Stat Computing > SAS > FAQ
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Stat Computing > SAS > FAQ
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Proc transreg performs transformation regression in which both the outcome and predictor(s) can be transformed and splines can be fit. Splines are piecewise polynomials that can be used to estimate relationships that are difficult to fit with a single function.
In this page, we will walk through an example proc transreg with the bspline option and explore its defaults. The bspline, spline, and pspline options, when similarly specified, yield the same results. Their differences lie in the number and type of transformed variables generated for estimation. For more information on the other options available, see the SAS Online Documentation.
We can begin by creating a dataset with an outcome Y and a predictor X. This example data is generated in the SAS examples for proc transreg.
data a; x=-0.000001; do i=0 to 199; if mod(i,50)=0 then do; c=((x/2)-5)**2; if i=150 then c=c+5; y=c; end; x=x+0.1; y=y-sin(x-c); output; end; run; proc gplot data = a; plot y*x; run;
Clearly, there is not a single, continuous function relating Y to X. The relationship does not appear random, but it does appear to change with X. Thus it makes sense to try to fit this with splines. Before running the proc transreg, we can see that our data contains four variables:
proc print data = a (obs = 5); run; Obs X I C Y 1 0.10000 0 25.0000 24.7694 2 0.20000 1 25.0000 24.4427 3 0.30000 2 25.0000 24.0234 4 0.40000 3 25.0000 23.5155 5 0.50000 4 25.0000 22.9241
In the proc transreg command, we indicate in the model line that we wish to predict variable y without transformation with identity(y). If we wished to model a transformed version of y (the log or rank of y, for example), we would indicate the transformation here. To predict y, we indicate that we wish to expand x into a b-spline basis with bspline(x). We also opted to output a dataset, a2, containing predicted values from the model.
proc transreg data=a;
model identity(y) = bspline(x);
output out = a2 predicted;
run;
The TRANSREG Procedure
TRANSREG Univariate Algorithm Iteration History for Identity(Y)
Iteration Average Maximum Criterion
Number Change Change R-Square Change Note
-------------------------------------------------------------------------
1 0.00000 0.00000 0.46884 Converged
We can see in the outcome above that the model converged and has an R-squared value of 0.47. Let's look at the dataset output by proc transreg.
proc print data = a2 (obs = 5); run; Obs _TYPE_ _NAME_ Y TY PY Intercept X_0 X_1 X_2 1 SCORE ROW1 24.7694 24.7694 24.1144 . 1.00000 0.000000 7.5759E-27 2 SCORE ROW2 24.4427 24.4427 23.4722 . 0.98500 0.014924 .000075375 3 SCORE ROW3 24.0234 24.0234 22.8424 . 0.97015 0.029548 .000299977 4 SCORE ROW4 23.5155 23.5155 22.2249 . 0.95545 0.043873 .000671523 5 SCORE ROW5 22.9241 22.9241 21.6195 . 0.94090 0.057902 .001187727 Obs X_3 TIntercept TX_0 TX_1 TX_2 TX_3 X 1 1.269E-40 . 1.00000 0.000000 0 0 0.10000 2 .000000127 . 0.98500 0.014924 .000075375 .000000127 0.20000 3 .000001015 . 0.97015 0.029548 .000299977 .000001015 0.30000 4 .000003426 . 0.95545 0.043873 .000671523 .000003426 0.40000 5 .000008121 . 0.94090 0.057902 .001187727 .000008121 0.50000
In addition to adding the predicted values, py, to the dataset, we can see that a new variable, ty, has been added for the "transformed" value of y (since our transformation was the identity, these values are the same as y); our predictor variable x has been expanded into four variables (x_0, x_1, x_2, x_3) that form the basis. Within an observation, the sum of these four values is one. The number of variables in the basis is determined by the polynomial degree or number of knots indicated. SAS generates a basis of (#degrees + #knots + 1) variables and, by default, assumes degree = 3 and zero knots.
Transformations of these basis variables are also included and indicated with a 't'. We can plot the predicted values to see how closely they match the original data.
legend label=none value=('y' 'predicted y') position=(bottom left inside) mode=share down = 2; proc gplot data = a2; plot (y py)*x / overlay legend = legend; run;
Using the basis variables generated by proc transreg to predict y using an ordinary least squares regression model will result in the same R-Squared value as that shown in the proc transreg output. We can choose any three of the four variables for the model since we know they always sum to one.
proc reg data = a2; model y = x_1 x_2 x_3; run;
The REG Procedure
Model: MODEL1
Dependent Variable: Y
Number of Observations Read 200
Number of Observations Used 200
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 7955.26078 2651.75359 57.67 <.0001
Error 196 9012.65604 45.98294
Corrected Total 199 16968
Root MSE 6.78107 R-Square 0.4688
Dependent Mean 12.04335 Adj R-Sq 0.4607
Coeff Var 56.30551
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 24.11444 1.88257 12.81 <.0001
X_1 X 1 1 -43.01079 5.44815 -7.89 <.0001
X_2 X 2 1 -3.82721 3.47800 -1.10 0.2725
X_3 X 3 1 -1.67330 2.96550 -0.56 0.5732
It is important to note that these basis variables x_1, x_2, and x_3 are not powers of x as we would see when using the pspline option.
Note that the default settings for bspline, spline, and pspline yield identical fitted values.
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