In this situation, it was shown by Gnanadesikan and Kettenring (1972), using a result of Wilks (1962), that is exactly distributed as a multiple of a variable with a beta distribution. Specifically,

Tracy, Young, and Mason (1992) used this result to derive initial control limits for a multivariate chart based on three quality measures from a chemical process in the start-up phase: percent of impurities, temperature, and concentration. The remainder of this section describes the construction of a multivariate control chart using their data, which are given here by the data set Startup.

data Startup;
   input Sample Impure Temp Conc;
   label Sample = 'Sample Number'
         Impure = 'Impurities'
         Temp   = 'Temperature'
         Conc   = 'Concentration' ;
   datalines;
 1  14.92  85.77  42.26
 2  16.90  83.77  43.44
 3  17.38  84.46  42.74
 4  16.90  86.27  43.60
 5  16.92  85.23  43.18
 6  16.71  83.81  43.72
 7  17.07  86.08  43.33
 8  16.93  85.85  43.41
 9  16.71  85.73  43.28
10  16.88  86.27  42.59
11  16.73  83.46  44.00
12  17.07  85.81  42.78
13  17.60  85.92  43.11
14  16.90  84.23  43.48
;

In preparation for the computation of the control limits, the sample size is calculated and parameter variables are defined.

proc means data=Startup noprint ;
   var Impure Temp Conc;
   output out=means n=n;
run;

data Startup;
   if _n_ = 1 then set means;
   set Startup;
   p        = 3;
   _subn_   = 1;
   _limitn_ = 1;
run;

Next, the PRINCOMP procedure is used to compute the principal components of the variables and save them in an output data set named Prin.

proc princomp data=Startup out=Prin outstat=scores std cov;
   var Impure Temp Conc;
run;

The following statements compute and its exact control limits, using the fact that is the sum of squares of the principal components.^[111] Note that these statements create several special SAS variables so that the data set Prin can subsequently be read as a TABLE= input data set by the SHEWHART procedure. These special variables begin and end with an underscore character. The data set Prin is listed in Figure 17.220.

data Prin (rename=(tsquare=_subx_));
   length _var_ $ 8 ;
   drop prin1 prin2 prin3 _type_ _freq_;
   set Prin;
   comp1   = prin1*prin1;
   comp2   = prin2*prin2;
   comp3   = prin3*prin3;
   tsquare = comp1 + comp2 + comp3;
   _var_   = 'tsquare';
   _alpha_ = 0.05;
   _lclx_  = ((n-1)*(n-1)/n)*betainv(_alpha_/2, p/2, (n-p-1)/2);
   _mean_  = ((n-1)*(n-1)/n)*betainv(0.5, p/2, (n-p-1)/2);
   _uclx_  = ((n-1)*(n-1)/n)*betainv(1-_alpha_/2, p/2, (n-p-1)/2);
   label tsquare = 'T Squared'
         comp1   = 'Comp 1'
         comp2   = 'Comp 2'
         comp3   = 'Comp 3';
run;

You can now use the data set Prin as input to the SHEWHART procedure to create the multivariate control chart displayed in Figure 17.221.

title 'T' m=(+0,+0.5) '2'
          m=(+0,-0.5) ' Chart For Chemical Example';
proc shewhart table=Prin;
   xchart tsquare*Sample /
      xsymbol  = mu
      nolegend ;
run;

The methods used in this example easily generalize to other types of multivariate control charts. You can create charts using the and F distributions by using the appropriate CINV or FINV function in place of the BETAINV function. For details, refer to Alt (1985), Jackson (1980, 1991), and Ryan (1989).