Example 43.3: Working with Unequal Subgroup Sample Sizes

XRCHART Statement

Example 43.3: Working with Unequal Subgroup Sample Sizes

See SHWXR4 in the SAS/QC Sample Library

The following data set (WIRE) contains breaking strength measurements recorded in pounds per inch for 25 samples from a metal wire manufacturing process. The subgroup sample sizes vary between 3 and 7.

   data wire;
      input day size @;
      informat day date7.;
      format day date7.;
      do i=1 to size;
         input brstr @@;
         output;
         end;
      drop i size;
      label brstr  ='Breaking Strength (lb/in)';
      datalines;
   20JUN94 5 60.6 62.3 62.0 60.4 59.9
   21JUN94 5 61.9 62.1 60.6 58.9 65.3
   22JUN94 4 57.8 60.5 60.1 57.7
   23JUN94 5 56.8 62.5 60.1 62.9 58.9
   24JUN94 5 63.0 60.7 57.2 61.0 53.5
   25JUN94 7 58.7 60.1 59.7 60.1 59.1 57.3 60.9
   26JUN94 5 59.3 61.7 59.1 58.1 60.3
   27JUN94 5 61.3 58.5 57.8 61.0 58.6
   28JUN94 6 59.5 58.3 57.5 59.4 61.5 59.6
   29JUN94 5 61.7 60.7 57.2 56.5 61.5
   30JUN94 3 63.9 61.6 60.9
   01JUL94 5 58.7 61.4 62.4 57.3 60.5
   02JUL94 5 56.8 58.5 55.7 63.0 62.7
   03JUL94 5 62.1 60.6 62.1 58.7 58.3
   04JUL94 5 59.1 60.4 60.4 59.0 64.1
   05JUL94 5 59.9 58.8 59.2 63.0 64.9
   06JUL94 6 58.8 62.4 59.4 57.1 61.2 58.6
   07JUL94 5 60.3 58.7 60.5 58.6 56.2
   08JUL94 5 59.2 59.8 59.7 59.3 60.0
   09JUL94 5 62.3 56.0 57.0 61.8 58.8
   10JUL94 4 60.5 62.0 61.4 57.7
   11JUL94 4 59.3 62.4 60.4 60.0
   12JUL94 5 62.4 61.3 60.5 57.7 60.2
   13JUL94 5 61.2 55.5 60.2 60.4 62.4
   14JUL94 5 59.0 66.1 57.7 58.5 58.9
   ;

The following statements request $\bar{X}$ and R charts, shown in Output 43.3.1, for the strength measurements:

   title 'Mean and Range Charts for Breaking Strengths';
   symbol v=dot c=salmon;
   proc shewhart data=wire;
      xrchart brstr*day / nohlabel
                          cframe   = bigb
                          cinfill  = ywh
                          cconnect = salmon;
   run;

Output 43.3.1: $\bar{X}$ and R Charts with Varying Subgroup Sample Sizes

Note that the central line on the R chart and the control limits on both charts vary with the subgroup sample size. The sample size legend in the lower left corner displays the minimum and maximum subgroup sample sizes.

The XRCHART statement provides various options for working with unequal subgroup sample sizes. For example, you can use the LIMITN= option to specify a fixed (nominal) sample size for computing control limits, as illustrated by the following statements:

   proc shewhart data=wire;
      xrchart brstr*day / nohlabel
                          limitn   = 5
                          cframe   = bigb
                          cinfill  = ywh
                          cconnect = salmon;
   run;

The resulting charts are shown in Output 43.3.2.

Output 43.3.2: Control Limits Based on Fixed Sample Size

Note that the only points displayed on the chart are those corresponding to subgroups whose sample sizes match the nominal sample size of five. To plot points for all subgroups (regardless of subgroup sample size), you can specify the ALLN option, as follows:

   proc shewhart data=wire;
      xrchart brstr*day / nohlabel
                          limitn   = 5
                          alln
                          nmarkers
                          cframe   = bigb
                          cinfill  = ywh
                          cconnect = salmon;
   run;

The charts are shown in Output 43.3.3. The NMARKERS option requests special symbols to identify points for which the subgroup sample size differs from the nominal sample size.

Output 43.3.3: Displaying All Subgroups Regardless of Sample Size

You can use the SMETHOD= option to determine how the process standard deviation $\sigma$ is to be estimated when the subgroup sample sizes vary. The default method computes $\hat{\sigma}$ as an unweighted average of subgroup estimates of $\sigma$ .Specifying SMETHOD=MVLUE requests an estimate that assigns greater weight to estimates of $\sigma$ from subgroups with larger sample sizes. For more information, see "Methods for Estimating the Standard Deviation" .

The following statements apply both methods:

   proc shewhart data=wire;
      xrchart brstr*day / outlimits = wlim1
                          outindex  = 'Default'
                          nochart;
      xrchart brstr*day / smethod   = mvlue
                          outlimits = wlim2
                          outindex  = 'MVLUE'
                          nochart;
   run;


   data wlimits;
      set wlim1 wlim2;
   run;

The data set WLIMITS is listed in Output 43.3.4.

Output 43.3.4: Listing of the Data Set WLIMITS

The WLIMITS Data Set

_VAR_	_SUBGRP_	_INDEX_	_TYPE_	_LIMITN_	_ALPHA_	_SIGMAS_	_LCLX_	_MEAN_	_UCLX_	_LCLR_	_R_	_UCLR_	_STDDEV_
brstr	day	Default	ESTIMATE	V	.002699796	3	V	59.9766	V	V	V	V	2.11146
brstr	day	MVLUE	ESTIMATE	V	.002699796	3	V	59.9766	V	V	V	V	2.11240

The variables in an OUTLIMITS= data set whose values vary with subgroup sample size are assigned the special missing value V. Consequently, the control limit variables (_LCLX_, _UCLX_, _LCLR_, and _UCLR_), as well as the variables _R_ and _LIMITN_, have this value.

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