This script illustrates the ability of the GAMPL procedure to fit non-parametric Tweedie regression models. It starts by simulating a dataset (compound Poisson distribution) with nonlinear relationships between predictors and the response. It then compares the results of a standard generalized linear model (PROC GENMOD) with two GAMPL models: one using parametric linear terms and the other using smoothing splines.
Data Analysis
Type : INTERNAL_CREATION
Data is artificially generated in the DATA step 'one'. Random variables (x1-x4) are transformed via nonlinear functions (sine, exponential, polynomials) to create a mean 'mu', which is then used to simulate a response 'y' following a Tweedie distribution.
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DATA STEP Data
Explanation : Synthetic data generation. The macro variables 'phi' and 'power' control the Tweedie distribution. The code simulates 1000 observations with random predictors and a response variable 'y' constructed from complex nonlinear transformations and a compound Poisson process.
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title 'Nonparametric Tweedie Model';
%let phi=0.4;
%let power=1.5;
data one;
do i=1 to 1000;
/* Sample the predictors */
x1=ranuni(1);
x2=ranuni(1);
x3=ranuni(1);
x4=ranuni(1);
/* Apply nonlinear transformations to predictors */
f1=2*sin(3.14159265*x1);
f2=exp(2*x2)*0.8;
f3=0.2*x3**11*(10*(1-x3))**6+10*(10*x3)**3*(1-x3)**10;
xb=f1+f2+f3;
xb=xb/20;
mu=exp(xb);
/* Compute parameters of compound Poisson distribution */
lambda=mu**(2-&power)/(&phi*(2-&power));
alpha=(2-&power)/(&power-1);
gamma=&phi*(&power-1)*(mu**(&power-1));
/* Simulate the response */
rpoi=ranpoi(1,lambda);
if rpoi=0 then y=0;
else do;
y=0;
do j=1 to rpoi;
y=y+rangam(1,alpha);
end;
y=y*gamma;
end;
output;
end;
run;
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title 'Nonparametric Tweedie Model';
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%let phi=0.4;
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%let power=1.5;
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DATA one;
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DO i=1 to 1000;
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/* Sample the predictors */
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x1=ranuni(1);
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x2=ranuni(1);
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x3=ranuni(1);
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x4=ranuni(1);
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/* Apply nonlinear transformations to predictors */
/* Compute parameters of compound Poisson distribution */
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lambda=mu**(2-&power)/(&phi*(2-&power));
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alpha=(2-&power)/(&power-1);
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gamma=&phi*(&power-1)*(mu**(&power-1));
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/* Simulate the response */
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rpoi=ranpoi(1,lambda);
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IF rpoi=0 THEN y=0;
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ELSEDO;
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y=0;
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DO j=1 to rpoi;
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y=y+rangam(1,alpha);
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END;
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y=y*gamma;
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END;
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OUTPUT;
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END;
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RUN;
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PROC GENMOD
Explanation : Fitting a reference generalized linear model (GLM) using the Tweedie distribution. This model assumes a linear relationship between predictors and the response's link function, which may be insufficient given the nonlinear nature of the generated data.
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proc genmod data=one;
model y=x1 x2 x3 x4/dist=tweedie;
run;
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PROC GENMOD
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DATA=one;
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model y=x1 x2 x3 x4/dist=tweedie;
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RUN;
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PROC GAMPL
Explanation : Using PROC GAMPL to fit a model similar to GLM (parametric linear terms only) with the Tweedie distribution. This allows comparing the basic performance of GAMPL with GENMOD.
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proc gampl data=one seed=1234;
model y=param(x1 x2 x3 x4)/dist=tweedie;
run;
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PROC GAMPL
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DATA=one seed=1234;
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model y=param(x1 x2 x3 x4)/dist=tweedie;
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RUN;
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PROC GAMPL
Explanation : Fitting a complete generalized additive model (GAM). The 'spline()' terms allow modeling nonlinear relationships for each predictor. The 'plots' option generates graphs to visualize the splines fitted against the data.
model y=spline(x1) spline(x2) spline(x3) spline(x4)/dist=tweedie;
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RUN;
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