Modern Approaches to Predictive Modeling and Covariable Adjustment in
Randomized Clinical Trials
Frank E Harrell Jr
University of Virginia
- Audience: Statisticians and persons from other quantitative
disciplines who are interested in multivariable regression analysis of
univariate responses, in developing, validating, and graphically
describing multivariable predictive models, and in
covariable adjustment in randomized
clinical trials .
The course will be of particular interest to:
Applied statisticians who want to learn new methodology for
flexibly fitting all types of multivariable regression models while
making estimation of optimal covariable
transformations an explicit part of the modeling process.
- Those who want to learn how to develop models that are likely to
predict future observations as accurately as they predicted responses
from the data used to fit the models.
- Statisticians who want to learn how to graphically present
complex regression models to non-statisticians.
- Analysts who would like to
be introduced to multiple
imputation with regression models to handle missing and incomplete
- Quantitatively-minded epidemiologists and others who need to use
models for analyzing and predicting outcomes in observational studies.
- Biostatisticians, health services and outcomes researchers, and
economists who need to study or predict health outcomes or
- Biostatisticians working in clinical trials
who would like to learn about the need for adjusting for covariables
in perfectly balanced randomized trials and to be introduced to
developing analytic plans for such adjustment.
- Prerequisites: A good general knowledge of statistical estimation
and inference methods and a good command of ordinary linear
Participants are encouraged to read
references [1, 2, 4] in advance.
Those interested in covariable adjustment in
randomized clinical trials may also want to read .
- Course Description:
The first part of the course presents the following
elements of multivariable predictive modeling for a single response
variable: using regression splines to relax linearity assumptions,
perils of variable selection and overfitting, where to spend degrees
of freedom, shrinkage, imputation of missing data, data reduction,
and interaction surfaces. Then a
default overall modeling strategy will be described. This is followed
by methods for graphically understanding models (e.g., using
nomograms) and using re-sampling to estimate a model's likely
performance on new data. Then the freely available S-Plus
Design library will be overviewed. Design
facilitates most of the steps of the modeling process.
A case study exploring voting
tendencies over U.S. counties in the 1992 presidential election will
This short course will survey the advantages
of modeling in randomized trials and will provide some guidance in
developing a prospective statistical plan for use in a Phase III
The methods covered in this course will apply to almost any regression
model, including ordinary least squares, logistic regression models,
and survival models.
Be familiar with modern methods for fitting multivariable regression models:
- in a way the sample size will allow, without overfitting
- uncovering complex non-linear or non-additive relationships
- testing for and quantifying the association between one or
more predictors and the response, with possible adjustment
for other factors
- Be able to validate models for predictive accuracy and to detect
- Be able to interpret fitted models using both parameter estimates
- Be able to critique the literature to detect models that are likely
to be unreliable
- Understand benefits of covariable adjustment in randomized
Planning for Modeling
- Covariable Adjustment in Randomized Clinical Trials
- Reducing bias even with perfect balance
- Notation for Regression Models
- Interpreting Model Parameters
- Relaxing Linearity Assumption for Continuous Predictors
Simple Nonlinear Terms
- Splines for Estimating Shape of Regression Function and
Determining Predictor Transformations
- Cubic Spline Functions
- Restricted Cubic Splines
- Nonparametric regression
- Advantages of Splines over Other Methods
- Tests of Association
- Assessment of Model Fit
- Modeling and Testing Interactions
- Missing Data
Types of Missingness
- Understanding Patterns of Missing Values
- Problems with Simple Alternatives to Imputation
- Strategies for Developing Imputation Algorithms
- Single Conditional Mean Imputation
- Multiple Imputation
- S-Plus Software for Fitting Models and Adjusting Variances for
- Multivariable Modeling Strategy
Pre-Specification of Predictor Complexity
- Variable Selection
- Overfitting and Limits on Number of Predictors
- Data Reduction
- Resampling, Validating, Describing, and Simplifying the Model
- Model Validation
- Graphically Describing the Fitted Model
- Simplifying the Model by Approximating It
- S-Plus Design library
- Case Study using Least Squares Multiple Regression: Voting
Patterns in U.S. Counties
Dr. Harrell is Professor of Biostatistics and Statistics and Chief
of the Division of Biostatistics and Epidemiology, Department of
Health Evaluation Sciences, University of Virginia School of Medicine,
Charlottesville. He received his Ph.D. in biostatistics
from the University of North Carolina, Chapel Hill in 1979, where he
studied under P.K. Sen. Dr. Harrell has been involved in statistical
computing since 1969 and is the author of many S-Plus functions and SAS
procedures. Since 1973 he has been involved in medical applications
of statistics, especially in the area of survival analysis and
clinical prediction modeling. He is an editorial consultant for the
Journal of Clinical Epidemiology, is on the editorial board of
Statistics in Medicine, is co-managing editor of the new
journal Health Services and Outcomes Research Methodology, and is
a consultant to FDA.
- Handouts: Participants will receive copies of the
185 slides that will be presented
Regression models are frequently used to develop diagnostic,
prognostic, and health resource utilization models in clinical, health
services, outcomes, pharmacoeconomic, and epidemiologic research, and
in a multitude of non-health-related areas. Regression models
are also used to adjust for patient heterogeneity in randomized
clinical trials, to obtain tests that are more powerful and valid than
unadjusted treatment comparisons.
Models must be flexible enough to fit nonlinear and non-additive
relationships, but unless the sample size is enormous, the approach to
modeling must avoid common problems with data mining or data dredging
that result in overfitting and a failure of the predictive model to
validate on new subjects.
All standard regression models have assumptions that must be verified
for the model to have power to test hypotheses and for it to be able
to predict accurately. Of the principal assumptions (linearity,
additivity, distributional), this short course will emphasize methods for
assessing and satisfying the first two. Practical but powerful tools
are presented for validating model assumptions and presenting model
results. This course provides methods for estimating the shape of the
relationship between predictors and response using the widely
applicable method of augmenting the design matrix using restricted
F. E. Harrell, K. L. Lee, and D. B. Mark.
Multivariable prognostic models: Issues in developing models,
evaluating assumptions and adequacy, and measuring and reducing errors.
Statistics in Medicine, 15:361--387, 1996.
F. E. Harrell, P. A. Margolis, S. Gove, K. E. Mason, E. K. Mulholland,
D. Lehmann, L. Muhe, S. Gatchalian, and H. F. Eichenwald.
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.
Statistics in Medicine, 17:909--944, 1998.
W. W. Hauck, S. Anderson, and S. M. Marcus.
Should we adjust for covariates in nonlinear regression analyses of
Controlled Clinical Trials, 19:249--256, 1998.
A. Spanos, F. E. Harrell, and D. T. Durack.
Differential diagnosis of acute meningitis: An analysis of the
predictive value of initial observations.
Journal of the American Medical Association, 262:2700--2707,