General Statistics Examples

Example 8.7: Response Surface Methodology

A regression model with a complete quadratic set of regressions across several factors can be processed to yield the estimated critical values that can optimize a response. First, the regression is performed for two variables according to the model

The estimates are then divided into a vector of linear coefficients (estimates) and a matrix of quadratic coefficients . The solution for critical values is

The following program creates a module to perform quadratic response surface regression.

  
    /*          Quadratic Response Surface Regression            */ 
    /* This matrix routine reads in the factor variables and     */ 
    /* the response, forms the quadratic regression model and    */ 
    /* estimates the parameters, and then solves for the optimal */ 
    /* response, prints the optimal factors and response, and    */ 
    /* displays the eigenvalues and eigenvectors of the          */ 
    /* matrix of quadratic parameter estimates to determine if   */ 
    /* the solution is a maximum or minimum, or saddlepoint, and */ 
    /* which direction has the steepest and gentlest slopes.     */ 
    /*                                                           */ 
    /* Given that d contains the factor variables,               */ 
    /* and y contains the response.                              */ 
    /*                                                           */ 
    start rsm; 
       n=nrow(d); 
       k=ncol(d);                                  /* dimensions */ 
       x=j(n,1,1)||d;                    /* set up design matrix */ 
       do i=1 to k; 
          do j=1 to i; 
             x=x||d[,i] #d[,j]; 
          end; 
       end; 
       beta=solve(x`*x,x`*y);       /* solve parameter estimates */ 
       print "Parameter Estimates" , beta; 
       c=beta[1];                          /* intercept estimate */ 
       b=beta[2:(k+1)];                      /* linear estimates */ 
       a=j(k,k,0); 
       L=k+1;                     /* form quadratics into matrix */ 
       do i=1 to k; 
          do j=1 to i; 
             L=L+1; 
             a[i,j]=beta [L,]; 
          end; 
       end; 
       a=(a+a`)*.5;                                /* symmetrize */ 
       xx=-.5*solve(a,b);            /* solve for critical value */ 
       print , "Critical Factor Values" , xx; 
          /* Compute response at critical value */ 
       yopt=c + b`*xx + xx`*a*xx; 
       print , "Response at Critical Value" yopt; 
       call eigen(eval,evec,a); 
       print , "Eigenvalues and Eigenvectors", eval, evec; 
       if min(eval)>0 then print , "Solution Was a Minimum"; 
       if max(eval)<0 then print , "Solution Was a Maximum"; 
    finish rsm;
 

Running the module with the following sample data produces the following results and Output 8.7.1:

  
     /* Sample Problem with Two Factors */ 
     d={-1 -1, -1  0, -1  1, 
         0 -1,  0  0,  0  1, 
         1 -1,  1  0,  1  1}; 
     y={ 71.7, 75.2, 76.3, 79.2, 81.5, 80.2, 80.1, 79.1, 75.8}; 
     run rsm;
 

Output 8.7.1: Response Surface Regression: Results

PARAMETER ESTIMATES BETA 81.222222 1.9666667 0.2166667 -3.933333 -2.225 -1.383333 CRITICAL FACTOR VALUES XX 0.2949376 -0.158881   YOPT Response at Critical Value 81.495032 EIGENVALUES AND EIGENVECTORS EVAL -0.96621 -4.350457 EVEC -0.351076 0.9363469 0.9363469 0.3510761