vignettes/high-quality-processes.Rmd
high-quality-processes.RmdHigh-quality processes are processes in which nonconformities are rare. In that setting, attribute data are often highly discrete, bounded below by zero, and strongly asymmetric. Classical normal-approximation control limits may therefore be poorly calibrated, even when the usual chart is familiar and easy to apply.
This is one of the main motivations for IQCC. The package includes methods for improving the calibration of attribute charts and now also includes the numerical core for double-sampling np charts.
A standard Shewhart p chart usually relies on a normal approximation to the binomial distribution. This can work well when the subgroup size is large and the proportion of nonconforming items is not too close to zero or one.
In high-quality processes, however, the in-control proportion
p0 can be very small. Then the number of nonconforming
units in a subgroup is better understood as a binomial count with a
skewed and discrete distribution. A symmetric three-sigma approximation
can produce limits with an actual false alarm risk that differs from the
intended one.
IQCC implements a Cornish-Fisher corrected p chart through
cchart.p() with type = "CF". The goal is to
keep a familiar p-chart interface while using a limit calculation that
better reflects the distributional asymmetry of the binomial
statistic.
library(IQCC)
data(binomdata)
cchart.p(
x1 = binomdata$Di[1:12],
n1 = binomdata$ni[1:12],
type = "CF",
x2 = binomdata$Di[13:25],
n2 = binomdata$ni[13:25]
)Use this option when the data are proportions of nonconforming items and the ordinary normal approximation is questionable because the estimated proportion is small.
Double-sampling np charts use two possible sampling stages. A first sample is inspected. If the count of nonconforming items is clearly acceptable, the process is not signaled. If it is clearly large, the chart signals. If the count lies in an intermediate region, a second sample is inspected and the combined count is used for the final decision.
The DS-np method is useful because it can reduce the average sample size while maintaining a desired monitoring performance. This is particularly relevant when inspection is expensive and nonconformities are rare.
IQCC currently provides the numerical core for DS-np charts:
dsnp_prob_accept() computes the total acceptance
probability for a proposed plan.dsnp_arl() computes the average run length for a
proposed plan.dsnp_ass() computes the average sample size for a
proposed plan.The automatic limit search and the plotting interface are planned as subsequent steps.
The following plan is written in the fractional-limit convention used by the DS-np numerical core. Fractional limits avoid ambiguity when the monitored statistic is an integer count.
library(IQCC)
n1 <- 34
n2 <- 162
wl <- 1.5
ucl1 <- 2.5
ucl2 <- 4.5
# In-control proportion
p0 <- 0.005
# Alternative proportion
p1 <- 0.0075
dsnp_prob_accept(p0, n1, n2, wl, ucl1, ucl2)
dsnp_arl(p0, n1, n2, wl, ucl1, ucl2)
dsnp_ass(p0, n1, n2, wl, ucl1)
dsnp_arl(p1, n1, n2, wl, ucl1, ucl2)This separates the numerical design problem from the eventual control-chart plot. Before plotting a chart, it should be possible to inspect the operating characteristics of the proposed limits.
For a DS-np plan with first-stage count D1 and
second-stage count D2, IQCC uses the following
interpretation:
D1 <= floor(wl);D1 >= floor(ucl1) + 1;D1 + D2 <= floor(ucl2).For example, with wl = 1.5, ucl1 = 2.5, and
ucl2 = 4.5:
D1 <= 1;D1 = 2;D1 >= 3;D1 + D2 <= 4.A practical high-quality-process workflow in IQCC is:
cchart.p(type = "CF") when monitoring the
proportion of nonconforming items and subgroup sizes are known.dsnp_limits() and
cchart.DSnp() interfaces once they are implemented.The high-quality-process functionality is being expanded incrementally. The current development version includes the DS-np numerical core. Future PRs should add automatic limit search, a plotting wrapper, and examples that connect the DS-np functions to a complete monitoring workflow.