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Specialized Control Charts 
The OUTLIMITS= option saves the control limits from the chart in Figure 49.25 in a SAS data set named DELAYLIM, which is listed in Figure 49.26.

The control limits can be replaced with the corresponding percentiles from a fitted lognormal distribution. The equation for the lognormal density function is
where denotes the shape parameter and denotes the scale parameter.
The following statements use the CAPABILITY procedure to fit a lognormal model and superimpose the fitted density on a histogram of the data, shown in Figure 49.27:
title 'Lognormal Fit for Delay Distribution'; proc capability data=calls noprint; histogram time / lognormal(threshold=2.3 color=yellow w=2) cfill = vigb cframe = ligr outfit = lnfit nolegend ; inset n = 'Number of Calls' lognormal( sigma = 'Shape' (4.2) zeta = 'Scale' (5.2) theta ) / pos = ne ctext = black cfill = ywh; label time = 'Answering Delay (minutes)'; run;
Parameters of the fitted distribution and results of goodnessoffit tests are saved in the data set LNFIT, which is listed in Figure 49.28. The large pvalues for the goodnessoffit tests are evidence that the lognormal model provides a good fit.

The following statements replace the control limits
in DELAYLIM with limits computed from percentiles
of the fitted lognormal model. The 100percentile of the lognormal distribution is
,where denotes the inverse standard normal
cumulative distribution function. The SHEWHART procedure
constructs an X chart with the modified limits,
displayed in Figure 49.29.
data delaylim; merge delaylim lnfit; drop _sigmas_ ; _lcli_ = _locatn_ + exp(_scale_+probit(0.5*_alpha_)*_shape1_); _ucli_ = _locatn_ + exp(_scale_+probit(10.5*_alpha_)*_shape1_); _mean_ = _locatn_ + exp(_scale_+0.5*_shape1_*_shape1_);
symbol v=dot c=yellow; title 'Lognormal Control Limits for Delays'; proc shewhart data=calls limits=delaylim; irchart time*recnum / rtmplot = schematic cboxfill = vlib cframe = vigb cinfill = vlib cboxes = yellow cconnect = yellow nochart2 ; label recnum = 'Record Number' time = 'Delay in minutes' ; run;
Clearly the process is in control, and the control limits (particularly the lower limit) are appropriate for the data. The particular probability level associated with these limits is somewhat immaterial, and other values of such as 0.001 or 0.01 could be specified with the ALPHA= option in the original IRCHART statement.
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