Example 12.1: Cusum and Standard Deviation Charts

XCHART Statement

Example 12.1: Cusum and Standard Deviation Charts

See CUSXS in the SAS/QC Sample Library

When you are working with subgrouped data, it can be helpful to accompany a cusum chart for means with a Shewhart s chart for monitoring the variability of the process. This example creates this combination for the variable WEIGHT in the data set OIL (see "Creating a V-Mask Cusum Chart from Raw Data" ).

The first step is to create a one-sided cusum chart for means that detects a shift of one standard error ( $\delta=-1$ )below the target mean.

   proc cusum data=oil;
      xchart weight*hour /
         nochart                    /* suppress display         */
         mu0    = 8.100             /* target mean              */
         sigma0 = 0.050             /* known standard deviation */
         delta  = -1                /* shift to be detected     */
         h      = 3                 /* decision interval        */
         k      = 0.5               /* reference value          */
         scheme=onesided            /* one-sided scheme         */
         outtable = tabcusum        /* save the results         */
            ( drop   = _var_ _subn_ _subx_ _exlim_               
              rename = ( _cusum_ = _subx_ _h_ = _uclx_ ) )
         ;
   run;

The results are saved in an OUTTABLE= data set named TABCUSUM. The cusum variable (_CUSUM_) and the decision interval variable (_H_) are renamed to _SUBX_ and _LCLX_ so that they can later be read by the SHEWHART procedure.

The next step is to construct a Shewhart $\bar{X}$ and s chart for WEIGHT and save the results in a data set named TABXSCHT.

   proc shewhart data=oil;
      xschart weight*hour /
         nochart                    /* suppress display */
         outtable = tabxscht        /* save the results */
            ( drop = _subx_  _uclx_ );
   run;

Note that the variables _SUBX_ and _UCLX_ are dropped from TABXSCHT.

The third step is to merge the data sets TABCUSUM and TABXSCHT.

   data taball;
      merge tabxscht tabcusum; by hour;
      _mean_ = _uclx_ * 0.5;
      _lclx_ = 0.0;
   run;

   proc print;
   run;

The variable _LCLX_ is assigned the role of the lower limit for the cusums, and the variable _MEAN_ is assigned a dummy value. Now, TABALL, which is listed in Output 12.1.1, has the structure required for a TABLE= data set used with the XSCHART statement in the SHEWHART procedure (see "TABLE= Data Set" in Chapter 44, "XSCHART Statement").

Output 12.1.1: Listing of the Data Set TABALL

Obs	_VAR_	hour	_SIGMAS_	_LIMITN_	_SUBN_	_LCLX_	_MEAN_	_EXLIM_	_LCLS_	_SUBS_	_S_	_UCLS_	_EXLIMS_	_subx_	_uclx_
1	weight	1	3	4	4	0	1.5		0	0.059640	0.049943	0.11317		0.00	3
2	weight	2	3	4	4	0	1.5		0	0.090220	0.049943	0.11317		0.00	3
3	weight	3	3	4	4	0	1.5		0	0.076346	0.049943	0.11317		0.00	3
4	weight	4	3	4	4	0	1.5		0	0.025552	0.049943	0.11317		0.00	3
5	weight	5	3	4	4	0	1.5		0	0.026500	0.049943	0.11317		0.00	3
6	weight	6	3	4	4	0	1.5		0	0.075617	0.049943	0.11317		0.30	3
7	weight	7	3	4	4	0	1.5		0	0.037242	0.049943	0.11317		0.00	3
8	weight	8	3	4	4	0	1.5		0	0.059290	0.049943	0.11317		0.18	3
9	weight	9	3	4	4	0	1.5		0	0.005737	0.049943	0.11317		1.21	3
10	weight	10	3	4	4	0	1.5		0	0.046522	0.049943	0.11317		0.62	3
11	weight	11	3	4	4	0	1.5		0	0.040542	0.049943	0.11317		0.00	3
12	weight	12	3	4	4	0	1.5		0	0.056103	0.049943	0.11317		0.00	3

The final step is to use the SHEWHART procedure to read TABALL as a TABLE= data set and to display the cusum and s charts.

   title 'Cusum Chart for Mean and s chart';
   symbol v=dot c=salmon;
   proc shewhart table=taball;
      xschart weight * hour /
         ucllabel = 'h=3.0'
         nolimitslegend
         noctl
         split    = '/'
         nolegend
         cinfill  = ywh
         coutfill = yellow
         cconnect = salmon
         cframe   = bigb;
      label _subx_ = 'Lower Cusum/Std Dev';
   run;

The central line for the primary (cusum) chart is suppressed with the NOCTL option, and the default 3 $\sigma$ Limits legend is suppressed with the NOLIMITLEGEND option. The charts are shown in Output 12.1.2.

Output 12.1.2: Combined Cusum Chart and s Chart

The process variability is stable, and there is no signal of a downward shift in the process mean.

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