The HPLOGISTIC Procedure

The following DATA step contains 100 observations on a dichotomous response variable (y), a character variable (C), and 10 continuous variables (x1–x10):

data getStarted;
  input C$ y x1-x10;
  datalines;
  D  0  10.2  6  1.6  38  15  2.4  20  0.8  8.5  3.9
  F  1  12.2  6  2.6  42  61  1.5  10  0.6  8.5  0.7
  D  1   7.7  1  2.1  38  61    1  90  0.6  7.5  5.2
  J  1  10.9  7  3.5  46  42  0.3   0  0.2    6  3.6
  E  0  17.3  6  3.8  26  47  0.9  10  0.4  1.5  4.7
  A  0  18.7  4  1.8   2  34  1.7  80    1  9.5  2.2
  B  0   7.2  1  0.3  48  61  1.1  10  0.8  3.5    4
  D  0   0.1  3  2.4   0  65  1.6  70  0.8  3.5  0.7
  H  1   2.4  4  0.7  38  22  0.2  20    0    3  4.2
  J  0  15.6  7  1.4   0  98  0.3   0    1    5  5.2
  J  0  11.1  3  2.4  42  55  2.2  60  0.6  4.5  0.7
  F  0     4  6  0.9   4  36  2.1  30  0.8    9  4.6
  A  0   6.2  2  1.8  14  79  1.1  70  0.2    0  5.1
  H  0   3.7  3  0.8  12  66  1.3  40  0.4  0.5  3.3
  A  1   9.2  3  2.3  48  51  2.3  50    0    6  5.4
  G  0    14  3    2  18  12  2.2   0    0    3  3.4
  E  1  19.5  6  3.7  26  81  0.1  30  0.6    5  4.8
  C  0    11  3  2.8  38   9  1.7  50  0.8  6.5  0.9
  I  0  15.3  7  2.2  20  98  2.7 100  0.4    7  0.8
  H  1   7.4  4  0.5  28  65  1.3  60  0.2  9.5  5.4
  F  0  11.4  2  1.4  42  12  2.4  10  0.4    1  4.5
  C  1  19.4  1  0.4  42   4  2.4  10    0  6.5  0.1
  G  0   5.9  4  2.6  12  57  0.8  50  0.4    2  5.8
  G  1  15.8  6  3.7  34   8  1.3  90  0.6  2.5  5.7
  I  0    10  3  1.9  16  80    3  90  0.4  9.5  1.9
  E  0  15.7  1  2.7  32  25  1.7  20  0.2  8.5    6
  G  0    11  5  2.9  48  53  0.1  50    1  3.5  1.2
  J  1  16.8  0  0.9  14  86  1.4  40  0.8    9    5
  D  1    11  4  3.2  48  63  2.8  90  0.6    0  2.2
  J  1   4.8  7  3.6  24   1  2.2  20    1  8.5  0.5
  J  1  10.4  5    2  42  56    1  20    0  3.5  4.2
  G  0  12.7  7  3.6   8  56  2.1  70    1  4.5  1.5
  G  0   6.8  1  3.2  30  27  0.6   0  0.8    2  5.6
  E  0   8.8  0  3.2   2  67  0.7  10  0.4    1    5
  I  1   0.2  0  2.9  10  41  2.3  60  0.2    9  0.3
  J  1   4.6  7  3.9  50  61  2.1  50  0.4    3  4.9
  J  1   2.3  2  3.2  36  98  0.1  40  0.6  4.5  4.3
  I  0  10.8  3  2.7  28  58  0.8  80  0.8    3    6
  B  0   9.3  2  3.3  44  44  0.3  50  0.8  5.5  0.4
  F  0   9.2  6  0.6   4  64  0.1   0  0.6  4.5  3.9
  D  0   7.4  0  2.9  14   0  0.2  30  0.8  7.5  4.5
  G  0  18.3  3  3.1   8  60  0.3  60  0.2    7  1.9
  F  0   5.3  4  0.2  48  63  2.3  80  0.2    8  5.2
  C  0   2.6  5  2.2  24   4  1.3  20    0    2  1.4
  F  0  13.8  4  3.6   4   7  1.1  10  0.4  3.5  1.9
  B  1  12.4  6  1.7  30  44  1.1  60  0.2    6  1.5
  I  0   1.3  1  1.3   8  53  1.1  70  0.6    7  0.8
  F  0  18.2  7  1.7  26  92  2.2  30    1  8.5  4.8
  J  0   5.2  2  2.2  18  12  1.4  90  0.8    4  4.9
  G  1   9.4  2  0.8  22  86  0.4  30  0.4    1  5.9
  J  1  10.4  2  1.7  26  31  2.4  10  0.2    7  1.6
  J  0    13  1  1.8  14  11  2.3  50  0.6  5.5  2.6
  A  0  17.9  4  3.1  46  58  2.6  90  0.6  1.5  3.2
  D  1  19.4  6    3  20  50  2.8 100  0.2    9  1.2
  I  0  19.6  3  3.6  22  19  1.2   0  0.6    5  4.1
  I  1     6  2  1.5  30  30  2.2  20  0.4  8.5  5.3
  G  0  13.8  1  2.7   0  52  2.4  20  0.8    6    2
  B  0  14.3  4  2.9  30  11  0.6  90  0.6  0.5  4.9
  E  0  15.6  0  0.4  38  79  0.4  80  0.4    1  3.3
  D  0    14  2    1  22  61    3  90  0.6    2  0.1
  C  1   9.4  5  0.4  12  53  1.7  40    0    3  1.1
  H  0  13.2  1  1.6  40  15  0.7  40  0.2    9  5.5
  A  0  13.5  5  2.4  18  89  1.6  20  0.4  9.5  4.7
  E  0   2.6  4  2.3  38   6  0.8  20  0.4    5  5.3
  E  0  12.4  3  1.3  26   8  2.8  10  0.8    6  5.8
  D  0   7.6  2  0.9  44  89  1.3  50  0.8    6  0.4
  I  0  12.7  1  2.3  42   6  2.4  10  0.4    1    3
  C  1  10.7  4  3.2  28  23  2.2  90  0.8  5.5  2.8
  H  0  10.1  2  2.3  10  62  0.9  50  0.4  2.5  3.7
  C  1  16.6  1  0.5  12  88  0.1  20  0.6  5.5  1.8
  I  1   0.2  3  2.2   8  71  1.7  80  0.4  0.5  5.5
  C  0  10.8  4  3.5  30  70  2.3  60  0.4  4.5  5.9
  F  0   7.1  4    3  14  63  2.4  70    0    7  3.1
  D  0  16.5  1  3.3  30  80  1.6  40    0  3.5  2.7
  H  0  17.1  7  2.1  30  45  1.5  60  0.6  0.5  2.8
  D  0   4.3  1  1.5  24  44    0  70    0    5  0.5
  H  0    15  2  0.2  14  87  1.8  50    0  4.5  4.7
  G  0  19.7  3  1.9  36  99  1.5  10  0.6    3  1.7
  H  1   2.8  6  0.6  34  21    2  60    1    9  4.7
  G  0  16.6  3  3.3  46   1  1.4  70  0.6  1.5  5.3
  E  0  11.7  5  2.7  48   4  0.9  60  0.8  4.5  1.6
  F  0  15.6  3  0.2   4  79  0.5   0  0.8  1.5  2.9
  C  1   5.3  6  1.4   8  64    2  80  0.4    9  4.2
  B  1   8.1  7  1.7  40  36  1.4  60  0.6    6  3.9
  I  0  14.8  2  3.2   8  37  0.4  10    0  4.5    3
  D  0   7.4  4    3  12   3  0.6  60  0.6    7  0.7
  D  0   4.8  3  2.3  44  41  1.9  60  0.2    3  3.1
  A  0   4.5  0  0.2   4  48  1.7  80  0.8    9  4.2
  D  0   6.9  6  3.3  14  92  0.5  40  0.4  7.5    5
  B  0   4.7  4  0.9  14  99  2.4  80    1  0.5  0.7
  I  1   7.5  4  2.1  20  79  0.4  40  0.4  2.5  0.7
  C  0   6.1  0  1.4  38  18  2.3  60  0.8  4.5  0.7
  C  0  18.3  1    1  26  98  2.7  20    1  8.5  0.5
  F  0  16.4  7  1.2  32  94  2.9  40  0.4  5.5  2.1
  I  0   9.4  2  2.3  32  42  0.2  70  0.4  8.5  0.3
  F  1  17.9  4  1.3  32  42    2  40  0.2    1  5.4
  H  0  14.9  3  1.6  36  74  2.6  60  0.2    1  2.3
  C  0  12.7  0  2.6   0  88  1.1  80  0.8  0.5  2.1
  F  0   5.4  4  1.5   2   1  1.8  70  0.4  5.5  3.6
  J  1  12.1  4  1.8  20  59  1.3  60  0.4    3  3.8
;

The following statements fit a logistic model to these data by using a classification effect for variable C and 10 regressor effects for x1–x10:

proc hplogistic data=getStarted;
   class C;
   model y = C x1-x10;
run;

The default output from this analysis is presented in Figure 9.1 through Figure 9.11.

The “Performance Information” table in Figure 9.1 shows that the procedure executes in single-machine mode—that is, the model is fit on the machine where the SAS session executes. This run of the HPLOGISTIC procedure was performed on a multicore machine with the same number of CPUs as there are threads; that is, one computational thread was spawned per CPU.

Figure 9.2 displays the “Model Information” table. The HPLOGISTIC procedure uses a Newton-Raphson algorithm to model a binary distribution for the variable y with a logit link function. The CLASS variable C is parameterized using the GLM parameterization, which is the default.

The CLASS variable C has 10 unique formatted levels, and these are displayed in the “Class Level Information” table in Figure 9.3.

Figure 9.4 displays the “Number of Observations” table. All 100 observations in the data set are used in the analysis.

The “Response Profile” table in Figure 9.5 is produced by default for binary and multinomial response variables. It shows the breakdown of the response variable levels by frequency. By default for binary data, the HPLOGISTIC procedure models the probability of the event with the lower-ordered value in the “Response Profile” table—this is indicated by the note that follows the table. In this example, the values represented by y = '0' are modeled as the “successes” in the Bernoulli experiments.

You can use the response-variable options in the MODEL statement to affect which value of the response variable is modeled.

Figure 9.6 displays the “Dimensions” table for this model. This table summarizes some important sizes of various model components. For example, it shows that there are 21 columns in the design matrix , which correspond to one column for the intercept, 10 columns for the effect associated with the classification variable C, and one column each for the continuous variables x1–x10. However, the rank of the crossproducts matrix is only 20. Because the classification variable C uses GLM parameterization and because the model contains an intercept, there is one singularity in the crossproducts matrix of the model. Consequently, only 20 parameters enter the optimization.

The “Iteration History” table is shown in Figure 9.7. The Newton-Raphson algorithm with ridging converged after four iterations, not counting the initial setup iteration.

Figure 9.8 displays the final convergence status of the Newton-Raphson algorithm. The GCONV= relative convergence criterion is satisfied.

The “Fit Statistics” table is shown in Figure 9.9. The –2 log likelihood at the converged estimates is 88.7007. You can use this value to compare the model to nested model alternatives by means of a likelihood-ratio test. To compare models that are not nested, information criteria such as AIC (Akaike’s information criterion), AICC (Akaike’s bias-corrected information criterion), and BIC (Schwarz’ Bayesian information criterion) are used. These criteria penalize the –2 log likelihood for the number of parameters. Because of the large number of parameters relative to the number of observations, the discrepancy between the –2 log likelihood and, say, AIC, is substantial in this case.

Figure 9.10 shows the global test for the null hypothesis that all model effects jointly do not affect the probability of success of the binary response. The test is significant (p-value = 0.0135). One or more of the model effects thus significantly affects the probability of observing an event.

However, a look at the “Parameter Estimates” table in Figure 9.11 shows that many parameters have fairly large p-values, indicating that one or more of the model effects might not be necessary.