Adaptive Fractional Polynomial Modeling in SAS

Transcription

1 Adaptive Fractional Polynomial Modeling in SAS George J. Knafl, PhD Professor University of North Carolina at Chapel Hill School of Nursing

2 Overview of Topics properties of the genreg macro for adaptive fractional polynomial modeling adaptive in the sense of adapting models to the data adaptive regression modeling of means (expected values) adaptive moderation adding interactions to models adaptive variance modeling adaptive outcome (dependent, response) transformation using the ypower macro 2

3 Properties of the Genreg Macro about 90,000 lines of code including comments interface of about 200 macro parameters on-line documentation starts by checking for errors in macro parameters produces formatted output documenting macro parameter settings and analysis 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 variables adaptive linear regression of continuous outcomes assuming normality adaptive logistic regression of dichotomous outcomes adaptive ordinal/multinomial regression of polytomous outcomes adaptive Poisson regression of count outcomes, possibly with offset variables multivariate/repeated outcome variables maximum likelihood estimation for continuous outcomes for continuous, dichotomous, polytomous, count outcomes transition (Markov) modeling of outcome measurements as functions of prior outcome measurements generalized estimation equations (GEE) parameter estimation 4

5 Support for Regression Modeling modeling of means continuous outcomes with identity link function dichotomous outcomes with logit link function polytomous outcomes with cumulative or generalized logit link functions count outcomes with log link function modeling of variances/dispersions all four outcome cases with log link function using fractional polynomial models based on one or more possibly non-integer power transforms 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=5 for reported analyses get similar results for other values of k 6

7 Model Selection Process model selection occurs in 2-3 phases expansion starting from base model, add power transforms x p of predictors x contraction remove any extraneous transforms from the expanded model remaining powers adjusted after each removal conditional transformation but only if the contraction leaves the expanded model unchanged search process controlled by tolerance parameters how much of a decrease in LCV scores can be tolerated continue as long as decrease in LCV score not too large produces parsimonious model with nearly optimal LCV score usually with all coefficients significant 7

8 Setup and Data load in macro assuming it is in the folder c:\macros version indicated by date in file name %include "c:\macros\genreg sas"; assume data set called MERCURY has been loaded into the default library with the following variables MERCURY with mercury levels for fish in parts per million (ppm) WEIGHT with weights for fish in kg RIVER the indicator for fish being caught in the Waccamaw River rather than the Lumber River in North Carolina for 169 largemouth bass collected by the NC Department of Environment and Natural Resources 8

9 Standard Regression Model fit a straight line model for means in WEIGHT to MERCURY with constant variances %genreg(modtype=norml,datain=mercury, yvar=mercury,xvars=weight,foldcnt=5); by default get an intercept (xintrcpt=y) power 1 transforms (xpowers=) constant variances (vvars=,vintrcpt=y) to get standard degree 2 polynomial in WEIGHT: %genreg(,xvars=weight WEIGHT, xpowers=1 2, ); 9

10 Example Output - Straight Line Model Analysis of Mercury Level Data 3 base model model correlation structure: IND GEE: N # of matched sets: 169 maximum # of distinct values within matched sets: 1 m, the number of measurements: 169 base expectation component predictor power estimate XINTRCPT weight base log variance component predictor power estimate VINTRCPT iterations to compute model: 5 condition # for model hessian matrix: fixed correlation: 0 MLE of outcome variance: MLE of outcome standard deviation: log likelihood: log likelihood: average log likelihood: mth root of the likelihood: maximum condition # for cross-validated hessian matrices: 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:

11 Adaptive Regression Model fit a nonlinear model in WEIGHT to MERCURY %genreg(modtype=norml,datain=mercury, yvar=mercury,foldcnt=5, expand=y,expxvars=weight,contract=y); expansion considers multiple transforms of each of the expxvars variables (can be more than 1) get two transforms WEIGHT 0.2 and WEIGHT 10 contraction adjusts remaining transforms with each removal of a transform or the intercept final model has WEIGHT 0.47 without an intercept 11

12 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 2 -based LCV ratio test (analogous to a likelihood ratio test) to decide when to stop based on tolerable percent decrease (PD) in LCV score changing with the sample size in this case, it is 1.13% (reported in the genreg output) a PD>1.13% is substantial, otherwise insubstantial LCV for nonlinear model is for straight-line model with PD of 2.86% mean mercury level distinctly nonlinear in weight 12

13 Adaptive Moderation is there an interaction effect? possible difference for fish caught in the two rivers %genreg(,expand=y,geomcmbm=y, expxvars=weight RIVER, ); do not need to create the interaction variable "geomcmbn=y" means have genreg adaptively generate geometric combinations, i.e., products of power transforms of expxvars variables generalizing interactions model contains WEIGHT 0.36, representing the common dependence on WEIGHT for fish caught in both rivers RIVER WEIGHT 1.4, how the dependence on WEIGHT changes for fish caught in the Waccamaw River without an intercept 13

14 Adaptive Moderation even though selected model contains an interaction term, there might not be substantial moderation need to compare to the adaptive additive model in WEIGHT and RIVER generated by changing to "geomcmbn=n", the default based on WEIGHT 0.52 and RIVER without an intercept LCV for moderation model is LCV for additive model is with substantial PD 3.94% mean mercury level changes with weight distinctly differently for fish caught in the two rivers 14

15 Adaptive Variance Modeling to also model the variance, run %genreg(,expxvars=weight RIVER, expvvars=weight RIVER,geomcmbn=Y, ); model contains for means (X): WEIGHT 0.26, (RIVER WEIGHT 1.5 ) 0.8, and (RIVER WEIGHT 1.3 ) 0.7 without an intercept for variances (V): WEIGHT 0.53 without an intercept LCV is , variances distinctly non-constant since PD for constant variance model is 7.28% the standard assumption of constant variances can be distinctly incorrect also model for means is distinctly changed (see paper) 15

16 mercury level in ppm mecury level in ppm Estimated Means weight in kg Estimated Standard Deviations weight in kg Estimated Model Waccamaw River Lumber River mean mercury levels increase with weight of the fish starting at about the same place for low weights but to a distinctly higher level for fish caught in the Waccamaw River SDs increase nonlinearly with weight of the fish in the same way for fish caught in the two rivers 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 variables can be set with macro parameters including estimated means/variances can be exported to a graphics tool and used to generate plots of the results 16

17 Normal (Probability) Plot there is a distinct amount of skew in untransformed mercury levels after modeling means & variances and an extreme outlier (standardized residual = 4.19) two possible alternatives for resolving these problems transform the mercury levels as well as the weights dichotomize mercury levels into high and low values natural cutoff of 1 ppm, the FDA limit for human consumption conduct an adaptive logistic regression both covered in the paper will only address outcome transformation in the presentation standardized residual normal score 17

18 Outcome Transformation assuming the ypower macro also loaded generate adaptive models in WEIGHT for MERCURY transformed by powers starting at 2.5 (yfst) by steps of size 0.5 (ystp) for 11 powers (ycnt) so stop after power 2.5 (0 power is the natural log) %ypower(datain=mercury,yvar=mercury, foldcnt=5,yfst=-2.5,ycnt=11,ystp=0.5, expand=y,expxvars=weight,contract=y); ypower transforms MERCURY for each requested power, invokes genreg to generate the adaptive model in WEIGHT for that transform, and computes power-adjusted LCV scores that can be used to adaptively determine an appropriate power 18

19 Outcome Transformation best LCV score with steps of 0.5 at power p=0 so then search around 0 by steps of size 0.1 best power-adjusted LCV score is at p=0.2 with substantial improvement over the score for p=1 using this power and the model in WEIGHT, RIVER, and geometric combinations for means and variances variances are now reasonably treated as constant standardized residuals are no longer skewed with no 3 outliers 2 adaptive outcome 1 0 transformation can -1 resolve model assumption -2-3 problems standardized residual normal score 19

20 Summary have demonstrated the genreg macro for adaptive regression modeling based on fractional polynomials including adaptive modeling of means and variances and adaptive moderation and the ypower macro for adaptive outcome transformation along with predictor transformation genreg supports modeling in a variety of other cases these are described in a book in press Adaptive Regression for Modeling Nonlinear Relationship to be published by Springer Verlag see 20