Multivariate Quality Control Charts

Create an object of class 'mqcc' to perform multivariate statistical quality control.

mqcc(data, type = c("T2", "T2.single"), center, cov,
     limits = TRUE, pred.limits = FALSE,
     data.name, labels, newdata, newlabels, 
     confidence.level = (1 - 0.0027)^p,
     plot = TRUE, ...)

# S3 method for mqcc
print(x, digits = getOption("digits"), ...)

# S3 method for mqcc
plot(x, 
     add.stats = qcc.options("add.stats"), 
     chart.all = qcc.options("chart.all"), 
     fill = qcc.options("fill"),
     label.limits = c("LCL", "UCL"), 
     label.pred.limits = c("LPL", "UPL"),
     title, xlab, ylab, ylim, axes.las = 0, 
     digits = getOption("digits"), 
     restore.par = TRUE, ...)

Arguments

data

For subgrouped data, a list with a data frame or a matrix for each variable to monitor. Each row of the data frame or matrix refers to a sample or ''rationale'' group. For individual observations, where each sample has a single observation, users can provide a list with a data frame or a matrix having a single column, or a data frame or a matrix where each rows refer to samples and columns to variables. See examples.

type

a character string specifying the type of chart:

	Chart description
`"T2"`	Hotelling $T^2$ chart for subgrouped data
`"T2.single"`	Hotelling $T^2$ chart for individual observations

center

a vector of values to use for center of input variables.

cov

a matrix of values to use for the covariance matrix of input variables.

limits

a logical indicating if control limits (Phase I) must be computed (by default using limits.T2 or limits.T2.single) and plotted, or a two-values vector specifying control limits.

pred.limits

a logical indicating if prediction limits (Phase II) must be computed (by default using limits.T2 or limits.T2.single) and plotted, or a two-values vector specifying prediction limits.

data.name

a string specifying the name of the variable which appears on the plots. If not provided is taken from the object given as data.

labels

a character vector of labels for each group.

newdata

a data frame, matrix or vector, as for the data argument, providing further data to plot but not included in the computations.

newlabels

a character vector of labels for each new group defined in the argument newdata.

confidence.level

a numeric value between 0 and 1 specifying the confidence level of the computed probability limits. By default is set at $(1 - 0.0027)^p$ where $p$ is the number of variables, and $0.0027$ is the probability of Type I error for a single Shewhart chart at the usual 3-sigma control level.

plot

logical. If TRUE a quality chart is plotted.

add.stats

a logical value indicating whether statistics and other information should be printed at the bottom of the chart.

chart.all

a logical value indicating whether both statistics for data and for newdata (if given) should be plotted.

fill

a logical value specifying if the in-control area should be filled with the color specified in qcc.options("zones")$fill.

label.limits

a character vector specifying the labels for control limits (Phase I).

label.pred.limits

a character vector specifying the labels for prediction control limits (Phase II).

title

a character string specifying the main title. Set title = FALSE or title = NA to remove the title.

xlab

a string giving the label for the x-axis.

ylab

a string giving the label for the y-axis.

ylim

a numeric vector specifying the limits for the y-axis.

axes.las

numeric in {0,1,2,3} specifying the style of axis labels. See help(par).

digits

the number of significant digits to use when add.stats = TRUE.

restore.par

a logical value indicating whether the previous par settings must be restored. If you need to add points, lines, etc. to a control chart set this to FALSE.

x

an object of class 'mqcc'.

...

additional arguments to be passed to the generic function.

Value

Returns an object of class 'mqcc'.

References

Mason, R.L. and Young, J.C. (2002) Multivariate Statistical Process Control with Industrial Applications, SIAM.

Montgomery, D.C. (2013) Introduction to Statistical Quality Control, 7th ed. New York: John Wiley & Sons.

Ryan, T. P. (2011), Statistical Methods for Quality Improvement, 3rd ed. New York: John Wiley & Sons, Inc.

Scrucca, L. (2004). qcc: an R package for quality control charting and statistical process control. R News 4/1, 11-17.

Wetherill, G.B. and Brown, D.W. (1991) Statistical Process Control. New York: Chapman & Hall.

Author

Luca Scrucca

Examples

##
##  Subgrouped data
##

data(RyanMultivar)
str(RyanMultivar)
#> List of 2
#>  $ X1: num [1:20, 1:4] 72 56 55 44 97 83 47 88 57 26 ...
#>  $ X2: num [1:20, 1:4] 23 14 13 9 36 30 12 31 14 7 ...

q <- mqcc(RyanMultivar, type = "T2")

summary(q)
#> ── Multivariate Quality Control Chart ──────────── 
#> 
#> Chart type                 = T2 
#> Data (phase I)             = RyanMultivar 
#> Number of groups           = 20 
#> Group sample size          = 4 
#> Center = 
#>      X1      X2 
#> 60.3750 18.4875 
#> Covariance matrix = 
#>          X1        X2
#> X1 222.0333 103.11667
#> X2 103.1167  56.57917
#> |S| = 1929.414 
#> 
#> Control limits:
#>  LCL      UCL
#>    0 11.03976
ellipseChart(q)

ellipseChart(q, show.id = TRUE)

q <- mqcc(RyanMultivar, type = "T2", pred.limits = TRUE)


# Xbar-charts for single variables computed adjusting the 
# confidence level of the T^2 chart:
q1 <- with(RyanMultivar, 
           qcc(X1, type = "xbar", confidence.level = q$confidence.level^(1/2)))
summary(q1)
#> ── Quality Control Chart ───────────────────────── 
#> 
#> Chart type                 = xbar 
#> Data (phase I)             = X1 
#> Number of groups           = 20 
#> Group sample size          = 4 
#> Center of group statistics = 60.375 
#> Standard deviation         = 14.93443 
#> 
#> Control limits at nsigmas  = 2.999977 
#>       LCL      UCL
#>  37.97352 82.77648
q2 <- with(RyanMultivar,
           qcc(X2, type = "xbar", confidence.level = q$confidence.level^(1/2)))
summary(q2)
#> ── Quality Control Chart ───────────────────────── 
#> 
#> Chart type                 = xbar 
#> Data (phase I)             = X2 
#> Number of groups           = 20 
#> Group sample size          = 4 
#> Center of group statistics = 18.4875 
#> Standard deviation         = 7.139388 
#> 
#> Control limits at nsigmas  = 2.999977 
#>     LCL     UCL
#>  7.7785 29.1965

require(MASS)
#> Loading required package: MASS
# generate new "in control" data
Xnew <- list(X1 = matrix(NA, 10, 4), X2 =  matrix(NA, 10, 4))
for(i in 1:4)
   { x <- mvrnorm(10, mu = q$center, Sigma = q$cov)
     Xnew$X1[,i] <- x[,1]
     Xnew$X2[,i] <- x[,2] 
   }
qq <- mqcc(RyanMultivar, type = "T2", newdata = Xnew, pred.limits = TRUE)

summary(qq)
#> ── Multivariate Quality Control Chart ──────────── 
#> 
#> Chart type                 = T2 
#> Data (phase I)             = RyanMultivar 
#> Number of groups           = 20 
#> Group sample size          = 4 
#> Center = 
#>      X1      X2 
#> 60.3750 18.4875 
#> Covariance matrix = 
#>          X1        X2
#> X1 222.0333 103.11667
#> X2 103.1167  56.57917
#> |S| = 1929.414 
#> 
#> New data (phase II)        = Xnew 
#> Number of groups           = 10 
#> Group sample size          = 4 
#> 
#> Control limits:
#>  LCL      UCL
#>    0 11.03976
#> 
#> Prediction limits:
#>  LPL      UPL
#>    0 12.20184

# generate new "out of control" data
Xnew <- list(X1 = matrix(NA, 10, 4), X2 =  matrix(NA, 10, 4))
for(i in 1:4)
   { x <- mvrnorm(10, mu = 1.2*q$center, Sigma = q$cov)
     Xnew$X1[,i] <- x[,1]
     Xnew$X2[,i] <- x[,2] 
   }
qq <- mqcc(RyanMultivar, type = "T2", newdata = Xnew, pred.limits = TRUE)

summary(qq)
#> ── Multivariate Quality Control Chart ──────────── 
#> 
#> Chart type                 = T2 
#> Data (phase I)             = RyanMultivar 
#> Number of groups           = 20 
#> Group sample size          = 4 
#> Center = 
#>      X1      X2 
#> 60.3750 18.4875 
#> Covariance matrix = 
#>          X1        X2
#> X1 222.0333 103.11667
#> X2 103.1167  56.57917
#> |S| = 1929.414 
#> 
#> New data (phase II)        = Xnew 
#> Number of groups           = 10 
#> Group sample size          = 4 
#> 
#> Control limits:
#>  LCL      UCL
#>    0 11.03976
#> 
#> Prediction limits:
#>  LPL      UPL
#>    0 12.20184

##
## Individual observations data
##

data(boiler)
str(boiler)
#> 'data.frame':	25 obs. of  8 variables:
#>  $ t1: int  507 512 520 520 530 528 522 527 533 530 ...
#>  $ t2: int  516 513 512 514 515 516 513 509 514 512 ...
#>  $ t3: int  527 533 537 538 542 541 537 537 528 538 ...
#>  $ t4: int  516 518 518 516 525 524 518 521 529 524 ...
#>  $ t5: int  499 502 503 504 504 505 503 504 508 507 ...
#>  $ t6: int  512 510 512 517 512 514 512 508 512 512 ...
#>  $ t7: int  472 476 480 480 481 482 479 478 482 482 ...
#>  $ t8: int  477 475 477 479 477 480 477 472 477 477 ...

q <- mqcc(boiler, type = "T2.single", confidence.level = 0.999)

summary(q)
#> ── Multivariate Quality Control Chart ──────────── 
#> 
#> Chart type                 = T2.single 
#> Data (phase I)             = boiler 
#> Number of groups           = 25 
#> Group sample size          = 1 
#> Center = 
#>     t1     t2     t3     t4     t5     t6     t7     t8 
#> 525.00 513.56 538.92 521.68 503.80 512.44 478.72 477.24 
#> Covariance matrix = 
#>            t1        t2        t3         t4         t5         t6         t7
#> t1 54.0000000 0.9583333 20.583333 31.2916667 20.3333333 -2.2916667 20.3750000
#> t2  0.9583333 4.8400000  2.963333  2.6866667  0.3250000  3.0766667  0.7050000
#> t3 20.5833333 2.9633333 22.993333 10.0566667  4.9833333  2.0366667  6.4766667
#> t4 31.2916667 2.6866667 10.056667 22.3100000 13.5583333 -2.2700000 12.6983333
#> t5 20.3333333 0.3250000  4.983333 13.5583333 11.4166667 -1.6583333 10.6500000
#> t6 -2.2916667 3.0766667  2.036667 -2.2700000 -1.6583333  4.5066667 -0.7466667
#> t7 20.3750000 0.7050000  6.476667 12.6983333 10.6500000 -0.7466667 11.6266667
#> t8  0.2083333 3.4433333  2.770000  0.1633333 -0.2833333  3.7233333  0.5283333
#>            t8
#> t1  0.2083333
#> t2  3.4433333
#> t3  2.7700000
#> t4  0.1633333
#> t5 -0.2833333
#> t6  3.7233333
#> t7  0.5283333
#> t8  3.8566667
#> |S| = 8313.241 
#> 
#> Control limits:
#>  LCL      UCL
#>    0 17.41705

# generate new "in control" data
boilerNew <- mvrnorm(10, mu = q$center, Sigma = q$cov)
qq <- mqcc(boiler, type = "T2.single", confidence.level = 0.999, 
           newdata = boilerNew, pred.limits = TRUE)

summary(qq)
#> ── Multivariate Quality Control Chart ──────────── 
#> 
#> Chart type                 = T2.single 
#> Data (phase I)             = boiler 
#> Number of groups           = 25 
#> Group sample size          = 1 
#> Center = 
#>     t1     t2     t3     t4     t5     t6     t7     t8 
#> 525.00 513.56 538.92 521.68 503.80 512.44 478.72 477.24 
#> Covariance matrix = 
#>            t1        t2        t3         t4         t5         t6         t7
#> t1 54.0000000 0.9583333 20.583333 31.2916667 20.3333333 -2.2916667 20.3750000
#> t2  0.9583333 4.8400000  2.963333  2.6866667  0.3250000  3.0766667  0.7050000
#> t3 20.5833333 2.9633333 22.993333 10.0566667  4.9833333  2.0366667  6.4766667
#> t4 31.2916667 2.6866667 10.056667 22.3100000 13.5583333 -2.2700000 12.6983333
#> t5 20.3333333 0.3250000  4.983333 13.5583333 11.4166667 -1.6583333 10.6500000
#> t6 -2.2916667 3.0766667  2.036667 -2.2700000 -1.6583333  4.5066667 -0.7466667
#> t7 20.3750000 0.7050000  6.476667 12.6983333 10.6500000 -0.7466667 11.6266667
#> t8  0.2083333 3.4433333  2.770000  0.1633333 -0.2833333  3.7233333  0.5283333
#>            t8
#> t1  0.2083333
#> t2  3.4433333
#> t3  2.7700000
#> t4  0.1633333
#> t5 -0.2833333
#> t6  3.7233333
#> t7  0.5283333
#> t8  3.8566667
#> |S| = 8313.241 
#> 
#> New data (phase II)        = boilerNew 
#> Number of groups           = 10 
#> Group sample size          = 1 
#> 
#> Control limits:
#>  LCL      UCL
#>    0 17.41705
#> 
#> Prediction limits:
#>  LPL      UPL
#>    0 70.02943

# generate new "out of control" data
boilerNew <- mvrnorm(10, mu = 1.01*q$center, Sigma = q$cov)
qq <- mqcc(boiler, type = "T2.single", confidence.level = 0.999, 
           newdata = boilerNew, pred.limits = TRUE)

summary(qq)
#> ── Multivariate Quality Control Chart ──────────── 
#> 
#> Chart type                 = T2.single 
#> Data (phase I)             = boiler 
#> Number of groups           = 25 
#> Group sample size          = 1 
#> Center = 
#>     t1     t2     t3     t4     t5     t6     t7     t8 
#> 525.00 513.56 538.92 521.68 503.80 512.44 478.72 477.24 
#> Covariance matrix = 
#>            t1        t2        t3         t4         t5         t6         t7
#> t1 54.0000000 0.9583333 20.583333 31.2916667 20.3333333 -2.2916667 20.3750000
#> t2  0.9583333 4.8400000  2.963333  2.6866667  0.3250000  3.0766667  0.7050000
#> t3 20.5833333 2.9633333 22.993333 10.0566667  4.9833333  2.0366667  6.4766667
#> t4 31.2916667 2.6866667 10.056667 22.3100000 13.5583333 -2.2700000 12.6983333
#> t5 20.3333333 0.3250000  4.983333 13.5583333 11.4166667 -1.6583333 10.6500000
#> t6 -2.2916667 3.0766667  2.036667 -2.2700000 -1.6583333  4.5066667 -0.7466667
#> t7 20.3750000 0.7050000  6.476667 12.6983333 10.6500000 -0.7466667 11.6266667
#> t8  0.2083333 3.4433333  2.770000  0.1633333 -0.2833333  3.7233333  0.5283333
#>            t8
#> t1  0.2083333
#> t2  3.4433333
#> t3  2.7700000
#> t4  0.1633333
#> t5 -0.2833333
#> t6  3.7233333
#> t7  0.5283333
#> t8  3.8566667
#> |S| = 8313.241 
#> 
#> New data (phase II)        = boilerNew 
#> Number of groups           = 10 
#> Group sample size          = 1 
#> 
#> Control limits:
#>  LCL      UCL
#>    0 17.41705
#> 
#> Prediction limits:
#>  LPL      UPL
#>    0 70.02943

# provides "robust" estimates of means and covariance matrix
rob <- cov.rob(boiler)
qrob <- mqcc(boiler, type = "T2.single", center = rob$center, cov = rob$cov)

summary(qrob)
#> ── Multivariate Quality Control Chart ──────────── 
#> 
#> Chart type                 = T2.single 
#> Data (phase I)             = boiler 
#> Number of groups           = 25 
#> Group sample size          = 1 
#> Center = 
#>     t1     t2     t3     t4     t5     t6     t7     t8 
#> 525.90 513.45 539.50 521.90 503.70 512.45 478.85 477.25 
#> Covariance matrix = 
#>           t1       t2        t3          t4         t5        t6          t7
#> t1 31.147368 3.731579  7.000000 25.09473684 15.8631579 -3.005263 14.98421053
#> t2  3.731579 5.523684  4.815789  4.04736842  1.1421053  3.839474  1.75526316
#> t3  7.000000 4.815789 17.315789  5.31578947  1.4736842  2.236842  2.18421053
#> t4 25.094737 4.047368  5.315789 20.93684211 13.1789474 -1.531579 12.35263158
#> t5 15.863158 1.142105  1.473684 13.17894737 11.0631579 -2.121053 10.21578947
#> t6 -3.005263 3.839474  2.236842 -1.53157895 -2.1210526  4.155263 -1.50789474
#> t7 14.984211 1.755263  2.184211 12.35263158 10.2157895 -1.507895 10.97631579
#> t8 -1.184211 4.197368  2.500000  0.02631579 -0.7105263  4.039474  0.09210526
#>             t8
#> t1 -1.18421053
#> t2  4.19736842
#> t3  2.50000000
#> t4  0.02631579
#> t5 -0.71052632
#> t6  4.03947368
#> t7  0.09210526
#> t8  4.40789474
#> |S| = 207.7032 
#> 
#> Control limits:
#>  LCL      UCL
#>    0 14.26225