Pregibon Test for Goodness of Link in SAS

This article is originally published at https://statcompute.wordpress.com

When estimating generalized linear models for binary outcomes, we often choose the logit link function by default and seldom consider other alternatives such as probit or cloglog. The Pregibon test (Pregibon, 1980) provides a mean to check the goodness of link with a simple logic outlined below.

1. First of all, we can estimate the regression model with the hypothesized link function, e.g. logit;
2. After the model estimation, we calculate yhat and yhat ^ 2 and then estimate a secondary regression with the identical response variable Y and link function but with yhat and yhat ^ 2 as model predictors (with the intercept).
3. If the link function is correctly specified, then the t-value of yaht ^2 should be insignificant.

The SAS macro shown below is the implementation of Pregibon test in the context of logistic regressions. However, the same idea can be generalized to any GLM.

%macro pregibon(data = , y = , x = );
***********************************************************;
* SAS MACRO PERFORMING PREGIBON TEST FOR GOODNESS OF LINK *;
* ======================================================= *;
* INPUT PAREMETERS:                                       *;
*  DATA : INPUT SAS DATA TABLE                            *;
*  Y    : THE DEPENDENT VARIABLE WITH 0 / 1 VALUES        *;
*  X    : MODEL PREDICTORS                                *;
* ======================================================= *;
* AUTHOR: [email protected]                               *;
***********************************************************;
options mprint mlogic nocenter;

%let links = logit probit cloglog;
%let loop = 1;

proc sql noprint;
  select n(&data) - 3 into :df from &data;
quit; 

%do %while (%scan(&links, &loop) ne %str());

  %let link = %scan(&links, &loop);
  
  proc logistic data = &data noprint desc;
    model &y = &x / link = &link;
    score data = &data out = _out1;
  run;
  
  data _out2;
    set _out1(rename = (p_1 = p1));
    p2 = p1 * p1;
  run;
  
  ods listing close;
  ods output ParameterEstimates = _parm;  
  proc logistic data = _out2 desc;
    model &y = p1 p2 /  link = &link ;
  run;
  ods listing;
    
  %if &loop = 1 %then %do;
    data _parm1;
      format link $10.;
      set _parm(where = (variable = "p2"));
      link = upcase("&link");
    run;
  %end;
  %else %do;
    data _parm1;
      set _parm1 _parm(where = (variable = "p2") in = new);
      if new then link = upcase("&link");
    run;
  %end;
  
  data _parm2(drop = variable);
    set _parm1;
    _t = estimate / stderr;
    _df = &df;
    _p = (1 - probt(abs(_t), _df)) * 2;
  run;
  
  %let loop = %eval(&loop + 1);

%end;

title;
proc report data = _last_ spacing = 1 headline nowindows split = "*";
  column(" * PREGIBON TEST FOR GOODNESS OF LINK
           * H0: THE LINK FUNCTION IS SPECIFIED CORRECTLY * "
         link _t _df _p);
  define link / "LINK FUNCTION" width = 15 order order = data;          
  define _t   / "T-VALUE"       width = 15 format = 12.4;
  define _df  / "DF"            width = 10;
  define _p   / "P-VALUE"       width = 15 format = 12.4;
run;

%mend;

After applying the macro to the kyphosis data (https://stat.ethz.ch/R-manual/R-devel/library/rpart/html/kyphosis.html), we can see that both logit and probit can be considered appropriate link functions in this specific case and cloglog might not be a good choice.

             PREGIBON TEST FOR GOODNESS OF LINK
        H0: THE LINK FUNCTION IS SPECIFIED CORRECTLY

 LINK FUNCTION           T-VALUE         DF         P-VALUE
-----------------------------------------------------------
 LOGIT                   -1.6825         78          0.0965
 PROBIT                  -1.7940         78          0.0767
 CLOGLOG                 -2.3632         78          0.0206

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