Description

rms is the package that goes along with the book Regression Modeling Strategies. rms does regression modeling, testing, estimation, validation, graphics, prediction, and typesetting by storing enhanced model design attributes in the fit. rms is a re-written version of the Design package that has improved graphics and duplicates very little code in the survival package.

The package is a collection of about 180 functions that assist and streamline modeling, especially for biostatistical and epidemiologic applications. It also contains functions for binary and ordinal logistic regression models and the Buckley-James multiple regression model for right-censored responses, and implements penalized maximum likelihood estimation for logistic and ordinary linear models. rms works with almost any regression model, but it was especially written to work with logistic regression, Cox regression, accelerated failure time models, ordinary linear models, the Buckley-James model, generalized lease squares for longitudinal data (using the nlme package), generalized linear models, and quantile regression (using the quantreg package). rms requires the Hmisc package to be installed. Note that Hmisc has several functions useful for data analysis (especially data reduction and imputation).

Older references below pertaining to the Design package are relevant to rms.

Details

To make use of automatic typesetting features you must have LaTeX or one of its variants installed.

Some aspects of rms (e.g., latex) will not work correctly if options(contrasts=) other than c("contr.treatment", "contr.poly") are used.

rms relies on a wealth of survival analysis functions written by Terry Therneau of Mayo Clinic. Front-ends have been written for several of Therneau's functions, and other functions have been slightly modified.

Statistical Methods Implemented

• Ordinary linear regression models

• Binary and ordinal logistic models (proportional odds and continuation ratio models, probit, log-log, complementary log-log including ordinal cumulative probability models for continuous Y, efficiently handling thousands of distinct Y values using full likelihood methods)

• Bayesian binary and ordinal regression models, partial proportional odds model, and random effects

• Cox model

• Parametric survival models in the accelerated failure time class

• Buckley-James least-squares linear regression model with possibly right-censored responses

• Generalized linear model

• Quantile regression

• Generalized least squares

• Bootstrap model validation to obtain unbiased estimates of model performance without requiring a separate validation sample

• Automatic Wald tests of all effects in the model that are not parameterization-dependent (e.g., tests of nonlinearity of main effects when the variable does not interact with other variables, tests of nonlinearity of interaction effects, tests for whether a predictor is important, either as a main effect or as an effect modifier)

• Graphical depictions of model estimates (effect plots, odds/hazard ratio plots, nomograms that allow model predictions to be obtained manually even when there are nonlinear effects and interactions in the model)

• Various smoothed residual plots, including some new residual plots for verifying ordinal logistic model assumptions

• Composing S functions to evaluate the linear predictor (X*beta hat), hazard function, survival function, quantile functions analytically from the fitted model

• Typesetting of fitted model using LaTeX

• Robust covariance matrix estimation (Huber or bootstrap)

• Cubic regression splines with linear tail restrictions (natural splines)

• Tensor splines

• Interactions restricted to not be doubly nonlinear

• Penalized maximum likelihood estimation for ordinary linear regression and logistic regression models. Different parts of the model may be penalized by different amounts, e.g., you may want to penalize interaction or nonlinear effects more than main effects or linear effects

• Estimation of hazard or odds ratios in presence of nolinearity and interaction

• Sensitivity analysis for an unmeasured binary confounder in a binary logistic model

Motivation

rms was motivated by the following needs:

• need to automatically print interesting Wald tests that can be constructed from the design

  • tests of linearity with respect to each predictor

  • tests of linearity of interactions

  • pooled interaction tests (e.g., all interactions involving race)

  • pooled tests of effects with higher order effects

    • test of main effect not meaningful when effect in interaction

    • pooled test of main effect + interaction effect is meaningful

    • test of 2nd-order interaction + any 3rd-order interaction containing those factors is meaningful

• need to store transformation parameters with the fit

  • example: knot locations for spline functions

  • these are "remembered" when getting predictions, unlike standard S or R

  • for categorical predictors, save levels so that same dummy variables will be generated for predictions; check that all levels in out-of-data predictions were present when model was fitted

• need for uniform re-insertion of observations deleted because of NAs when using predict without newdata or when using resid

• need to easily plot the regression effect of any predictor

  • example: age is represented by a linear spline with knots at 40 and 60y plot effect of age on log odds of disease, adjusting interacting factors to easily specified constants

  • vary 2 predictors: plot x1 on x-axis, separate curves for discrete x2 or 3d perspective plot for continuous x2

  • if predictor is represented as a function in the model, plots should be with respect to the original variable:
    f <- lrm(y ~ log(cholesterol)+age)
plot(Predict(f, cholesterol)) # cholesterol on x-axis, default range
ggplot(Predict(f, cholesterol)) # same using ggplot2 plotp(Predict(f, cholesterol)) # same directly using plotly

• need to store summary of distribution of predictors with the fit

  • plotting limits (default: 10th smallest, 10th largest values or %-tiles)

  • effect limits (default: .25 and .75 quantiles for continuous vars.)

  • adjustment values for other predictors (default: median for continuous predictors, most frequent level for categorical ones)

  • discrete numeric predictors: list of possible values example: x=0,1,2,3,5 -> by default don't plot prediction at x=4

  • values are on the inner-most variable, e.g. cholesterol, not log(chol.)

  • allows estimation/plotting long after original dataset has been deleted

  • for Cox models, underlying survival also stored with fit, so original data not needed to obtain predicted survival curves

• need to automatically print estimates of effects in presence of non- linearity and interaction

  • example: age is quadratic, interacting with sex default effect is inter-quartile-range hazard ratio (for Cox model), for sex=reference level

  • user-controlled effects: summary(fit, age=c(30,50), sex="female") -> odds ratios for logistic model, relative survival time for accelerated failure time survival models

  • effects for all variables (e.g. odds ratios) may be plotted with multiple-confidence-level bars

• need for prettier and more concise effect names in printouts, especially for expanded nonlinear terms and interaction terms

  • use inner-most variable name to identify predictors

  • e.g. for pmin(x^2-3,10) refer to factor with legal S-name x

• need to recognize that an intercept is not always a simple concept

  • some models (e.g., Cox) have no intercept

  • some models (e.g., ordinal logistic) have multiple intercepts

• need for automatic high-quality printing of fitted mathematical model (with dummy variables defined, regression spline terms simplified, interactions "factored"). Focus is on regression splines instead of nonparametric smoothers or smoothing splines, so that explicit formulas for fit may be obtained for use outside S. rms can also compose S functions to evaluate X*Beta from the fitted model analytically, as well as compose SAS code to do this.

• need for automatic drawing of nomogram to represent the fitted model

• need for automatic bootstrap validation of a fitted model, with only one S command (with respect to calibration and discrimination)

• need for robust (Huber sandwich) estimator of covariance matrix, and be able to do all other analysis (e.g., plots, C.L.) using the adjusted covariances

• need for robust (bootstrap) estimator of covariance matrix, easily used in other analyses without change

• need for Huber sandwich and bootstrap covariance matrices adjusted for cluster sampling

• need for routine reporting of how many observations were deleted by missing values on each predictor (see na.delete in Hmisc)

• need for optional reporting of descriptive statistics for Y stratified by missing status of each X (see na.detail.response)

• need for pretty, annotated survival curves, using the same commands for parametric and Cox models

• need for ordinal logistic model (proportional odds model, continuation ratio model)

• need for estimating and testing general contrasts without having to be conscious of variable coding or parameter order

Fitting Functions Compatible with rms

rms will work with a wide variety of fitting functions, but it is meant especially for the following:

Methods in rms

The following generic functions work with fits with rms in effect:

Background for Examples

The following programs demonstrate how the pieces of the rms package work together. A (usually) one-time call to the function datadist requires a pass at the entire data frame to store distribution summaries for potential predictor variables. These summaries contain (by default) the .25 and .75 quantiles of continuous variables (for estimating effects such as odds ratios), the 10th smallest and 10th largest values (or .1 and .9 quantiles for small n) for plotting ranges for estimated curves, and the total range. For discrete numeric variables (those having <=10 unique values), the list of unique values is also stored. Such summaries are used by the summary.rms, Predict, and nomogram.rms functions. You may save time and defer running datadist. In that case, the distribution summary is not stored with the fit object, but it can be gathered before running summary, plot, ggplot, or plotp.

d <- datadist(my.data.frame) # or datadist(x1,x2)
options(datadist="d") # omit this or use options(datadist=NULL)
# if not run datadist yet
cf <- ols(y ~ x1 * x2)
anova(f)
fastbw(f)
Predict(f, x2) predict(f, newdata)

In the Examples section there are three detailed examples using a fitting function designed to be used with rms, lrm (logistic regression model). In Detailed Example 1 we create 3 predictor variables and a two binary response on 500 subjects. For the first binary response, dz, the true model involves only sex and age, and there is a nonlinear interaction between the two because the log odds is a truncated linear relationship in age for females and a quadratic function for males. For the second binary outcome, dz.bp, the true population model also involves systolic blood pressure (sys.bp) through a truncated linear relationship. First, nonparametric estimation of relationships is done using the Hmisc package's plsmo function which uses lowess with outlier detection turned off for binary responses. Then parametric modeling is done using restricted cubic splines. This modeling does not assume that we know the true transformations for age or sys.bp but that these transformations are smooth (which is not actually the case in the population).

For Detailed Example 2, suppose that a categorical variable treat has values "a", "b", and "c", an ordinal variable num.diseases has values 0,1,2,3,4, and that there are two continuous variables, age and cholesterol. age is fitted with a restricted cubic spline, while cholesterol is transformed using the transformation log(cholesterol - 10). Cholesterol is missing on three subjects, and we impute these using the overall median cholesterol. We wish to allow for interaction between treat and cholesterol. The following S program will fit a logistic model, test all effects in the design, estimate effects, and plot estimated transformations. The fit for num.diseases really considers the variable to be a 5-level categorical variable. The only difference is that a 3 d.f. test of linearity is done to assess whether the variable can be re-modeled "asis". Here we also show statements to attach the rms package and store predictor characteristics from datadist.

Detailed Example 3 shows some of the survival analysis capabilities of rms related to the Cox proportional hazards model. We simulate data for 2000 subjects with 2 predictors, age and sex. In the true population model, the log hazard function is linear in age and there is no age x sex interaction. In the analysis below we do not make use of the linearity in age. rms makes use of many of Terry Therneau's survival functions that are builtin to S.

The following is a typical sequence of steps that would be used with rms in conjunction with the Hmisc transcan function to do single imputation of all NAs in the predictors (multiple imputation would be better but would be harder to do in the context of bootstrap model validation), fit a model, do backward stepdown to reduce the number of predictors in the model (with all the severe problems this can entail), and use the bootstrap to validate this stepwise model, repeating the variable selection for each re-sample. Here we take a short cut as the imputation is not repeated within the bootstrap.

In what follows we (atypically) have only 3 candidate predictors. In practice be sure to have the validate and calibrate functions operate on a model fit that contains all predictors that were involved in previous analyses that used the response variable. Here the imputation is necessary because backward stepdown would otherwise delete observations missing on any candidate variable.

Note that you would have to define x1, x2, x3, y to run the following code.

xt <- transcan(~ x1 + x2 + x3, imputed=TRUE)
impute(xt) # imputes any NAs in x1, x2, x3
# Now fit original full model on filled-in data
f <- lrm(y ~ x1 + rcs(x2,4) + x3, x=TRUE, y=TRUE) #x,y allow boot.
fastbw(f)
# derives stepdown model (using default stopping rule)
validate(f, B=100, bw=TRUE) # repeats fastbw 100 times
cal <- calibrate(f, B=100, bw=TRUE) # also repeats fastbw
plot(cal)

Common Problems to Avoid

1. Don't have a formula like y ~ age + age^2. In S you need to connect related variables using a function which produces a matrix, such as pol or rcs. This allows effect estimates (e.g., hazard ratios) to be computed as well as multiple d.f. tests of association.

2. Don't use poly or strata inside formulas used in rms. Use pol and strat instead.

3. Almost never code your own dummy variables or interaction variables in S. Let S do this automatically. Otherwise, anova can't do its job.

4. Almost never transform predictors outside of the model formula, as then plots of predicted values vs. predictor values, and other displays, would not be made on the original scale. Use instead something like y ~ log(cell.count+1), which will allow cell.count to appear on x-axes. You can get fancier, e.g., y ~ rcs(log(cell.count+1),4) to fit a restricted cubic spline with 4 knots in log(cell.count+1). For more complex transformations do something like f <- function(x) {
... various 'if' statements, etc.
log(pmin(x,50000)+1)
}
fit1 <- lrm(death ~ f(cell.count))
fit2 <- lrm(death ~ rcs(f(cell.count),4))
}

5. Don't put $ inside variable names used in formulas. Either attach data frames or use data=.

6. Don't forget to use datadist. Try to use it at the top of your program so that all model fits can automatically take advantage if its distributional summaries for the predictors.

7. Don't validate or calibrate models which were reduced by dropping "insignificant" predictors. Proper bootstrap or cross-validation must repeat any variable selection steps for each re-sample. Therefore, validate or calibrate models which contain all candidate predictors, and if you must reduce models, specify the option bw=TRUE to validate or calibrate.

8. Dropping of "insignificant" predictors ruins much of the usual statistical inference for regression models (confidence limits, standard errors, P-values, chi-squares, ordinary indexes of model performance) and it also results in models which will have worse predictive discrimination.

Accessing the Package

Use require(rms).

Published Applications of rms and Regression Splines

• Spline fits

  1. Spanos A, Harrell FE, Durack DT (1989): Differential diagnosis of acute meningitis: An analysis of the predictive value of initial observations. JAMA 2700-2707.

  2. Ohman EM, Armstrong PW, Christenson RH, et al. (1996): Cardiac troponin T levels for risk stratification in acute myocardial ischemia. New Eng J Med 335:1333-1341.

• Bootstrap calibration curve for a parametric survival model:

  1. Knaus WA, Harrell FE, Fisher CJ, Wagner DP, et al. (1993): The clinical evaluation of new drugs for sepsis: A prospective study design based on survival analysis. JAMA 270:1233-1241.

• Splines, interactions with splines, algebraic form of fitted model from latex.rms

  1. Knaus WA, Harrell FE, Lynn J, et al. (1995): The SUPPORT prognostic model: Objective estimates of survival for seriously ill hospitalized adults. Annals of Internal Medicine 122:191-203.

• Splines, odds ratio chart from fitted model with nonlinear and interaction terms, use of transcan for imputation

  1. Lee KL, Woodlief LH, Topol EJ, Weaver WD, Betriu A. Col J, Simoons M, Aylward P, Van de Werf F, Califf RM. Predictors of 30-day mortality in the era of reperfusion for acute myocardial infarction: results from an international trial of 41,021 patients. Circulation 1995;91:1659-1668.

• Splines, external validation of logistic models, prediction rules using point tables

  1. Steyerberg EW, Hargrove YV, et al (2001): Residual mass histology in testicular cancer: development and validation of a clinical prediction rule. Stat in Med 2001;20:3847-3859.

  2. van Gorp MJ, Steyerberg EW, et al (2003): Clinical prediction rule for 30-day mortality in Bjork-Shiley convexo-concave valve replacement. J Clinical Epidemiology 2003;56:1006-1012.

• Model fitting, bootstrap validation, missing value imputation

  1. Krijnen P, van Jaarsveld BC, Steyerberg EW, Man in 't Veld AJ, Schalekamp, MADH, Habbema JDF (1998): A clinical prediction rule for renal artery stenosis. Annals of Internal Medicine 129:705-711.

• Model fitting, splines, bootstrap validation, nomograms

  1. Kattan MW, Eastham JA, Stapleton AMF, Wheeler TM, Scardino PT. A preoperative nomogram for disease recurrence following radical prostatectomy for prostate cancer. J Natl Ca Inst 1998; 90(10):766-771.

  2. Kattan, MW, Wheeler TM, Scardino PT. A postoperative nomogram for disease recurrence following radical prostatectomy for prostate cancer. J Clin Oncol 1999; 17(5):1499-1507

  3. Kattan MW, Zelefsky MJ, Kupelian PA, Scardino PT, Fuks Z, Leibel SA. A pretreatment nomogram for predicting the outcome of three-dimensional conformal radiotherapy in prostate cancer. J Clin Oncol 2000; 18(19):3252-3259.

  4. Eastham JA, May R, Robertson JL, Sartor O, Kattan MW. Development of a nomogram which predicts the probability of a positive prostate biopsy in men with an abnormal digital rectal examination and a prostate specific antigen between 0 and 4 ng/ml. Urology. (In press).

  5. Kattan MW, Heller G, Brennan MF. A competing-risk nomogram fir sarcoma-specific death following local recurrence. Stat in Med 2003; 22; 3515-3525.

• Penalized maximum likelihood estimation, regression splines, web site to get predicted values

  1. Smits M, Dippel DWJ, Steyerberg EW, et al. Predicting intracranial traumatic findings on computed tomography in patients with minor head injury: The CHIP prediction rule. Ann Int Med 2007; 146:397-405.

• Nomogram with 2- and 5-year survival probability and median survival time (but watch out for the use of univariable screening)

  1. Clark TG, Stewart ME, Altman DG, Smyth JF. A prognostic model for ovarian cancer. Br J Cancer 2001; 85:944-52.

• Comprehensive example of parametric survival modeling with an extensive nomogram, time ratio chart, anova chart, survival curves generated using survplot, bootstrap calibration curve

  1. Teno JM, Harrell FE, Knaus WA, et al. Prediction of survival for older hospitalized patients: The HELP survival model. J Am Geriatrics Soc 2000; 48: S16-S24.

• Model fitting, imputation, and several nomograms expressed in tabular form

  1. Hasdai D, Holmes DR, et al. Cardiogenic shock complicating acute myocardial infarction: Predictors of death. Am Heart J 1999; 138:21-31.

• Ordinal logistic model with bootstrap calibration plot

  1. Wu AW, Yasui U, Alzola CF et al. Predicting functional status outcomes in hospitalized patients aged 80 years and older. J Am Geriatric Society 2000; 48:S6-S15.

• Propensity modeling in evaluating medical diagnosis, anova dot chart

  1. Weiss JP, Gruver C, et al. Ordering an echocardiogram for evaluation of left ventricular function: Level of expertise necessary for efficient use. J Am Soc Echocardiography 2000; 13:124-130.

• Simulations using rms to study the properties of various modeling strategies

  1. Steyerberg EW, Eijkemans MJC, Habbema JDF. Stepwise selection in small data sets: A simulation study of bias in logistic regression analysis. J Clin Epi 1999; 52:935-942.

  2. Steyerberg WE, Eijekans MJC, Harrell FE, Habbema JDF. Prognostic modeling with logistic regression analysis: In search of a sensible strategy in small data sets. Med Decision Making 2001; 21:45-56.

• Statistical methods and references related to rms, along with case studies which includes the rms code which produced the analyses

  1. Harrell FE, Lee KL, Mark DB (1996): Multivariable prognostic models: Issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors. Stat in Med 15:361-387.

  2. Harrell FE, Margolis PA, Gove S, Mason KE, Mulholland EK et al. (1998): Development of a clinical prediction model for an ordinal outcome: The World Health Organization ARI Multicentre Study of clinical signs and etiologic agents of pneumonia, sepsis, and meningitis in young infants. Stat in Med 17:909-944.

  3. Bender R, Benner, A (2000): Calculating ordinal regression models in SAS and S-Plus. Biometrical J 42:677-699.

Bug Reports

The author is willing to help with problems. Send E-mail to fh@fharrell.com. To report bugs, please do the following:

1. If the bug occurs when running a function on a fit object (e.g., anova), attach a dump'd text version of the fit object to your note. If you used datadist but not until after the fit was created, also send the object created by datadist. Example: save(myfit,"/tmp/myfit.rda") will create an R binary save file that can be attached to the E-mail.

2. If the bug occurs during a model fit (e.g., with lrm, ols, psm, cph), send the statement causing the error with a save'd version of the data frame used in the fit. If this data frame is very large, reduce it to a small subset which still causes the error.

Copyright Notice

GENERAL DISCLAIMER This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. In short: you may use this code any way you like, as long as you don't charge money for it, remove this notice, or hold anyone liable for its results. Also, please acknowledge the source and communicate changes to the author.

If this software is used is work presented for publication, kindly reference it using for example: Harrell FE (2009): rms: S functions for biostatistical/epidemiologic modeling, testing, estimation, validation, graphics, and prediction. Programs available from https://hbiostat.org/R/rms/. Be sure to reference other packages used as well as R itself.

Author(s)

Frank E Harrell Jr
Professor of Biostatistics
Vanderbilt University School of Medicine
Nashville, Tennessee
fh@fharrell.com

References

The primary resource for the rms package is Regression Modeling Strategies, second edition by FE Harrell (Springer-Verlag, 2015) and the web page https://hbiostat.org/R/rms/. See also the Statistics in Medicine articles by Harrell et al listed below for case studies of modeling and model validation using rms.

Several datasets useful for multivariable modeling with rms are found at https://hbiostat.org/data/.

Examples