This example shows how you can use PROC GAMPL to build a nonparametric logistic regression model for a data set that contains a binary response and then use that model to classify observations.
The example uses the Pima Indian Diabetes data set, which can be obtained from the UCI Machine Learning Repository (Asuncion
and Newman 2007). It is extracted from a larger database that was originally owned by the National Institute of Diabetes and Digestive and
Kidney Diseases. Data are for female patients who are at least 21 years old, are of Pima Indian heritage, and live near Phoenix,
Arizona. The objective of this study is to investigate the relationship between a diabetes diagnosis and variables that represent
physiological measurements and medical attributes. Some missing or invalid observations are removed from the analysis. The
reduced data set contains 532 records. The following DATA step creates the data set Pima
:
title 'Pima Indian Diabetes Study'; data Pima; input NPreg Glucose Pressure Triceps BMI Pedigree Age Diabetes Test@@; datalines; 6 148 72 35 33.6 0.627 50 1 1 1 85 66 29 26.6 0.351 31 0 1 1 89 66 23 28.1 0.167 21 0 0 3 78 50 32 31 0.248 26 1 0 2 197 70 45 30.5 0.158 53 1 0 5 166 72 19 25.8 0.587 51 1 1 0 118 84 47 45.8 0.551 31 1 1 1 103 30 38 43.3 0.183 33 0 1 3 126 88 41 39.3 0.704 27 0 0 9 119 80 35 29 0.263 29 1 0 ... more lines ... 1 128 48 45 40.5 0.613 24 1 0 2 112 68 22 34.1 0.315 26 0 0 1 140 74 26 24.1 0.828 23 0 1 2 141 58 34 25.4 0.699 24 0 1 7 129 68 49 38.5 0.439 43 1 1 0 106 70 37 39.4 0.605 22 0 0 1 118 58 36 33.3 0.261 23 0 1 8 155 62 26 34 0.543 46 1 0 ;
The data set contains nine variables, including the binary response variable Diabetes
. Table 7.15 describes the variables.
Table 7.15: Variables in the Pima
Data Set
Variable 
Description 


Number of pregnancies 

Twohour plasma glucose concentration in an oral glucose tolerance test 

Diastolic blood pressure (mm Hg) 

Triceps skin fold thickness (mm) 

Body mass index (weight in kg/(height in m)) 

Diabetes pedigree function 

Age (years) 

0 if test negative for diabetes, 1 if test positive 

0 for training role, 1 for test 
The Test
variable splits the data set into training and test subsets. The training observations (whose Test
value is 0) hold approximately 59.4% of the data. To build a model that is based on the training data and evaluate its performance
by predicting the test data, you use the following statements to create a new variable, Result
, whose value is the same as that of the Diabetes
variable for a training observation and is missing for a test observation:
data Pima; set Pima; Result = Diabetes; if Test=1 then Result=.; run;
As a starting point of your analysis, you can build a parametric logistic regression model on the training data and predict the test data. The following statements use PROC HPLOGISTIC to perform the analysis:
proc hplogistic data=Pima; model Diabetes(event='1') = NPreg Glucose Pressure Triceps BMI Pedigree Age; partition role=Test(test='1' train='0'); run;
Output 7.2.1 shows the summary statistics from the parametric logistic regression model.
Output 7.2.1: Fit Statistics
Output 7.2.2 shows fit statistics for both training and test subsets of the data, including the misclassification error for the test data.
Output 7.2.2: Partition Fit Statistics
Partition Fit Statistics  

Statistic  Training  Testing 
Area under the ROCC  0.8607  0.8547 
Average Square Error  0.1452  0.1401 
HosmerLemeshow Test  0.9726  0.1487 
Misclassification Error  0.2061  0.2178 
RSquare  0.3438  0.2557 
Maxrescaled RSquare  0.4695  0.3706 
McFadden's RSquare  0.3197  0.2522 
Mean Difference  0.3780  0.3605 
Somers' D  0.7213  0.7093 
True Negative Fraction  0.8702  0.8435 
True Positive Fraction  0.6639  0.6182 
The parametric logistic regression model is restricted in the sense that all variables affect the response in strictly linear fashion. If you are uncertain that a variable is an important factor and its contribution is linear in predicting the response, you might want to choose a nonparametric logistic regression model to fit the data. You can use PROC GAMPL to form a nonparametric model by including the spline transformation of each explanatory variable, as shown in the following statements:
proc gampl data=Pima seed=12345; model Result(event='1') = spline(NPreg) spline(Glucose) spline(Pressure) spline(Triceps) spline(BMI) spline(Pedigree) spline(Age) / dist=binary; run;
Because the response variable Result
is binary, the DIST=BINARY option in the MODEL statement specifies a binary distribution for the response variable. By default,
PROC GAMPL models the probability of the first ordered response category, which is a negative diabetes testing result in this
case. The EVENT=
option specifically requests that PROC GAMPL model the probability of positive diabetes testing results. The “Response Profile”
table in Output 7.2.3 shows the frequencies of the response in both categories.
Output 7.2.3: Response Profile
Output 7.2.4 lists the summary statistics from the nonparametric logistic regression model, which include spline transformations of all variables.
Output 7.2.4: Fit Statistics
The “Tests for Smoothing Components” table in Output 7.2.5 shows approximate tests results. Although some spline terms are significant, others are not. The null testing hypothesis
is whether the total contribution from a variable is 0. So you can form a reduced model by removing those nonsignificant spline
terms from the model. In this case, spline transformations for NPreg, Pressure, BMI
, and Triceps
are dropped from the model because their pvalues are larger than the 0.1 nominal level.
Output 7.2.5: Tests for Smoothing Components
Tests for Smoothing Components  

Component  Effective DF 
Effective DF for Test 
ChiSquare  Pr > ChiSq 
Spline(NPreg)  1.00001  1  0.0758  0.7831 
Spline(Glucose)  1.00000  1  38.4662  <.0001 
Spline(Pressure)  7.80438  8  10.2605  0.2472 
Spline(Triceps)  1.00000  1  0.6343  0.4258 
Spline(BMI)  8.00000  8  11.8047  0.1601 
Spline(Pedigree)  1.00002  1  8.5323  0.0035 
Spline(Age)  5.05456  7  15.7671  0.0273 
The following statements use PROC GAMPL to fit a reduced nonparametric logistic regression model. The OUTPUT statement requests
predicted probabilities for both training and test data sets. The ID statement requests that the Diabetes
and Test
variables also be included in the output data set so that you can use them to identify test observations and compute misclassification
errors.
ods graphics on; proc gampl data=Pima plots seed=12345; model Result(event='1') = spline(Glucose) spline(Pedigree) spline(Age) / dist=binary; output out=PimaOut; id Diabetes Test; run;
Output 7.2.6 shows the summary statistics from the reduced nonparametric logistic regression model. The values of the information criteria are better than of the parametric logistic regression model.
Output 7.2.6: Fit Statistics
Pima Indian Diabetes Study 
Fit Statistics  

Penalized Log Likelihood  149.85765 
Roughness Penalty  2.85613 
Effective Degrees of Freedom  8.05242 
Effective Degrees of Freedom for Error  320.61181 
AIC (smaller is better)  312.96402 
AICC (smaller is better)  313.41826 
BIC (smaller is better)  343.55593 
UBRE (smaller is better)  0.00230 
In the “Estimates for Smoothing Components” table in Output 7.2.7, PROC GAMPL reports that the effective degrees of freedom value for spline transformations of Glucose
is quite close to 1. This suggests strictly linear form for Glucose
. For Pedigree
, the degrees of freedom value demonstrates a moderate amount of nonlinearity. For Age
, the degrees of freedom value is much larger than 1. The measure suggests a nonlinear pattern in the dependency of the response
on Age
.
Output 7.2.7: Estimates for Smoothing Components
The “Tests for Smoothing Components” table in Output 7.2.8 shows that all spline transformations are significant in predicting diabetes testing results.
Output 7.2.8: Tests for Smoothing Components
The smoothing component panel (which is produced by the PLOTS option and is shown in Output 7.2.9) visualizes the spline transformations for the four variables in addition to 95% Bayesian curvewise confidence bands. For
Glucose
, the spline transformation is almost a straight line. For Pedigree
, the spline transformation shows a slightly nonlinear trend. For Age
, the dependency is obviously nonlinear.
Output 7.2.9: Smoothing Components Panel
The following statements compute the misclassification error on the test data set from the reduced nonparametric logistic regression model that PROC GAMPL produces:
data test; set PimaOut(where=(Test=1)); if ((Pred>0.5 & Diabetes=1)  (Pred<0.5 & Diabetes=0)) then Error=0; else Error=1; run; proc freq data=test; tables Diabetes*Error/nocol norow; run;
Output 7.2.10 shows the misclassification errors for observations in the test set and observations of each response category. The error is smaller than the error from the parametric logistic regression model.
Output 7.2.10: Crosstabulation Table for Test Set Prediction