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Modeling with Observational Data

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Modeling with Observational Data Michael Babyak, PhD All models are wrong, some are useful -- George Box A useful model is Not very biased Interpretable ... – PowerPoint PPT presentation

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Title: Modeling with Observational Data


1
Modeling with Observational Data
  • Michael Babyak, PhD

2
What is a model ?
Y f(x1, x2, x3xn)
Y a b1x1 b2x2bnxn
Y e a b1x1 b2x2bnxn
3
All models are wrong, some are useful --
George Box
  • A useful model is
  • Not very biased
  • Interpretable
  • Replicable (predicts in a new sample)

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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.
  • Theres no substitute for thinking about the
    problem

6
Observational Studies
  • Underserved reputation
  • Especially if conducted and analyzed wisely
  • Biggest threats
  • Third Variable
  • Selection Bias (see above)
  • Poor Planning

7
Correlation between results of randomized trials
and observational studies http//www.epidemiologic
.org/2006/11/agreement-of-observational-and.html
8
Mean of Estimates
9
Head-to-head comparisons
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Statistics is a cumulative, evolving field How
do we know this stuff?
  • Theory
  • Simulation

12
Concept of Simulation
Y b X error
bs1
bs2
bsk-1
bsk
bs3
bs4
.
13
Concept of Simulation
Y b X error
bs1
bs2
bsk-1
bsk
bs3
bs4
.
Evaluate
14
Simulation Example
Y .4 X error
bs1
bs2
bsk-1
bsk
bs3
bs4
.
15
Simulation Example
Y .4 X error
bs1
bs2
bsk-1
bsk
bs3
bs4
.
Evaluate
16
True Model Y .4x1 e
17
Ingredients of a Useful Model
Correct probability model
Based on theory
Good measures/no loss of information
Useful Model
Comprehensive
Parsimonious
Tested fairly
Flexible
18
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

19
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)
20
Factor Analytic Family
Structural Equation Models
Partial Least Squares
Latent Variable Models (Confirmatory Factor
Analysis)
Multiple regression
Common Factor Analysis
Principal Components
21
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

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Measure well
  • Adequate range
  • Representative values
  • Watch for ceiling/floor effects

29
Using all the information
  • 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

30
Missing Data
  • Imputation or related approaches are almost
    ALWAYS better than deleting incomplete cases
  • Multiple Imputation
  • Full Information Maximum Likelihood

31
Multiple Imputation
32
http//www.lshtm.ac.uk/msu/missingdata/mi_web/node
5.html
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Modern Missing Data Techniques
  • Preserve more information from original sample
  • Incorporate uncertainty about missingness into
    final estimates
  • Produce better estimates of population (true)
    values

35
Dont waste information from variables
  • Use all the information about the variables of
    interest
  • Dont create clinical cutpoints before modeling
  • Model with ALL the data first, then use
    prediction to make decisions about cutpoints

36
Dichotomizing for Convenience Dubious
Practice (C.R.A.P.)
  • Convoluted Reasoning and Anti-intellectual
    Pomposity
  • Streiner Norman Biostatistics The Bare
    Essentials

37
Implausible measurement assumption
not depressed
depressed
A
B
C
Depression score
38
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.ht
ml
39
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
40
Power to detect non-zero b-weight when x is
continuous versus dichotomized
True model y .4x e
41
Dichotomizing will obscure non-linearity
Low
High
CESD Score
42
Dichotomizing will obscure non-linearity Same
data as previous slide modeled continuously
43
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.
44
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

45
CONCLUSIONS
  • Cutting
  • Doesnt 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.

46
Statistical Adjustment/Control
  • What does it mean to adjust or control for
    another variable?

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Y
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Difficulties
  • What if lines arent parallel?
  • What if poor overlap between groups?

60
A Note on Mediation vs Confounding
  • Mathematically identical no test can tell you
    which is which
  • Depends on YOUR causal hypothesis
  • Criteria for either
  • All three variables, predictor,
    confounder/mediator, outcome must be related

61
Possible Models
Initial condition all related
A
C
B
62
Possible Models
Initial condition all related
A
C
C
B
B
A
63
Possible Models
Typical regression result
A
C
B
64
Possible Models
Mediational relation between A and C
A
C
B
65
Possible Models
Spurious relation between A and C
A
C
B
66
Possible Models
Or worse
A
C
U
B
67
  • With cross-sectional design, best you can do is
    say that observed relations are consistent/not
    consistent with hypothesized relation
  • Prospective better but still vulnerable to
    outside variables
  • Interpretation of mediator/confounding
    distinction is entirely substantive

68
Not always clear difference between mediator and
confounder
  • Beware that adjustment for confounder might
    actually be modeling an explanatory mechanism
  • E.g., relation between depression and mortality
  • Often adjust for medical comorbidity
  • Comorbidity however, might be a proxy for poor
    self-care, which in turn is linked to depression

69
Sample size and the problem of underfitting vs
overfitting
  • Model assumption is that ALL relevant variables
    be includedthe antiparsimony principle or As
    big as a house.
  • Tempered by fact that estimating too many
    unknowns with too little data will yield junk.
  • In other words, cant build a mansion with a
    shantys worth of wood.

70
Sample Size Requirements
  • Linear regression
  • minimum of N 50 8/predictor (Green, 1990)or
    maybe more? (Kelley Maxwell, 2003)
  • 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)

71
Consequences of inadequate sample size
  • Lack of power for individual tests
  • Unstable estimates
  • Spurious good fitlots of unstable estimates will
    produce spurious good-looking (big) regression
    coefficients

72
All-noise, but good fit
R-squares from multivariable models where
population is completely random numbers
Events per predictor ratio
73
Simulation number of events/predictor ratio
Y .5x1 0x2 .2x3 0x4 -- Where r x1 x4
.4 -- N/p 3, 5, 10, 20, 50
74
Parameter stability and n/p ratio
75
Peduzzis Simulation number of events/predictor
ratio
P(survival) a b1NYHA b2CHF b3VES b4DM
b5STD b6HTN b7LVC --Events/p 2, 5,
10, 15, 20, 25 -- relative bias
(estimated b true b/true b)100
76
Simulation results number of events/predictor
ratio
77
Simulation results number of events/predictor
ratio
78
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
  • Tibshiranis Lasso

79
Any variable selection technique based on looking
at the data first will likely be biased
80
  • 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

81
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

82
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.

83
Simulation results Number of noise variables
included
Sample Size
20 candidate predictors 100 samples
84
Simulation results R-square from noise variables
Sample Size
20 candidate predictors 100 samples
85
Simulation results R-square from noise variables
Sample Size
20 candidate predictors 100 samples
86
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
87
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.  
88
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

89
Phantom Degrees of Freedom
  • Therefore, if you transform, select, etc., you
    must include the DF in (i.e., penalize for) the
    Final Model

90
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 modelvariable not significant at
    univariate test may be very important in the
    presence of other variables

91
Conventional Univariate Pre-selection
  • Despite the convention, testing for confounding
    has not been systematically studiedin 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

92
Better approach
  • Pick variables a priori
  • Stick with them
  • Penalize appropriately for any data-driven
    decision about how to model a variable

93
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

94
What to do
  • Penalization/Random effects
  • Propensity Scoring
  • Matches individuals on multiple dimensions to
    improve baseline balance
  • Tibshiranis Lasso

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Propensity Score Example
  • Observational data on SSRI use in post myocardial
    infarction patients
  • Early use of SSRI as an adjustment covariate
    revealed excess risk for all-cause mortality
    among SSRI users
  • Can use Propensity Score to help rule out
    confounders

97
Step 1 Kitchen Sink Model predicting SSRI use
  • Why is it OK to use lots of predictors in this
    case?
  • Working strictly at the sample level

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Generate conditional probabilities of being on an
SSRI for each patient
ID probssri 1 0.07071829 2
0.10357308 3 0.08324767 4 0.09562251
5 0.10424651 6 0.28105882 7
0.09824793
100
Step 2 Remove non-overlapping cases
SSRI0
SSRI1
density
101
Perform primary analysis predicting survival
  • Surv ssri
  • Surv ssri logit(pssri)
  • Surv ssri logit(pssri) BDI
  • Surv ssri logit(pssri) BDI others

102
Step 3 Unadjusted estimate
Factor HR Lower 0.95 Upper
0.95 ssri 0.22 0.18 1.05
Hazard Ratio 1.85 1.20 2.86
103
Step 4 Adjusted for Propensity (linear)
Factor Effect S.E. Lower 0.95 Upper 0.95
ssri 0.61 0.24 0.15 1.08
Hazard Ratio 1.85 NA 1.16 2.95
LOGIT 0.00 0.14 -0.27 0.28
Hazard Ratio 1.00 NA 0.76 1.33
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Better Step 4 Adjusted for Propensity
(non-linear)
Factor Effect S.E. Lower 0.95 Upper 0.95
ssri 0.55 0.24 0.07 1.03
Hazard Ratio 1.73 NA 1.07 2.79
LOGIT 0.02 0.25 -0.47 0.51
Hazard Ratio 1.02 NA 0.62 1.67
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Limitations
  • Still may be differences/confounding not measured
    and therefore not captured by propensity score
  • If poor overlap, limited generalizability
  • Many reviewers not familiar with it

108
What to do about heterogeneous slopes?
  • We know there is always heterogeneity of slopes,
    perhaps even important
  • Proper test is product interaction termNOT
    within subgroups tests (see BMJ series)
  • Increased error rate
  • Differential power
  • Danger of Accepting the null
  • Sparse cells and unstable estimates
  • Tension between low power of interaction and high
    error rate/instability
  • Especially true in observational data
  • I honestly dont know what to doany ideas?

109
If you worry about Type I
  • Use pooled test (see, for example, Cohen Cohen
    or Harrell)
  • If pooled test not significant, stop there

110
If Type II is a bigger concern
  • Report non-significant effects, acknowledging the
    uncertainty, but conveying need to investigate
    more
  • C.F. HRT data was there an age X HRT
    interaction?

111
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

112
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

113
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

114
http//myspace.com/monkeynavigatedrobots
115
Advanced topics and examples
116
Bootstrap
My Sample
?1
?2
?3
?k-1
?k
?4
.
WITH REPLACEMENT
Evaluate
117
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
118
Can use data to determine where to spend DF
  • Use Spearmans 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

119
Example-Predict Survival from age, gender, and
fare on Titanic example using R software
120
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
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Non-linearity using splines
123
Linear Spline (piecewise regression)
Y a b1(xlt10) b2(10ltxlt20) b3 (x gt20)
124
Cubic Spline (non-linear piecewise regression)
knots
125
Logistic regression model
fitfarelt-lrm(survived(rcs(fare,3)agesex)2,xT,
yT) anova(fitfare)
Spline with 3 knots
126
Wald Statistics Response survived
Factor
Chi-Square d.f. P fare (FactorHigher
Order Factors) 55.1 6 lt.0001 All
Interactions 13.8 4
0.0079 Nonlinear (FactorHigher Order
Factors) 21.9 3 0.0001 age
(FactorHigher Order Factors) 22.2 4
0.0002 All Interactions
16.7 3 0.0008 sex (FactorHigher
Order Factors) 208.7 4 lt.0001
All Interactions 20.2
3 0.0002 fare age (FactorHigher 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 (FactorHigher 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 (FactorHigher 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 lt.0001 TOTAL
245.3 9 lt.0001
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Wald Statistics Response survived
Factor
Chi-Square d.f. P fare (FactorHigher
Order Factors) 55.1 6 lt.0001 All
Interactions 13.8 4
0.0079 Nonlinear (FactorHigher Order
Factors) 21.9 3 0.0001 age
(FactorHigher Order Factors) 22.2 4
0.0002 All Interactions
16.7 3 0.0008 sex (FactorHigher
Order Factors) 208.7 4 lt.0001
All Interactions 20.2
3 0.0002 fare age (FactorHigher 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 (FactorHigher 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 (FactorHigher 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 lt.0001 TOTAL
245.3 9 lt.0001
128
Wald Statistics Response survived
Factor
Chi-Square d.f. P fare (FactorHigher
Order Factors) 55.1 6 lt.0001 All
Interactions 13.8 4
0.0079 Nonlinear (FactorHigher Order
Factors) 21.9 3 0.0001 age
(FactorHigher Order Factors) 22.2 4
0.0002 All Interactions
16.7 3 0.0008 sex (FactorHigher
Order Factors) 208.7 4 lt.0001
All Interactions 20.2
3 0.0002 fare age (FactorHigher 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 (FactorHigher 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 (FactorHigher 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 lt.0001 TOTAL
245.3 9 lt.0001
129
Wald Statistics Response survived
Factor
Chi-Square d.f. P fare (FactorHigher
Order Factors) 55.1 6 lt.0001 All
Interactions 13.8 4
0.0079 Nonlinear (FactorHigher Order
Factors) 21.9 3 0.0001 age
(FactorHigher Order Factors) 22.2 4
0.0002 All Interactions
16.7 3 0.0008 sex (FactorHigher
Order Factors) 208.7 4 lt.0001
All Interactions 20.2
3 0.0002 fare age (FactorHigher 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 (FactorHigher 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 (FactorHigher 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 lt.0001 TOTAL
245.3 9 lt.0001
130
Wald Statistics Response survived
Factor
Chi-Square d.f. P fare (FactorHigher
Order Factors) 55.1 6 lt.0001 All
Interactions 13.8 4
0.0079 Nonlinear (FactorHigher Order
Factors) 21.9 3 0.0001 age
(FactorHigher Order Factors) 22.2 4
0.0002 All Interactions
16.7 3 0.0008 sex (FactorHigher
Order Factors) 208.7 4 lt.0001
All Interactions 20.2
3 0.0002 fare age (FactorHigher 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 (FactorHigher 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 (FactorHigher 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 lt.0001 TOTAL
245.3 9 lt.0001
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Bootstrap Validation
135
Summary
  • Think about your model
  • Collect enough data

136
Summary
  • Measure well
  • Dont destroy what youve measured

137
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

138
Summary
  • Use more df on important variables, fewer df on
    nuisance variables
  • Dont peek at Y to combine, discard, or transform
    variables

139
Summary
  • Estimate validity and shrinkage with bootstrap

140
Summary
  • By all means, tinker with the model later, but be
    aware of the costs of tinkering
  • Dont forget to say you tinkered
  • Go collect more data

141
Web links for references, software, and more
  • Harrells regression modeling text
  • http//hesweb1.med.virginia.edu/biostat/rms/
  • R software
  • http//cran.r-project.org/
  • SAS Macros for spline estimation
  • http//hesweb1.med.virginia.edu/biostat/SAS/survri
    sk.txt
  • Some results comparing validation methods
  • http//hesweb1.med.virginia.edu/biostat/reports/lo
    gistic.val.pdf
  • SAS code for bootstrap
  • ftp//ftp.sas.com/pub/neural/jackboot.sas
  • S-Plus home page
  • insightful.com
  • Mike Babyaks e-mail
  • michael.babyak_at_duke.edu
  • This presentation
  • http//www.duke.edu/mababyak

142
  • www.duke.edu/mababyak
  • michael.babyak _at_ duke.edu
  • symptomresearch.nih.gov/chapter_8/

143
Observational Data and Clinical
Trials http//www.epidemiologic.org/2006/11/agreem
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