A SAS Macro for Adaptive Regression Modeling

Transcription

1 A SAS Macro for Adaptive Regression Modeling George J. Knafl, PhD Professor University of North Carolina at Chapel Hill School of Nursing Supported in part by NIH Grants R01 AI57043 and R03 MH086132

2 Overview of Topics properties of the genreg macro adaptive regression modeling of expected values (means) adaptive interaction effects adaptive regression modeling of variances other analysis examples 2

3 Properties of the Macro over 20,000 lines of codes/comments interface of about 125 macro parameters has an extensive header comment describing the macro parameters starts by checking for errors in macro parameters produces formatted output documenting macro parameter settings and results written primarily in matrix language of PROC IML used so far only in Windows, but has no Windowsspecific code to our knowledge 3

4 Support for Regression Modeling univariate outcome (dependent, response) variables adaptive linear regression for continuous outcomes (e.g., normally distributed) adaptive logistic regression for dichotomous-valued or polytomous-valued outcomes adaptive Poisson regression for count outcomes, possibly with offset variables multivariate/repeated outcome variables continuous outcomes only, normally distributed with exchangeable (constant) correlations (EC) order 1 autoregressive correlations (AR1) will describe an analysis of this kind 4

5 Support for Regression Modeling modeling of expected values continuous outcomes with identity link function logistic outcomes with (generalized) logit link function count outcomes with log link function modeling of variances/dispersions all cases with log link function using fractional polynomial models based on one or more fractional power transform of one or more predictor variables 5

6 Model Evaluation k-fold likelihood cross-validation (LCV) randomly partition data into k disjoint sets called folds evaluate likelihood for a fold using parameter estimates computed from the remaining data outside of the fold multiply up these deleted likelihoods over folds and normalize by the total number of measurements so a geometric average deleted likelihood score larger LCV scores indicate better models, more compatible with the data k=10 for reported analyses similar results generated at alternative values of k 6

7 Model Selection Process model selection occurs in two phases expansion starting from base model, add power transforms x p of x contraction remove any extraneous transforms from expanded model remaining powers adjusted after each removal search process controlled by tolerance parameters how much of a decrease in LCV scores can be tolerated continue search as long as penalty in LCV not too high produces model with nearly optimal LCV score usually parsimonious with all coefficients significant 7

8 Setup assume the following options and title commands options linesize=76 pagesize=53 pageno=1 nodate; title "Analysis of Dental Measurements"; load in macro assuming it in folder c:\macros version indicated by date in file name %include "c:\macros\genreg sas"; setup data library assuming data in c:projects\dental libname datalib "c:\projects\dental"; 8

9 Example Data classic growth curve data of Porthoff & Roy (1964) how do dental measurements in mm for 16 boys and 11 girls change as they age from 8 to 14 years old? data DENTDATA; set datalib.dentdata; label SUBJECT="Subject ID" DENTMEAS="Dental Measurement (mm)" AGE="Age (8, 10, 12, or 14 years)" SEX="Boy (1) or Girl (0)"; run; 9

10 Standard Regression Model fit a straight-line model in AGE to DENTMEAS %genreg(modtype=norml,datain=dentdata, yvar=dentmeas,xvars=age, matchvar=subject,withinvar=age, covtype=ec,foldcnt=10); get an intercept by default (xintrcpt=y) get power 1 transforms by default (xpowers=) to request a standard degree 2 polynomial in AGE: %genreg(,xvars=age AGE,xpowers=1 2, ); 10

11 Example Output - Straight Line Model Analysis of Dental Measurements 2 base model model correlation structure: EC # of matched sets: 27 maximum # of distinct values within matched sets: 4 m, the number of measurements: 108 lower bound on correlation: -0.3 base expectation component predictor power estimate XINTRCPT AGE base log variance component predictor power estimate VINTRCPT iterations: 5 estimated correlation: MLE of response variance: MLE of response standard deviation: log likelihood: log likelihood: average log likelihood: mth root of the likelihood: average deleted square error: standard deleted prediction error: log likelihood using deleted predictions: log likelihood using deleted predictions: average log likelihood using deleted predictions: mth root of the likelihood using deleted predictions:

12 Adaptive Regression Model fit a nonlinear model in AGE to DENTMEAS %genreg(modtype=norml,datain=dentdata, yvar=dentmeas,matchvar=subject, withinvar=age,covtype=ec,foldcnt=10, expand=y,expxvars=age,contract=y); expansion considers multiple transforms of each of the expxvars variables (can be more than 1) get single transform AGE 2 contraction adjusts remaining transforms with each removal of a transform final model has AGE 0.31 without an intercept 12

13 LCV Ratio Tests contraction stopping tolerance is crucial if too small, extraneous terms left in model if too large, valuable terms removed from model genreg uses a LCV ratio test (analogous to likelihood ratio tests) to decide when to stop based on tolerable percent decrease (PD) in LCV score in this case, it is 1.76% (reported in contraction output) a PD>1.76% is substantial, otherwise insubstantial LCV is for EC, for AR1 substantial PD of 11.6% for straight-line model, PD of 1.34% 13

14 Adaptive Interaction Effects likely differences for boys compared to girls create interaction variable AGEBOYS (=AGE*SEX) in DENTDATA and run %genreg(,expxvars=age SEX AGEBOYS, ); model contains AGE 0.21, common dependence on AGE for girls and boys AGEBOYS 2, how dependence on AGE changes for boys LCV is , so interaction effect is substantial since PD for non-interaction model is 6.00% expected values change differently for boys and girls 14

15 Adaptive Variance Modeling to also model the variance, run %genreg(,expxvars=age SEX AGEBOYS, expvvars=age SEX AGEBOYS, ); model contains AGE 0.23 & AGEBOYS 2 for expected values (X) AGE 0.5 & AGEBOYS 0.8 for variances (V) LCV is , variances substantially nonconstant since PD for constant variance model is 5.56% the standard assumption of constant variances can be distinctly inferior 15

16 Selection of Both X and V 4 alternatives expand/contract X before/after V controlled by expord and contord macro parameters to expand X before V and contract X before V, use expord=xv,contord=xv default approach (expord=.,contord=.) adaptively selects expansion and contraction orderings in this case, adaptive approach produces same result as the alternative with the best LCV score of the 4 does not always hold, but so far the PD has not been substantial 16

17 Estimated Model dental measurements higher on the average for boys than for girls, and further apart as they age variability larger for boys than for girls, but decreases/increases for boys/girls as they age genreg creates a data set containing a copy of the datain data set along with a variety of output variables names of the data set and the variables can be set with associated macro parameters can be exported into a graphics tool and used to generate plots of the results see paper for residual analysis 17

18 HIV Viral Load Analysis log (base 10) viral loads for HIV+ subjects decreased nonlinearly on the average with improved adherence to antiretroviral medications with most of the effect at low adherence levels and with lower levels for subjects with lower baseline HIV severity high levels of adherence had a substantial effect on decreasing the variability in log viral loads, but only for subjects with low baseline HIV severity based on 638 measurements for 159 subjects at up to 7 time points 18

19 Change Point Analysis the standard of care for antiretroviral therapy changed in 1996 and so there should be an effect on CD4 cell counts for HIV+ subjects around that time this was supported, but starting in 1995 based on 13,855 measurements for 1,285 subjects of the Multicenter 19 AIDS Cohort Study

20 Summary have demonstrated the genreg macro for adaptive regression modeling need extensions to other covariance structures e.g., autoregressive moving average (ARMA) covariance need extension to adaptively search through fractional polynomials in random coefficients i.e., adaptive multilevel or hierarchical linear modeling need extensions to handle repeated measures in the logistic and Poisson regression settings i.e., for repeated categorical and count measurements 20