Example 48.3 Overcoming Convergence Problems by Specifying Initial Values

This example illustrates the use of parameter initial value specification to help overcome convergence difficulties.

The following statements create a SAS data set.

data raw;
   input censor x c1 @@;
   datalines;
0 16 0.00   0 17 0.00   0 18 0.00
0 17 0.04   0 18 0.04   0 18 0.04
0 23 0.40   0 22 0.40   0 22 0.40
0 33 4.00   0 34 4.00   0 35 4.00
1 54 40.00  1 54 40.00  1 54 40.00
1 54 400.00 1 54 400.00 1 54 400.00
;
run;

Output 48.3.1 shows the contents of the data set raw.

The following SAS statements request that a Weibull regression model be fit to the data:

title 'OLS (default) initial values';
proc lifereg data=raw;
   model x*censor(1) = c1 / distribution = Weibull itprint;
run;

Convergence was not attained in 50 iterations for this model, as the following messages to the log indicate:

WARNING: Convergence was not attained in 50 iterations. You might want to
         increase the maximum number of iterations (MAXITER= option) or
         change the convergence criteria (CONVERGE = value) in the MODEL
         statement.
WARNING: The procedure is continuing in spite of the above warning. Results
         shown are based on the last maximum likelihood iteration. Validity
         of the model fit is questionable.

The first line (iter=0) of the iteration history table, shown in Output 48.3.2, shows the default initial ordinary least squares (OLS) estimates of the parameters.

The log-logistic distribution is more robust to large values of the response than the Weibull distribution, so one approach to improving the convergence performance is to fit a log-logistic distribution, and if this converges, use the resulting parameter estimates as initial values in a subsequent fit of a model with the Weibull distribution.

The following statements fit a log-logistic distribution to the data:

proc lifereg data=raw;
   model x*censor(1) = c1 / distribution = llogistic;
run;

The algorithm converges, and the maximum likelihood estimates for the log-logistic distribution are shown in Output 48.3.3

The following statements refit the Weibull model by using the maximum likelihood estimates from the log-logistic fit as initial values:

proc lifereg data=raw outest=outest;
   model x*censor(1) = c1 / itprint distribution = weibull
                            intercept=2.898 initial=0.16 scale=0.05;
   output out=out xbeta=xbeta;
run;

Examination of the resulting output in Output 48.3.4 shows that the convergence problem has been solved by specifying different initial values.

As an example of an alternative way of specifying initial values, the following invocation of PROC LIFEREG, using the INEST= data set to provide starting values for the three parameters, is equivalent to the previous invocation:

data in;
   input  intercept c1 scale;
   datalines;
2.898 0.16 0.05
;
proc lifereg data=raw inest=in outest=outest;
   model x*censor(1) = c1 / itprint distribution = weibull;
   output out=out xbeta=xbeta;
run;