Example 60.1 Segmented Model

Suppose you are interested in fitting a model that consists of two segments that connect in a smooth fashion. For example, the following model states that for values of less than the mean of is a quadratic function in , and for values of greater than the mean of is constant:

In this model equation , , and are the coefficients of the quadratic segment, and is the plateau of the mean function. The NLIN procedure can fit such a segmented model even when the join point, , is unknown.

We also want to impose conditions on the two segments of the model. First, the curve should be continuous—that is, the quadratic and the plateau section need to meet at . Second, the curve should be smooth—that is, the first derivative of the two segments with respect to need to coincide at .

The continuity condition requires that

The smoothness condition requires that

If you solve for and substitute into the expression for , the two conditions jointly imply that

Although there are apparently four unknowns, the model contains only three parameters. The continuity and smoothness restrictions together completely determine one parameter given the other three.

The following DATA step creates the SAS data set for this example:

   data a;
      input y x @@;
      datalines;
   .46 1  .47  2 .57  3 .61  4 .62  5 .68  6 .69  7
   .78 8  .70  9 .74 10 .77 11 .78 12 .74 13 .80 13
   .80 15 .78 16
    ;

The following PROC NLIN statements fit this segmented model:

   title 'Quadratic Model with Plateau';
   proc nlin data=a;
      parms alpha=.45 beta=.05 gamma=-.0025;
      
      x0 = -.5*beta / gamma;
      
      if (x < x0) then 
           mean = alpha + beta*x  + gamma*x*x;
      else mean = alpha + beta*x0 + gamma*x0*x0;
      model y = mean;
   
      if _obs_=1 and _iter_ =.  then do;
         plateau =alpha + beta*x0 + gamma*x0*x0;
         put /  x0= plateau=  ;
      end;
      output out=b predicted=yp; 
   run;

The parameters of the model are , , and , respectively. They are represented in the PROC NLIN statements by the variables alpha, beta, and gamma, respectively. In order to model the two segments, a conditional statement is used that assigns the appropriate expression to the mean function depending on the value of . A PUT statement is used to print the constrained parameters every time the program is executed for the first observation. The OUTPUT statement computes predicted values for plotting and saves them to data set b.

Note that there are other ways in which you can write the conditional expressions for this model. For example, you could formulate a condition with two model statements, as follows:

   proc nlin data=a;
      parms alpha=.45 beta=.05 gamma=-.0025;
      x0 = -.5*beta / gamma;
      if (x < x0) then 
           model y = alpha+beta*x+gamma*x*x;
      else model y = alpha+beta*x0+gamma*x0*x0;
   run;

Or you could use a single expression with a conditional evaluation, as in the following statements:

   proc nlin data=a;
      parms alpha=.45 beta=.05 gamma=-.0025;
      x0 = -.5*beta / gamma;
      model y = (x <x0)*(alpha+beta*x +gamma*x*x) +
                (x>=x0)*(alpha+beta*x0+gamma*x0*x0);
   run;

The results from fitting this model with PROC NLIN are shown in Output 60.1.1–Output 60.1.3. The iterative optimization converges after six iterations (Output 60.1.1). Output 60.1.1 indicates that the join point is and the plateau value is .