Double-Sampling np Control Chart

Build and optionally plot a double-sampling np (DS-np) control chart for monitoring the nonconforming proportion in high-quality processes.

cchart.DSnp(
  x1,
  n1,
  n2,
  p0,
  x2 = NULL,
  wl = NULL,
  ucl1 = NULL,
  ucl2 = NULL,
  limits = NULL,
  alpha = 0.0027,
  p1 = NULL,
  plot = TRUE,
  ...
)

Arguments

x1: Integer vector of nonconforming counts from the first sample.
n1: First-stage sample size (positive integer).
n2: Second-stage sample size (positive integer).
p0: In-control nonconforming proportion.
x2: Optional integer vector of nonconforming counts from the second sample. Must have the same length as x1 when provided. Use NA for observations where no second sample was taken. Required for observations where x1 falls in the intermediate (warning) zone.
wl: Fractional warning limit. Must be less than ucl1.
ucl1: Fractional upper control limit for the first stage. Must be greater than wl.
ucl2: Fractional upper control limit for the combined samples.
limits: Optional object returned by dsnp_limits(). When provided, wl, ucl1, and ucl2 are taken from limits$best.
alpha: Maximum desired false alarm probability at p0. Used only when limits need to be computed via dsnp_limits().
p1: Optional out-of-control proportion. When provided, performance metrics at p1 are included in the returned object.
plot: Logical. If TRUE (default), draws the control chart. If FALSE, only returns the result object.
...: Additional arguments passed to plot() (currently unused).

Value

An object of class "cchart.DSnp", which is a list with the following elements:

call: The matched call.
data: A data.frame with columns index, x1, x2, total, stage, and signal.
limits: A list with the fractional and integer thresholds: wl, ucl1, ucl2, wl_accept, ucl1_reject, ucl2_accept.
parameters: A list with n1, n2, p0, alpha, and p1.
performance: A list with in-control performance metrics (arl0, ass0, p_signal0), and optionally out-of-control metrics (arl1, ass1, p_signal1) when p1 is provided.

Details

The DS-np chart uses two sampling stages. At the first stage, a sample of size n1 is inspected. If the count x1 is at or below the warning limit, the process is accepted. If x1 exceeds the first upper control limit, the process signals out-of-control immediately. Otherwise, a second sample of size n2 is inspected and the combined count x1 + x2 is compared to the second upper control limit.

Limits can be supplied manually via wl, ucl1, and ucl2, obtained from a pre-computed dsnp_limits() object via the limits argument, or computed automatically inside the function when neither is provided.

The fractional limits are converted to integer thresholds using the convention from the numerical core functions:

wl_accept = floor(wl): accept at first stage if x1 <= wl_accept.
ucl1_reject = floor(ucl1) + 1: signal at first stage if x1 >= ucl1_reject.
ucl2_accept = floor(ucl2): accept at second stage if x1 + x2 <= ucl2_accept.

References

Joekes, S., Smrekar, M. and Barbosa, E. (2015). Extending a double sampling control chart for non-conforming proportion in high quality processes to the case of small samples. Statistical Methodology.

Author

Daniela R. Recchia, Emanuel P. Barbosa

Examples


# Small example with manual limits
x1 <- c(0, 1, 2, 3, 1, 0, 2, 4, 1, 0)
x2 <- c(NA, NA, 2, NA, NA, NA, 3, NA, NA, NA)
res <- cchart.DSnp(x1, n1 = 10, n2 = 20, p0 = 0.05,
                   x2 = x2, wl = 1.5, ucl1 = 2.5, ucl2 = 4.5,
                   plot = FALSE)
res$limits
#> $wl
#> [1] 1.5
#> 
#> $ucl1
#> [1] 2.5
#> 
#> $ucl2
#> [1] 4.5
#> 
#> $wl_accept
#> [1] 1
#> 
#> $ucl1_reject
#> [1] 3
#> 
#> $ucl2_accept
#> [1] 4
#> 
res$performance
#> $arl0
#> [1] 58.35236
#> 
#> $ass0
#> [1] 11.4927
#> 
#> $p_signal0
#> [1] 0.01713727
#>