard models
  • It makes it easier to do the right thing
  • There are more extensible, general approaches, especially using Bayesian methods
    • brms package
    • rstanarm package
    • rstanarm survival analysis system

FAQ
  • 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 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, meant for discrete \(Y\) glm
orm Ordinal semi-parametric regression model for continuous \(Y\) and several link functions, penalization, left, right, and interval censoring 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)
adapt_orm Use AIC to find the best link function and number of knots for a single continuous predictor using orm; a replacement for nonparametric smoothers that handles censoring orm
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)
Ocens Used by orm to code left, right, interval censoring of \(Y\)

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 or Quarto html output. The ggplot and plotly (plotp; see later in this section) methods are recommended.

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 and LR tests of most meaningful hypotheses
plot.anova Graphical depiction of anova
contrast General contrasts, C.L., tests (both Wald- and likelihood-based)
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
impactPO Assess the impact of the proportional odds assumption for lrm ordParallel
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

The orm function has special after-fit functions:

orm After-fit Functions
Function Purpose Related Functions
intCalibration Internal calibration, useful for assessing goodness-of-fit calibrate
ordParallel Check parallelism assumption, e.g., proportional odds impactPO
plotIntercepts Line plot of intercept estimates
ordESS Effective sample size with special handling of censored observations
Olinks Compute deviance of model under four link functions
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

An extensive overview of Bayesian capabilities of the rmsb package may be found at hbiostat.org/R/rmsb/blrm.html.

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 available
options(prType='html')           # print, summary, anova, validate: html output
                                 # others: 'latex', 'plain'
options(grType='plotly')         # plotly graphics for ggplot, anova, summary
                                 # default is 'base' for static graphics
ddist <- datadist(cholesterol, treat, num.diseases, age)
# Could have used ddist <- datadist(data.frame.name)
options(datadist="ddist")        # defines data dist. to rms
cholesterol <- impute(cholesterol)
fit <- lrm(y ~ treat + scored(num.diseases) + rcs(age) +
                 log(cholesterol+10) + treat:log(cholesterol+10),
                     x=TRUE, y=TRUE)  # needed for robcov, anova test='LR'
fit    # outputs plain, LaTeX, or html markup
describe(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 characteristics
anova(fit)                       # Wald tests, plain, LaTex, or html
anova(fit, test='LR')            # Likelihood ratio tests
anova(fit, treat, cholesterol)   # Test these 2 by themselves
plot(anova(fit))                 # Summarize anova graphically
summary(fit)                     # Estimate effects using default ranges
                                 # prints plain, LaTeX, or html
plot(summary(fit))               # Graphical display of effects with C.I.
summary(fit, treat="b", age=60)  # Specify reference cell and adjustment val
summary(fit, age=c(50,70))       # Estimate effect of increasing age from
                                 # 50 to 70
summary(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 cholesterols
contrast(fit, list(treat='b', cholesterol=c(150,200,250)),
              list(treat='a', cholesterol=c(150,200,250)),
         type='average')
# Add conf.type='profile' got get more accurate profile likilihood CIs
# as well as likelihood ratio tests for contrasts.
# 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 panels
ggplot(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 variables
plot(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 limits
logit <- 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 <- 10:80
logit <- 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 values

plot(ag^.5, logit)               # try square root vs. spline transform.
plot(ag^1.5, logit)              # try 1.5 power

latex(fit)                       # fit in math notation
# Draw a nomogram for the model fit
plot(nomogram(fit))

# Compose R function to evaluate linear predictors analytically
g <- 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 input
fit <- lrm(y ~ age.tertile * rcs(cholesterol))

6.4 Other Functions

  • processMI: works with Hmisc::fit.mult.impute to process resampling validation statistics and likelihood ratio \(\chi^2\) statistics accounting for multiple imputation
  • 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 plots
ages <- 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