ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Control Charts for ¯ x and R Subsequent use of the charts The next 20 samples are added to the charts with the control limits derived from the first 25 samples. First the R chart: flowUse <- flow[26:45,] summary(qcc(flowDev, "R", newdata = flowUse)) Process variability looks stable, so continue with the ¯ x chart: summary(qcc(flowDev, "xbar", newdata = flowUse)) 1 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control The colored points indicate out-of-control conditions: Red points are outside the control limits; Orange points indicate a run of 7 or more points on the same side of the center line (use qcc.options(run.length = 8) to choose the more conventional rule). Red marking takes precedence over orange. Two-sigma warning limits are sometimes added to a control chart. The Western Electric rules add two-sigma and one-sigma limits, and four rules for taking action. 2 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Charts based on standard values In some situations, the process mean µ and standard deviation σ may be specified as standard values. Control limits for the ¯ x and R charts are then calculated from µ and x and ¯ σ instead of ¯ ¯ R ; necessary constants are again tabulated. This can improve the performance of the charts, but only if the standard values are actually appropriate for the process. 3 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Interpretation of ¯ x and R charts Possible patterns: Cyclic pattern; A mixture , or bimodal distribution; Shift in process level; Trend in process level; “Stratification”: all points close to the center line, usually because the control lines are too far out. 4 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Effect of nonnormality on ¯ x and R charts The α -risk of a chart is the probability of an out-of-control signal when the process is in control. For a normal distribution and known µ and σ , the α -risk of the ¯ x chart with three-sigma control limits is 0.0027. For a normal distribution and control limits constructed from ¯ ¯ x and ¯ R in m = 25 samples of size n = 5, the α -risk is around 0.0048. For a t -distribution with 10 degrees of freedom, and control limits x and ¯ constructed from ¯ ¯ R in m = 25 samples of size n = 5, the α -risk is around 0.0068 (0.0118 for df = 5). 5 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Operating Characteristic function As noted above, the α -risk of a chart is the probability of an out-of-control signal when the process is in control. The β -risk is the probability of not detecting a shift the process is not in control. The operating characteristic (OC) curve is a graph of β against the magnitude of the out-of-control change. 6 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control x chart ¯ Suppose the process mean shifts by k σ : In control: X ∼ ( µ 0 , σ 2 ). Control limits are µ 0 ± L σ/ √ n (typically L = 3). Out of control: X ∼ N ( µ 0 + k σ, σ 2 ). With samples of size n , when the mean has shifted, ¯ X ∼ N ( µ 0 + k σ, σ 2 / n ), so β = P (LCL ≤ ¯ X ≤ UCL | µ = µ 0 + k σ ) = . . . = Φ( L − k √ n ) − Φ( − L − k √ n ) . 7 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control In R: The qcc package provides a function oc.curves() that makes OC curves for an ¯ x chart made by qcc() : oc.curves(qcc(flowDev, type = "xbar", plot = FALSE)) Notes: The qcc object is used only to extract the values of the sigma-multiplier L , and the sample size n , which is emphasized on the graph; In version 2.6 of the qcc library, oc.curves does not work if you use the confidence.level option: oc.curves(qcc(flowDev, type = "xbar", plot = FALSE, confidence.level = 0.999)) but this does work in version 2.8. 8 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control R chart OC curves for the R chart are constructed similarly, using the distribution of the range instead of the normal distribution. In version 2.6 of the qcc library, oc.curves() can not make them, but in version 2.8 it can. It is convenient to modify the control limits on the R chart so that the α -risk is 0.0027, the same as for a three-sigma ¯ x chart, divided equally between the tails. In R: Specify the “confidence level” (1 − α ) instead of L : summary(qcc(flowDev, "R", confidence.level = pnorm(3) - pnorm(-3))) 9 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control In R, with version 2.6 of qcc: Graph the OC curve for the R chart with modified “three-sigma limits” and several values of sample size n , as a function of the scale change λ from 1 to 6: lambda <- seq(from = 1, to = 6, length = 100) n <- c(5, 2, 10, 15, 20) ptail <- pnorm(L, lower.tail = FALSE) beta <- outer(lambda, n, function(lambda, n) ptukey(qtukey(1 - ptail, n, Inf) / lambda, n, Inf) - ptukey(qtukey(ptail, n, Inf) / lambda, n, Inf)) matplot(lambda, beta, type = "l") legend("topright", legend = paste("n =", n), lty = 1:5, col = 1:6, inset = 0.05) title(paste("OC curves for R chart (L =", L, ")")) 10 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control In R, install version 2.8 of qcc: install.packages("githubinstall") library(githubinstall) githubinstall("qcc") If that fails: install.packages("devtools") library(devtools) install_github("luca-scr/qcc") Then: library(qcc) oc.curves(qcc(flowDev, type = "R", plot = FALSE, confidence.level = pnorm(3) - pnorm(-3))) 11 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Average run length The average run length (ARL) is the expected number of samples until the out-of-control flag is raised. As noted earlier, the in-control ARL is just ARL 0 = 1 α, and correspondingly the out-of-control ARL is 1 ARL 1 = 1 − β . 12 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control The β -risk for the ¯ x chart depends on k , the number of standard deviations of the shift, and n , the sample size. Corresponding to the OC curve, we can graph ARL 1 against k for various values of n . ARL curves can similarly be made for the R chart. Other measures related to ARL: Average time to signal (ATS) is ATS = ARL × sampling interval; Expected number of units sampled (I) is I = ARL × n . 13 / 14 Control Charts for Variables Control Charts for ¯ x and R
ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control In R: The oc.curves() function does not make ARL curves, but it does return the β values, so the ARL curve can be constructed from them: flowOc <- oc.curves(qcc(flowDev, type = "xbar", plot = FALSE)) matplot(x = seq(from = 0, to = 5, length = nrow(flowOc)), y = 1 / (1 - flowOc), type = "l", log = "y") title("ARL curves for xbar chart") legend("topright", legend = paste("n =", n), lty = 1:5, col = 1:6, inset = 0.05) 14 / 14 Control Charts for Variables Control Charts for ¯ x and R
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