building useful models: some new developments and easily avoidable errors michael babyak, phd
TRANSCRIPT
Building useful models: Some new developments and easily
avoidable errorsMichael Babyak, PhD
What is a model ?
Y = f(x1, x2, x3…xn)
Y = a + b1x1 + b2x2…bnxn
Y = e a + b1x1 + b2x2…bnxn
“All models are wrong, some are useful” -- George Box
• A useful model is– Not very biased– Interpretable– Replicable (predicts in a new sample)
Some Premises
• “Statistics” is a cumulative, evolving field• Newer is not necessarily better, but should be
entertained in the context of the scientific question at hand
• Data analytic practice resides along a continuum, from exploratory to confirmatory. Both are important, but the difference has to be recognized.
• There’s no substitute for thinking about the problem
Statistics is a cumulative, evolving field: How do we know this stuff?
• Theory
• Simulation
Y = b X + error
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bsk-1 bsk………………….
Concept of Simulation
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Y = b X + error
Evaluate
Concept of Simulation
Y = .4 X + error
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bsk-1 bsk………………….
Simulation Example
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Evaluate
Y = .4 X + error
Simulation Example
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Value of beta for x1
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True Model:Y = .4*x1 + e
Ingredients of a Useful Model
Correct probability model
Good measures/no loss of information
Based on theory
Comprehensive
Parsimonious
Flexible
Tested fairly
Useful Model
Correct Model
• Gaussian: General Linear Model• Multiple linear regression
• Binary (or ordinal): Generalized Linear Model• Logistic Regression• Proportional Odds/Ordinal Logistic
• Time to event: • Cox Regression or parametric survival
models
Generalized Linear Model
General Linear Model/Linear Regression
ANOVA/t-testANCOVA
Logistic Regression
Chi-square
Poisson, ZIP,negbin, gamma
Normal Binary/Binomial Count, heavy skew,Lots of zeros
Regression w/Transformed DV
Can be applied to clustered (e.g, repeated measures data)
Factor Analytic Family
Structural Equation Models
Partial Least SquaresLatent Variable Models
(Confirmatory Factor Analysis)
Multiple regression Principal
Components
Common FactorAnalysis
Use Theory
• Theory and expert information are critical in helping sift out artifact
• Numbers can look very systematic when the are in fact random– http://www.tufts.edu/~gdallal/multtest.htm
Measure well
Adequate rangeRepresentative valuesWatch for ceiling/floor effects
Using all the information
Preserving cases in data sets with missing dataConventional approaches:
Use only complete caseFill in with mean or medianUse a missing data indicator in the model
Missing Data
• Imputation or related approaches are almost ALWAYS better than deleting incomplete cases
• Multiple Imputation
• Full Information Maximum Likelihood
Multiple Imputation
Modern Missing Data Techniques
Preserve more information from original sample
Incorporate uncertainty about missingness into final estimates
Produce better estimates of population (true) values
Don’t throw waste information from variables
• Use all the information about the variables of interest
• Don’t create “clinical cutpoints” before modeling
• Model with ALL the data first, then use prediction to make decisions about cutpoints
Dichotomizing for Convenience = Dubious Practice
(C.R.A.P.*)
•Convoluted Reasoning and Anti-intellectual Pomposity •Streiner & Norman: Biostatistics: The Bare Essentials
0 4 8 12 16 20 24 28 32 36 40 44
Depression score
AB C
Implausible measurement assumption
“not depressed” “depressed”
http://psych.colorado.edu/~mcclella/MedianSplit/
http://www.bolderstats.com/jmsl/doc/medianSplit.html
Loss of power
Sometimes through sampling errorYou can get a ‘lucky cut.’
Dichotomization, by definition, reduces the magnitude of the estimate
by a minimum of about 30%
Dear Project Officer,
In order to facilitate analysis and interpretation, we have decided to throw away about 30% of our data. Even though this will waste about 3 or 4 hundred thousand dollars worth of subject recruitment and testing money, we are confident that you will understand.
Sincerely,
Dick O. Tomi, PhDProf. Richard Obediah Tomi, PhD
Power to detect non-zero b-weight when x is continuous versus
dichotomized
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0.85 0.75 0.65Reliability of x
% c
orr
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t re
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tio
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Continuous xDichotomized x
True model: y =.4x + e
Dichotomizing will obscure non-linearity
Dichotomized at Median (CES-D = 7)
Perc
ent w
ith W
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Motio
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bnorm
alit
y
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Not Depressed Depressed
Low HighCESD Score
WMA on at Least 1 TaskUsing Cubic Spline
CES-D Score
Pro
babi
lity
of W
MA
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Dichotomizing will obscure non-linearity:Same data as previous slide modeled
continuously
Type I error rates for the relation between x2 and y after dichotomizing two continuous predictors.
Maxwell and Delaney calculated the effect of dichotomizing two continuous predictors as a function of the correlation between them. The true model is
y = .5x1 + 0x2, where all variables are continuous. If x1 and x2 are dichotomized, the error rate for the relation between x2 and y increases as the
correlation between x1 and x2 increases.
Correlation between x1 and x2
N 0 .3 .5 .7
50 .05 .06 .08 .10
100 .05 .08 .12 .18
200 .05 .10 .19 .31
Is it ever a good idea to categorize quantitatively measured variables?
• Yes: – when the variable is truly categorical– for descriptive/presentational purposes– for hypothesis testing, if enough categories
are made.• However, using many categories can lead to problems of
multiple significance tests and still run the risk of misclassification
CONCLUSIONS• Cutting:
– Doesn’t always make measurement sense– Almost always reduces power– Can fool you with too much power in some
instances– Can completely miss important features of the
underlying function• Modern computing/statistical packages can
“handle” continuous variables
• Want to make good clinical cutpoints? Model first, decide on cuts afterward.
Sample size and the problem of underfitting vs overfitting
• Model assumption is that “ALL” relevant variables be included—the “antiparsimony principle”
• Tempered by fact that estimating too many unknowns with too little data will yield junk
Sample Size Requirements• Linear regression
– minimum of N = 50 + 8:predictor (Green, 1990)
• Logistic Regression– Minimum of N = 10-15/predictor among
smallest group (Peduzzi et al., 1990a)
• Survival Analysis– Minimum of N = 10-15/predictor (Peduzzi et
al., 1990b)
Consequences of inadequate sample size
• Lack of power for individual tests
• Unstable estimates
• Spurious good fit—lots of unstable estimates will produce spurious ‘good-looking’ (big) regression coefficients
All-noise, but good fit
R-Square from Full Model
De
nsi
ty
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n/p~3n/p~6.6n/p=10n/p~13.3
Events per predictor ratio
R-squares from a population model of completelyrandom variables
Simulation: number of events/predictor ratio
Y = .5*x1 + 0*x2 + .2*x3 + 0*x4
-- Where x1 x4 = .4
-- N/p = 3, 5, 10, 20, 50
Parameter stability and n/p ratiox1
Den
sity
-2.0 -1.0 0.0 0.5 1.0 1.5 2.0
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n/p=3n/p=5n/p=10n/p=20n/p=50
x2
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x3
Parameter Estimate
Den
sity
-2.0 -1.0 0.0 0.5 1.0 1.5 2.0
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Parameter Estimate
-2.0 -1.0 0.0 0.5 1.0 1.5 2.0
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Peduzzi’s Simulation: number of events/predictor ratio
P(survival) =a + b1*NYHA + b2*CHF + b3*VES+b4*DM + b5*STD + b6*HTN + b7*LVC
--Events/p = 2, 5, 10, 15, 20, 25
--% relative bias = (estimated b – true b/true b)*100
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-10
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Events per variable
% R
elat
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Simulation results: number of events/predictor ratio
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Events per variable
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port
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Simulation results: number of events/predictor ratio
Approaches to variable selection
• “Stepwise” automated selection• Pre-screening using univariate tests• Combining or eliminating redundant predictors• Fixing some coefficients• Theory, expert opinion and experience• Penalization/Random effects• Propensity Scoring
– “Matches” individuals on multiple dimensions to improve “baseline balance”
• Tibshirani’s “Lasso”
Any variable selection technique based on looking at the data first
will likely be biased
“I now wish I had never written the stepwise selection code for SAS.” --Frank Harrell, author of forward and
backwards selection algorithm for SAS PROC REG
Automated Selection: Derksen and Keselman (1992) Simulation Study
• Studied backward and forward selection
• Some authentic variables and some noise variables among candidate variables
• Manipulated correlation among candidate predictors
• Manipulated sample size
Automated Selection: Derksen and Keselman (1992) Simulation Study
• “The degree of correlation between candidate predictors affected the frequency with which the authentic predictors found their way into the model.”
• “The greater the number of candidate predictors, the greater the number of noise variables were included in the model.”
• “Sample size was of little practical importance in determining the number of authentic variables contained in the final model.”
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Variables in Final Model
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Simulation results: Number of noise variables included
20 candidate predictors; 100 samples
Sample Size
0102030405060708090
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% Variance Explained
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Simulation results: R-square from noise variables
20 candidate predictors; 100 samples
Sample Size
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quare
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Simulation results: R-square from noise variables
20 candidate predictors; 100 samples
Sample Size
1. It yields R-squared values that are badly biased high 2. The F and chi-squared tests quoted next to each variable on the
printout do not have the claimed distribution 3. The method yields confidence intervals for effects and predicted
values that are falsely narrow (See Altman and Anderson Stat in Med)
4. It yields P-values that do not have the proper meaning and the proper correction for them is a very difficult problem
5. It gives biased regression coefficients that need shrinkage (the coefficients for remaining variables are too large; see Tibshirani, 1996).
6. It has severe problems in the presence of collinearity 7. It is based on methods (e.g. F tests for nested models) that were
intended to be used to test pre-specified hypotheses. 8. Increasing the sample size doesn't help very much (see Derksen
and Keselman) 9. It allows us to not think about the problem 10. It uses a lot of paper
SOME of the problems with stepwise variable selection.
author ={Chatfield, C.}, title = {Model uncertainty, data mining and statistical inference (with discussion)}, journal = JRSSA, year = 1995, volume = 158, pages = {419-466}, annote =
--bias by selecting model because it fits the data well; bias in standard errors; P. 420: ... need for a better balance in the literature and in statistical teaching between techniques and problem solving strategies}. P. 421: It is `well known' to be `logically unsound and practically misleading' (Zhang, 1992) to make inferences as if a model is known to be true when it has, in fact, been selected from the same data to be used for estimation purposes. However, although statisticians may admit this privately (Breiman (1992) calls it a `quiet scandal'), they (we) continue to ignore the difficulties because it is not clear what else could or should be done. P. 421: Estimation errors for regression coefficients are usually smaller than errors from failing to take into account model specification. P. 422: Statisticians must stop pretending that model uncertainty does not exist and begin to find ways of coping with it. P. 426: It is indeed strange that we often admit model uncertainty by searching for a best model but then ignore this uncertainty by making inferences and predictions as if certain that the best fitting model is actually true.
Phantom Degrees of Freedom
• Faraway (1992)—showed that any pre-modeling strategy cost a df over and above df used later in modeling.
• Premodeling strategies included: variable selection, outlier detection, linearity tests, residual analysis.
• Thus, although not accounted for in final model, these phantom df will render the model too optimistic
Phantom Degrees of Freedom
• Therefore, if you transform, select, etc., you must include the DF in (i.e., penalize for) the “Final Model”
Conventional Univariate Pre-selection
• Non-significant tests also cost a DF• Non-significance is NOT
necessarily related to importance• Variables may not behave the
same way in a multivariable model—variable “not significant” at univariate test may be very important in the presence of other variables
• Despite the convention, testing for confounding has not been systematically studied—in many cases leads to overadjustment and underestimate of true effect of variable of interest.
• At the very least, pulling variables in and out of models inflates the model fit, often dramatically
Conventional Univariate Pre-selection
Better approach
• Pick variables a priori• Stick with them• Penalize appropriately for any
data-driven decision about how to model a variable
Spending DF wisely
• If not enough N/predictor, combine covariates using techniques that do not look at Y in the sample, PCA, FA, conceptual clustering, collapsing, scoring, established indexes.
• Save DF for finer-grained look at variables of most interest, e.g, non-linear functions
Help is on the way?
• Penalization/Random effects
• Propensity Scoring– “Matches” individuals on multiple dimensions
to improve “baseline balance”
• Tibshirani’s Lasso
http://myspace.com/monkeynavigatedrobots
Validation• Apparent fit
• Usually too optimistic• Internal
• cross-validation, bootstrap• honest estimate for model
performance• provides an upper limit to what would
be found on external validation• External validation
• replication with new sample, different circumstances
Validation
• Steyerburg, et al. (1999) compared validation methods
• Found that split-half was far too conservative
• Bootstrap was equal or superior to all other techniques
Conclusions• Measure well• Use all the information• Recognize the limitations based on how much
data you actually have• In the confirmatory mode, be as explicit as
possible about the model a priori, test it, and live with it
• By all means, explore data, but recognize— and state frankly --the limits post hoc analysis places on inference
Advanced topics and examples
?1………………….
My Sample
Evaluate
Bootstrap
?2 ?3 ?4 ?k-1 ?k
WITH REPLACEMENT
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7114510
1032221
351427
211727
4414210
Can use data to determine where to spend DF
• Use Spearman’s Rho to test “importance”
• Not peeking because we have chosen to include the term in the model regardless of relation to Y
• Use more DF for non-linearity
Example-Predict Survival from age, gender, and fare on Titanic:
example using S-Plus (or R) software
If you have already decided to include them (and promise to keep them in the model) you can peek at predictors in order to see where to add complexity
Adjusted rho^2
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1046 1
1308 1
1309 1
N df
age
fare
sex
Spearman Test
Non-linearity using splines
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YLinear Spline
(piecewise regression)
Y = a + b1(x<10) + b2(10<x<20) + b3 (x >20)
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knots
fitfare<-lrm(survived~(rcs(fare,3)+age+sex)^2,x=T,y=T)
anova(fitfare)
Logistic regression model
Spline with 3 knots
Wald Statistics Response: survived
Factor Chi-Square d.f. P fare (Factor+Higher Order Factors) 55.1 6 <.0001 All Interactions 13.8 4 0.0079 Nonlinear (Factor+Higher Order Factors) 21.9 3 0.0001 age (Factor+Higher Order Factors) 22.2 4 0.0002 All Interactions 16.7 3 0.0008 sex (Factor+Higher Order Factors) 208.7 4 <.0001 All Interactions 20.2 3 0.0002 fare * age (Factor+Higher Order Factors) 8.5 2 0.0142 Nonlinear 8.5 1 0.0036 Nonlinear Interaction : f(A,B) vs. AB 8.5 1 0.0036 fare * sex (Factor+Higher Order Factors) 6.4 2 0.0401 Nonlinear 1.5 1 0.2153 Nonlinear Interaction : f(A,B) vs. AB 1.5 1 0.2153 age * sex (Factor+Higher Order Factors) 9.9 1 0.0016 TOTAL NONLINEAR 21.9 3 0.0001 TOTAL INTERACTION 24.9 5 0.0001 TOTAL NONLINEAR + INTERACTION 38.3 6 <.0001 TOTAL 245.3 9 <.0001
Wald Statistics Response: survived
Factor Chi-Square d.f. P fare (Factor+Higher Order Factors) 55.1 6 <.0001 All Interactions 13.8 4 0.0079 Nonlinear (Factor+Higher Order Factors) 21.9 3 0.0001 age (Factor+Higher Order Factors) 22.2 4 0.0002 All Interactions 16.7 3 0.0008 sex (Factor+Higher Order Factors) 208.7 4 <.0001 All Interactions 20.2 3 0.0002 fare * age (Factor+Higher Order Factors) 8.5 2 0.0142 Nonlinear 8.5 1 0.0036 Nonlinear Interaction : f(A,B) vs. AB 8.5 1 0.0036 fare * sex (Factor+Higher Order Factors) 6.4 2 0.0401 Nonlinear 1.5 1 0.2153 Nonlinear Interaction : f(A,B) vs. AB 1.5 1 0.2153 age * sex (Factor+Higher Order Factors) 9.9 1 0.0016 TOTAL NONLINEAR 21.9 3 0.0001 TOTAL INTERACTION 24.9 5 0.0001 TOTAL NONLINEAR + INTERACTION 38.3 6 <.0001 TOTAL 245.3 9 <.0001
Wald Statistics Response: survived
Factor Chi-Square d.f. P fare (Factor+Higher Order Factors) 55.1 6 <.0001 All Interactions 13.8 4 0.0079 Nonlinear (Factor+Higher Order Factors) 21.9 3 0.0001 age (Factor+Higher Order Factors) 22.2 4 0.0002 All Interactions 16.7 3 0.0008 sex (Factor+Higher Order Factors) 208.7 4 <.0001 All Interactions 20.2 3 0.0002 fare * age (Factor+Higher Order Factors) 8.5 2 0.0142 Nonlinear 8.5 1 0.0036 Nonlinear Interaction : f(A,B) vs. AB 8.5 1 0.0036 fare * sex (Factor+Higher Order Factors) 6.4 2 0.0401 Nonlinear 1.5 1 0.2153 Nonlinear Interaction : f(A,B) vs. AB 1.5 1 0.2153 age * sex (Factor+Higher Order Factors) 9.9 1 0.0016 TOTAL NONLINEAR 21.9 3 0.0001 TOTAL INTERACTION 24.9 5 0.0001 TOTAL NONLINEAR + INTERACTION 38.3 6 <.0001 TOTAL 245.3 9 <.0001
Wald Statistics Response: survived
Factor Chi-Square d.f. P fare (Factor+Higher Order Factors) 55.1 6 <.0001 All Interactions 13.8 4 0.0079 Nonlinear (Factor+Higher Order Factors) 21.9 3 0.0001 age (Factor+Higher Order Factors) 22.2 4 0.0002 All Interactions 16.7 3 0.0008 sex (Factor+Higher Order Factors) 208.7 4 <.0001 All Interactions 20.2 3 0.0002 fare * age (Factor+Higher Order Factors) 8.5 2 0.0142 Nonlinear 8.5 1 0.0036 Nonlinear Interaction : f(A,B) vs. AB 8.5 1 0.0036 fare * sex (Factor+Higher Order Factors) 6.4 2 0.0401 Nonlinear 1.5 1 0.2153 Nonlinear Interaction : f(A,B) vs. AB 1.5 1 0.2153 age * sex (Factor+Higher Order Factors) 9.9 1 0.0016 TOTAL NONLINEAR 21.9 3 0.0001 TOTAL INTERACTION 24.9 5 0.0001 TOTAL NONLINEAR + INTERACTION 38.3 6 <.0001 TOTAL 245.3 9 <.0001
Wald Statistics Response: survived
Factor Chi-Square d.f. P fare (Factor+Higher Order Factors) 55.1 6 <.0001 All Interactions 13.8 4 0.0079 Nonlinear (Factor+Higher Order Factors) 21.9 3 0.0001 age (Factor+Higher Order Factors) 22.2 4 0.0002 All Interactions 16.7 3 0.0008 sex (Factor+Higher Order Factors) 208.7 4 <.0001 All Interactions 20.2 3 0.0002 fare * age (Factor+Higher Order Factors) 8.5 2 0.0142 Nonlinear 8.5 1 0.0036 Nonlinear Interaction : f(A,B) vs. AB 8.5 1 0.0036 fare * sex (Factor+Higher Order Factors) 6.4 2 0.0401 Nonlinear 1.5 1 0.2153 Nonlinear Interaction : f(A,B) vs. AB 1.5 1 0.2153 age * sex (Factor+Higher Order Factors) 9.9 1 0.0016 TOTAL NONLINEAR 21.9 3 0.0001 TOTAL INTERACTION 24.9 5 0.0001 TOTAL NONLINEAR + INTERACTION 38.3 6 <.0001 TOTAL 245.3 9 <.0001
0.50 2.00 4.00 6.00 8.00 10.00 12.00
fare - 31:7.9
age - 39:21
0.95
sex - female:male
Adjusted to:fare=14 age=28 sex=male
Predictors of Survival on Titanic
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Fare10
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Fare and Age Interaction
Fare
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of
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rviv
al
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male
Adjusted to: age=28
Fare and Gender Interaction
Index Training Corrected
Dxy 0.6565 0.646
R2 0.4273 0.407
Intercept 0.0000 -0.011
Slope 1.0000 0.952
Bootstrap Validation
Summary
• Think about your model• Collect enough data
Summary
• Measure well• Don’t destroy what you’ve
measured
• Pick your variables ahead of time and collect enough data to test the model you want
• Keep all your variables in the model unless extremely unimportant
Summary
• Use more df on important variables, fewer df on “nuisance” variables
• Don’t peek at Y to combine, discard, or transform variables
Summary
• Estimate validity and shrinkage with bootstrap
Summary
• By all means, tinker with the model later, but be aware of the costs of tinkering
• Don’t forget to say you tinkered
• Go collect more data
Summary
Web links for references, software, and more
• Harrell’s regression modeling text– http://hesweb1.med.virginia.edu/biostat/rms/
• SAS Macros for spline estimation– http://hesweb1.med.virginia.edu/biostat/SAS/survrisk.txt
• Some results comparing validation methods– http://hesweb1.med.virginia.edu/biostat/reports/logistic.val.pdf
• SAS code for bootstrap– ftp://ftp.sas.com/pub/neural/jackboot.sas
• S-Plus home page– insightful.com
• Mike Babyak’s e-mail – [email protected]
• This presentation– http://www.duke.edu/~mbabyak
• www.duke.edu/~mababyak
• michael.babyak @ duke.edu
• symptomresearch.nih.gov/chapter_8/