Click SigmaXL > Templates & Calculators > Control
Chart Templates > Average Run Length (ARL)
Calculators >Attribute C ARL. This template is
also located at SigmaXL > Control Charts >
Control Chart Templates> Average Run Length (ARL)
Calculators >Attribute C ARL.
The default template settings are Specify =
Exact (Test 1 Only), In-Control Historical Mean
Count (CL) = 25.
Notes: Specify Exact (Test 1 Only) or
Monte Carlo using the drop-down list. Parameters to
be specified will be shown in yellow highlight,
otherwise they are hidden. Exact uses the Poisson
cumulative distribution function. Monte Carlo
simulation uses Poisson random data with specified
proportion and allows you to assess the ARL
performance of all 4 Tests for Special Causes. Test
1 - 1 point more than 3 standard deviations from the
center line (CL) is always applied. Monte Carlo
simulation also produces the table of Run Length
Standard Deviation and Percentiles (scroll right to
view).
Click the Calculate Attribute C ARL button to
reproduce the ARL table and chart.
The ARL0 (in-control ARL with 0 shift in
mean) for the Attribute C chart is 443.05. The ARL1
for a small positive 1 sigma shift in mean count is
30.94, so is slow to detect. On the other hand, a
large 3 sigma shift in mean count has an ARL = 2.18,
so is detected rapidly. Small negative shifts of
one-half or one sigma cannot be detected.
Note: The Process Mean Count
values are displayed on the ARL chart, but the shift
in mean count as a multiple of sigma is also
included in the table.
Now we will use Monte Carlo simulation to obtain
approximate Run Length standard deviation and
percentiles for the Attribute C chart. Select
Specify = Monte Carlo. Enter In-Control
Historical Mean Count (CL) = 25, Number of
Replications = 1e4, Test 2 = N/A, Test
3 = N/A, Test 4 = N/A.
Click the Calculate Attribute C ARL button to
produce the Monte Carlo approximate ARL table, ARL
chart and Run Length Standard Deviation and
Percentiles table (scroll right to view). Monte
Carlo simulation with 10,000 (1e4) replications will
take about a minute to run.
The additional run length statistics show the
large variation of run length values. The MRL0 = 308
(in-control median run length with 0 sigma shift in
process mean count).
Note: The results will vary
slightly since this is Monte Carlo simulation
Now we will use Monte Carlo simulation to assess the
Attribute C chart with all 4 Tests for Special
Causes applied. Enter Specify = Monte Carlo,
In-Control Historical Mean Count (CL) = 25.
Number of Replications = 1e4, Test 2 = 9,
Test 3 = 6, and Test 4 = 14.
Note: These are the test settings
used as defaults in SigmaXL > Control Charts >
‘Tests for Special Causes’ Defaults. Test 1 is
always applied.
Click the Calculate Attribute C ARL button to
produce the Monte Carlo approximate ARL table, ARL
chart and Run Length Standard Deviation and
Percentiles table.
ARL0 with all 4 tests for special causes is
approx. 151.4. This is a poor performance with a 2.9
x increase (443.1/151.4) in false alarms compared to
Exact Test 1 only. MRL0 is approx. 106. On the other
hand, ARL1 for a small 1 sigma shift in proportion
is approx. 15.1, so is much faster to detect than
the Exact Test 1 only ARL1 of 30.94. It is now
possible to detect a small negative one sigma shift
in mean count.
Template Notes:
Specify Exact (Test 1 Only) or Monte Carlo
using the drop-down list. Parameters to be specified
will be shown in yellow highlight, otherwise they are
hidden.
Exact uses the Poisson cumulative distribution
function. Monte Carlo simulation uses Poisson random
data with specified proportion and allows you to assess
the ARL performance of all 4 Tests for Special Causes.
It also produces the table of Run Length Standard
Deviation and Percentiles (scroll right to view).
Test 1 - 1 point more than 3 standard deviations
from the center line (CL) is always applied.
Enter the Subgroup Size.
Enter the In-Control Historical Mean Count (CL).
If applicable, enter Number of Replications.
1000 (1e3) replications will be fast, approx. 10
seconds, but will have an ARL0 error approx. = +/- 10%;
10,000 (1e4) replications will take about a minute, with
an ARL0 error = +/- 3.2%; 100,000 (1e5) replications
will take about ten minutes, with an ARL0 error = +/-
1%.
If applicable, select values for Tests 2 to 4 using
the drop-down list. "N/A" indicates that the test is not
applied. Tests 2 and 3 provide options that match those
provided in SigmaXL's 'Tests for Special Causes'
Defaults dialog.
Click the Calculate Attribute C ARL button to
produce the ARL table and chart. If Monte Carlo was
selected, the table of Run Length Standard Deviation and
Percentiles will also be produced.
The Attribute C ARL is for a two-sided chart with
zero-state, i.e., the shift is assumed to occur at the
start. The proportion is also assumed to be known. This
will not likely be the case in use, but is still useful
for determining parameter settings and comparison of ARL
across chart types.
The ARL Chart is similar to an Operating
Characteristic (OC) Curve, except that the Y axis is ARL
rather than Beta probability. Note that the C chart is
ARL biased for small mean counts (<40), with=with
maximum=maximum ARL=ARL occurring=occurring below=below
the=the CL.=CL.
Due to the complexity of calculations, SigmaXL must
be loaded and appear on the menu in order for this
template to function. Do not add or delete rows or
columns in this template.
