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 Specialized Control Charts

Calculating the Chart Statistic

In this situation, it was shown by Gnanadesikan and Kettenring (1972), using a result of Wilks (1962), that T2i 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;

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

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 T2i and its exact control limits, using the fact that T2i is the sum of squares of the principal components.* 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 49.30.
   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;


 T2 Chart For Chemical Example

 _var_ n sample impure temp conc p _subn_ _limitn_ comp1 comp2 comp3 _subx_ _alpha_ _lclx_ _mean_ _uclx_ tsquare 14 1 14.92 85.77 42.26 3 1 1 0.79603 10.1137 0.01606 10.9257 0.05 0.24604 2.44144 7.13966 tsquare 14 2 16.90 83.77 43.44 3 1 1 1.84804 0.0162 0.17681 2.0410 0.05 0.24604 2.44144 7.13966 tsquare 14 3 17.38 84.46 42.74 3 1 1 0.33397 0.1538 5.09491 5.5827 0.05 0.24604 2.44144 7.13966 tsquare 14 4 16.90 86.27 43.60 3 1 1 0.77286 0.3289 2.76215 3.8640 0.05 0.24604 2.44144 7.13966 tsquare 14 5 16.92 85.23 43.18 3 1 1 0.00147 0.0165 0.01919 0.0372 0.05 0.24604 2.44144 7.13966 tsquare 14 6 16.71 83.81 43.72 3 1 1 1.91534 0.0645 0.27362 2.2534 0.05 0.24604 2.44144 7.13966 tsquare 14 7 17.07 86.08 43.33 3 1 1 0.58596 0.4079 0.44146 1.4354 0.05 0.24604 2.44144 7.13966 tsquare 14 8 16.93 85.85 43.41 3 1 1 0.29543 0.1729 0.73939 1.2077 0.05 0.24604 2.44144 7.13966 tsquare 14 9 16.71 85.73 43.28 3 1 1 0.23166 0.0001 0.44483 0.6766 0.05 0.24604 2.44144 7.13966 tsquare 14 10 16.88 86.27 42.59 3 1 1 1.30518 0.0004 0.86364 2.1692 0.05 0.24604 2.44144 7.13966 tsquare 14 11 16.73 83.46 44.00 3 1 1 3.15791 0.0274 0.98639 4.1717 0.05 0.24604 2.44144 7.13966 tsquare 14 12 17.07 85.81 42.78 3 1 1 0.43819 0.0823 0.87976 1.4003 0.05 0.24604 2.44144 7.13966 tsquare 14 13 17.60 85.92 43.11 3 1 1 0.41494 1.6153 0.30167 2.3320 0.05 0.24604 2.44144 7.13966 tsquare 14 14 16.90 84.23 43.48 3 1 1 0.90302 0.0001 0.00010 0.9032 0.05 0.24604 2.44144 7.13966
Figure 49.30: The Data Set PRIN

You can now use the data set PRIN as input to the SHEWHART procedure to create the multivariate control chart displayed in Figure 49.31.

   symbol v=dot c=yellow;
title 'T' m=(+0,+0.5) '2'
m=(+0,-0.5) ' Chart For Chemical Example';
proc shewhart table=prin;
xchart tsquare*sample /
xsymbol  = mu
cframe   = vigb
cinfill  = vlib
cconnect = yellow
nolegend ;
run;


Figure 49.31: Multivariate Control Chart for Chemical Process

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 in the statements . For details, refer to Alt (1985), Jackson (1980, 1991), and Ryan (1989).

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