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Building useful models: Some new developments and easily avoidable errors

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Building useful models: Some new developments and easily avoidable errors

Michael Babyak, PhD

What is a model ?

Y = f(x1, x2, x3…xn)

Y = a + b1x1 + b2x2…bnxn

Y = e a + b1x1 + b2x2…bnxn

- A useful model is
- Not very biased
- Interpretable
- Replicable (predicts in a new sample)

- “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

- Theory
- Simulation

Concept of Simulation

Y = b X + error

bs1

bs2

bsk-1

bsk

bs3

bs4

………………….

Concept of Simulation

Y = b X + error

bs1

bs2

bsk-1

bsk

bs3

bs4

………………….

Evaluate

Simulation Example

Y = .4 X + error

bs1

bs2

bsk-1

bsk

bs3

bs4

………………….

Simulation Example

Y = .4 X + error

bs1

bs2

bsk-1

bsk

bs3

bs4

………………….

Evaluate

True Model:Y = .4*x1 + e

Correct probability model

Based on theory

Good measures/no loss of information

Useful Model

Comprehensive

Parsimonious

Tested fairly

Flexible

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

Normal

Binary/Binomial

Count, heavy skew,

Lots of zeros

Poisson, ZIP,

negbin, gamma

General Linear Model/

Linear Regression

Logistic Regression

ANOVA/t-test

ANCOVA

Chi-square

Regression w/

Transformed DV

Can be applied to clustered (e.g, repeated measures data)

Structural Equation Models

Partial Least Squares

Latent Variable

Models

(Confirmatory Factor Analysis)

Multiple

regression

Common Factor

Analysis

Principal

Components

- 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

- Adequate range
- Representative values
- Watch for ceiling/floor effects

- Preserving cases in data sets with missing data
- Conventional approaches:
- Use only complete case
- Fill in with mean or median
- Use a missing data indicator in the model

- Conventional approaches:

- Imputation or related approaches are almost ALWAYS better than deleting incomplete cases
- Multiple Imputation
- Full Information Maximum Likelihood

- 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

- Convoluted Reasoning and Anti-intellectual Pomposity
- Streiner & Norman: Biostatistics: The Bare Essentials

Implausible measurement

assumption

“not depressed”

“depressed”

A

B

C

Depression score

Loss of power

http://psych.colorado.edu/~mcclella/MedianSplit/

Sometimes through sampling error

You can get a ‘lucky cut.’

http://www.bolderstats.com/jmsl/doc/medianSplit.html

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, PhD

Prof. Richard Obediah Tomi, PhD

Power to detect non-zero b-weight when x is continuous versus dichotomized

True model: y =.4x + e

Low

High

CESD Score

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.

- 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

- 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.

- 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

- 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)

- 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-squares from a population model of completelyrandom variables

Events per predictor ratio

Simulation: number of events/predictor ratio

Y = .5*x1 + 0*x2 + .2*x3 + 0*x4

-- Where r x1 x4 = .4

-- N/p = 3, 5, 10, 20, 50

Parameter stability and n/p ratio

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

Simulation results: number of events/predictor ratio

- “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”

- “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.”

Simulation results: Number of noise variables included

Sample Size

20 candidate predictors; 100 samples

Simulation results: R-square from noise variables

Sample Size

20 candidate predictors; 100 samples

Simulation results: R-square from noise variables

Sample Size

20 candidate predictors; 100 samples

SOME of the problems with

stepwise variable selection.

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

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

Conventional Univariate Pre-selection

- 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

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

- 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

- 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

Bootstrap

My Sample

?1

?2

?3

?k-1

?k

?4

………………….

WITH REPLACEMENT

Evaluate

1, 3, 4, 5, 7, 10

7

1

1

4

5

10

10

3

2

2

2

1

3

5

1

4

2

7

2

1

1

7

2

7

4

4

1

4

2

10

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

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

Linear Spline

(piecewise regression)

Y = a + b1(x<10) + b2(10<x<20) + b3 (x >20)

Cubic Spline

(non-linear piecewise regression)

knots

Logistic regression model

fitfare<-lrm(survived~(rcs(fare,3)+age+sex)^2,x=T,y=T)

anova(fitfare)

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

Bootstrap Validation

Summary

- Think about your model
- Collect enough data

Summary

- Measure well
- Don’t destroy what you’ve measured

Summary

- 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

- 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
- michael.babyak@duke.edu

- This presentation
- http://www.duke.edu/~mbabyak

- www.duke.edu/~mababyak
- michael.babyak @ duke.edu
- symptomresearch.nih.gov/chapter_8/