What You See May Not Be What You Get: A Primer on Regression Artifacts

What You See May Not Be What You Get:

A Primer on Regression Artifacts

Michael A. Babyak, PhD

Duke University Medical Center

Topics to Cover

1. Models: what and why?2. Preliminaries—requirements for a good model3. Dichotomizing a graded or continuous variable

is dumb4. Using degrees of freedom wisely5. Covariate selection6. Transformations and smoothing techniques for

non-linear effects7. Resampling as a superior method of model

validation

What is a model ?

Y = f(x1, x2, x3…xn)

Y = a + b1x1 + b2x2…bnxn

Y = e a + b1x1 + b2x2…bnxn

Why Model? (instead of test)

1. Can capture theoretical/predictive system

2. Estimates of population parameters

3. Allows prediction as well as hypothesis testing

4. More information for replication

Preliminaries

1. Correct model2. Measure well and don’t throw

information away3. Adequate Sample Size

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

• Distribution of predictors generally not important

Measure well and don’t throw information away

• Reliable, interpretable• 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 Can Destroy a Model

0 4 8 12 16 20 24 28 32 36 40 44

Depression score

AB C

Implausible measurement assumption

“not depressed” “depressed”

Dichotomization, by definition, reduces power by a minimum of about

30%

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

Dichotomization, by definition, reduces power 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

Examples from the WCGS Study:Correlations with CHD Mortality (n = 750)

Continuous Dichotomizedat median

Reductionin r2

Variable r r2 r r2

SystolicBloodPressure

.15 .023 .12 .014 -39%

Hostility .15 .023 .08 .006 -74%

Dichotomizing does not reduce measurement error

Gustafson, P. and Le, N.D. (2001). A comparison of continuous and discrete measurement error: is it wise

to dichotomize imprecise covariates? Submitted. Available at http://www.stat.ubc.ca/people/gustaf.

Simulation: Dichotomizing makes matters worse when

measure is unreliable

X1 = .4

Y

True Model: X1 continuous



X1 = .4

Y

Same Model with X1 dichotomized



X1 = .4

Y

= .4YX1

Contin.

Dich.

Reliability=.65, .75., .85, 1.00

Models with reliability of X1 manipulated

Dichotomization of a variable measured with error (y = .4x + e)

50

60

70

80

90

100

1.00 0.85 0.75 0.65Reliability of x

% c

orr

ec

t re

jec

tio

ns

of

nu

ll h

yp

oth

es

is

Continuous x

Dichotomization of a variable measured with error (y = .4x + e)

50

60

70

80

90

100

1.00 0.85 0.75 0.65Reliability of x

% c

orr

ec

t re

jec

tio

ns

of

nu

ll h

yp

oth

es

is

Continuous xDichotomized x

Dichotomizing will obscure non-linearity

Dichotomized at Median (CES-D = 7)

Perc

ent w

ith W

all

Motio

n A

bnorm

alit

y

0

6

12

18

24

30

Not Depressed Depressed

WMA on at Least 1 TaskUsing Cubic Spline

CES-D Score

Pro

babi

lity

of W

MA

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30 35 40

Dichotomizing will obscure non-linearity

Simulation 2: Dichotomizing a continuous predictor that is

correlated with another predictor

X1 = .4

= .0X2

Y

X1 and X2 continuous

X1 = .4

= .0X2

Y

X1 dichotomized



X1 = .4

= .0

=

.0, .4, .7

X2

Y

X1 dichotomized; rho12 manipulated



00.5

11.5

22.5

33.5

44.5

0 0.4 0.7

Correlation between x1, x2

(%)

Incorr

ect

reje

cti

ons

of X2 =

0

X1 and X2 continuous



0

5

10

15

20

25

30

0 0.4 0.7

Correlation between x1, x2

(%)

Inco

rrec

t re

ject

ion

s

of

X2

= 0

Both continuous x1 dichotomous, x2 continuous



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, cut later.

Pro

b{

even

t}

Maximum Change in LVEF (%)

Clinical Events and LVEF Change during Mental Stress: 5 Year follow-upModel first, cut later

Requirements: Sample Size

• 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)

Y = b X + error

bs

1

bs

2

bs

3

bs

4

bsk-1 bsk………………….

Concept of Simulation

bs

1

bs

2

bs

3

bs

4

bsk-1 bsk………………….

Y = b X + error

Evaluate

Concept of Simulation

Y = .4 X + error

bs

1

bs

2

bs

3

bs

4

bsk-1 bsk………………….

Simulation Example

bs

1

bs

2

bs

3

bs

4

bsk-1 bsk………………….

Evaluate

Y = .4 X + error

Simulation Example

0.2 0.4 0.6

05

00

10

00

15

00

20

00

25

00

Value of beta for x1

Fre

qu

en

cy o

f b

eta

va

lue

True Model:Y = .4*x1 + e

Sample Size• 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)

All-noise, but good fit

R-Square from Full Model

De

nsi

ty

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

02

46

81

01

21

41

6

n/p~3n/p~6.6n/p=10n/p~13.3

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

01

23

45

67

8

n/p=3n/p=5n/p=10n/p=20n/p=50

x2

-2.0 -1.0 0.0 0.5 1.0 1.5 2.0

01

23

45

67

8

x3

Parameter Estimate

Den

sity

-2.0 -1.0 0.0 0.5 1.0 1.5 2.0

01

23

45

67

8

x4

Parameter Estimate

-2.0 -1.0 0.0 0.5 1.0 1.5 2.0

01

23

45

67

8

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

-20

-10

0

10

20

30

40

50

0 2 5 10 15 20 25

Events per variable

% R

elat

ive

Bia

s NYHACHFVESDMSTDHTNLVC

Simulation results: number of events/predictor ratio

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 5 10 15 20 25

Events per variable

Pro

port

ion w

/ B

ias

>

100%

NYHACHFVESDMSTDHTNLVC

Simulation results: number of events/predictor ratio

Predictor (covariate) selection

1. Theory, substantive knowledge, prior models

2. Testing for confounding

3. Univariate testing

4. Last (and least), automated methods, aka stepwise and best subset regression

Searching for Confounders

• Fundamental tension between underfitting and overfitting•Underfitting = not adjusting for

important confounders•Overfitting = capitalizing on

chance relations (sample fluctuation)

Covariate selection

• Overfitting has been studied extensively

• “Scariest” study is by 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.

Covariate selection

• Therefore, if you transform, select, etc., you must include the DF in (i.e., penalize for) the “Final Model”

Covariate selection: Univariate Testing

• Non-Significant tests also cost a DF• 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

Covariate selection

• Despite the convention, testing for confounding has not been systematically studied—likely 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 Type I error rate, sometimes dramatically

1. It yields R-squared values that are badly biased high

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


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)



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




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)





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






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






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)







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







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


“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.”

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7

Variables in Final Model

% o

f sa

mple

s

100200500100010000

Simulation results: Number of Noise Variables Included

20 candidate predictors; 100 samples

Sample Size

0102030405060708090

100

0 0-5 5-10 10-15 15-20 20-25 > 25

% Variance Explained

% o

f sa

mple

s

100200500100010000

Simulation results: R-Square From Noise Variables


Sample Size

0

0.05

0.1

0.15

0.2

0.25

0.3

Samples (Deciles)

R-S

quare

10,0001,000500200100

Simulation results: R-Square From Noise Variables


Sample Size

Variable Selection

• Pick variables a priori• Stick with them• Penalize appropriately for any

data-driven decision about how to model a variable

Spending DF wisely

• Select variables of most importance• Use DF to assess non-linearity using

flexible curve approach (more about this later)

• 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, propensity scores.

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

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

0.0 0.05 0.10 0.15 0.20 0.25

1046 1

1308 1

1309 1

N df

age

fare

sex

Spearman Test

Non-linearity using splines

0

0.5

1

1.5

2

2.5

0 0 5 10 15 20 25

X

YLinear Spline

(piecewise regression)

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

0

0.5

1

1.5

2

2.5

0 0

X

Y

Cubic Spline (non-linear piecewise

regression)

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

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

0

50

100150

200250

Fare10

20

30

40

50

60

age

00.

20.

40.

60.

81

Pro

b. o

f Sur

viva

l

Adjusted to: sex=male

Fare and Age Interaction

Fare

Pro

b.

of

Su

rviv

al

0 50 100 150 200 250 300

0.2

0.4

0.6

0.8

1.0

female

male

Adjusted to: age=28

Fare and Gender Interaction

Validation• Apparent

• 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

?1………………….

My Sample

Evaluate

Bootstrap

?2 ?3 ?4 ?k-1 ?k

WITH REPLACEMENT

1, 3, 4, 5, 7, 10

7114510

1032221

351427

211727

4414210

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

What You See May Not Be What You Get: A Primer on Regression Artifacts

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