Example 45.3: Probit Model with Likelihood function

The NLIN Procedure

Example 45.3: Probit Model with Likelihood function

The data, taken from Lee (1974), consist of patient characteristics and a variable indicating whether cancer remission occured. This example demonstrates how to use PROC NLIN with a likelihood function. In this case, the likelihood function to minimize is

$-2 \log L &=& -2 \sum_{i=1}^N wght_i \log( \hat{p}_i( y_i, x_i) )$

where

$\hat{p}_i(y_i, x_i) = \{ \Phi( \alpha + {\beta^'x}_i ) & y_i = 0 \ 1 - \Phi( \alpha + \beta' x_i ) & y_i = 1 .$

and $\Phi$ is the normal probability function. This is the likelihood function for a binary probit model. This likelihood is strictly positive so that you can take a square root of $\log( \hat{p}_i( y_i, x_i) )$ and use this as your residual in PROC NLIN. The DATA step also creates a zero-valued dummy variable, like, that is used as the dependent variable.

   Data remiss;
      input remiss cell smear infil li blast temp;
      label remiss = 'complete remission';
      like = 0;
      label like = 'dummy variable for nlin';
      datalines;
   1 .8 .83 .66 1.9 1.1 .996 
   1 .9 .36 .32 1.4 .74 .992 
   0 .8 .88 .7 .8 .176 .982 
   0 1 .87 .87 .7 1.053 .986 
   1 .9 .75 .68 1.3 .519 .98 
   0 1 .65 .65 .6 .519 .982 
   1 .95 .97 .92 1 1.23 .992 
   0 .95 .87 .83 1.9 1.354 1.02 
   0 1 .45 .45 .8 .322 .999 
   0 .95 .36 .34 .5 0 1.038 
   0 .85 .39 .33 .7 .279 .988 
   0 .7 .76 .53 1.2 .146 .982 
   0 .8 .46 .37 .4 .38 1.006 
   0 .2 .39 .08 .8 .114 .99 
   0 1 .9 .9 1.1 1.037 .99 
   1 1 .84 .84 1.9 2.064 1.02 
   0 .65 .42 .27 .5 .114 1.014 
   0 1 .75 .75 1 1.322 1.004 
   0 .5 .44 .22 .6 .114 .99 
   1 1 .63 .63 1.1 1.072 .986 
   0 1 .33 .33 .4 .176 1.01 0
   0 .9 .93 .84 .6 1.591 1.02 
   1 1 .58 .58 1 .531 1.002 
   0 .95 .32 .3 1.6 .886 .988 
   1 1 .6 .6 1.7 .964 .99 
   1 1 .69 .69 .9 .398 .986 
   0 1 .73 .73 .7 .398 .986 
   ;
   run;

   proc nlin data=remiss method=newton sigsq=1;
      parms a -2 b  -1 c 6 int -10;
      
           /* Linear portion of model ------*/
      eq1 = a*cell + b*li + c*temp +int;
      
             /* probit */
      p = probnorm(eq1);

      if ( remiss = 1 ) then p = 1-p;

      model.like = sqrt(- 2 * log( p));
      output out=p p=predict;
   run;

Note that the asymptotic standard errors of the parameters are computed under the least squares assumptions. The SIGSQ=1 option on the PROC NLIN statement forces PROC NLIN to replace the usual mean square error with 1. Also, METHOD=NEWTON is selected so the true Hessian of the likelihood function is used to calculate parameter standard errors rather than the crossproducts approximation to the Hessian.

Output 45.3.1: Nonlinear Least Squares Analysis from PROC NLIN

Beaton/Tukey Biweight Robust Regression using IRLS

The NLIN Procedure

NOTE:

An intercept was not specified for this model.

Source	DF	Sum of Squares	Mean Square	F Value	Approx Pr > F
Regression	4	-21.9002	-5.4750	-5.75	.
Residual	23	21.9002	0.9522
Uncorrected Total	27	0

Corrected Total	26	0

Parameter	Estimate	Approx Std Error	Approximate 95% Confidence Limits
a	-5.6298	4.6376	-15.2234	3.9638
b	-2.2513	0.9790	-4.2764	-0.2262
c	45.1815	34.9095	-27.0337	117.4
int	-36.7548	32.3607	-103.7	30.1879

The problem can be more simply solved using the following SAS statements.

proc probit data=remiss ;
   class remiss;
   model remiss=cell li temp ;
run;

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