attributes.chart.unbalanced: Attributes control chart with balanced/unbalanced samples

Description

Constructs the attributes control charts with balanced/unbalanced samples.

Usage

acc (x, n, type=c("p","np","c","u","g","h","t"), parameter, pEstimator=c("Wald","Wilson"),
     gEstimator = c("ML", "MVU"), tModel=c("E","W"), location.shift = 0,
     sigmaFactor=3, nk)

Value

acc returns an object of class "acc". The function summary is used to obtain and print a summary of the results and the function plot is used to plot the control chart.

Arguments

x: a numeric vector of the number of nonconforming units.
n: a numeric vector of sample sizes.
type: a character string specifying the type of control chart.
parameter: a known Bernoulli parameter value for the corresponding chart. If not known, it is estimated. For more details, refer to vignette("acc", package="rQCC").
pEstimator: a method for estimating the Bernoulli parameter for \(p\) and \(np\) charts.
gEstimator: a method for estimating the Bernoulli parameter for \(g\) and \(h\) charts.
tModel: Probability model for \(t\) chart. "E" for Exponential and "W" for Weibull.
location.shift: a known location shift parameter value for \(g\) and \(h\) charts.
sigmaFactor: a factor for the standard deviation (\(\sigma\)). For example, the American Standard uses "3*sigma" limits (0.27% false alarm rate), while the British Standard uses "3.09*sigma" limits (0.20% false alarm rate).
nk: sample size for Phase-II. If nk is missing, the average of the subsample sizes is used.

Author

Chanseok Park

Details

acc constructs the attributes control charts for nonconforming units (\(p\) and \(np\) charts) and for nonconformities per unit (\(c\) and \(u\) charts).

References

ASTM (2010). Manual on Presentation of Data and Control Chart Analysis (STP 15-D), 8th edition. American Society for Testing and Materials, West Conshohocken, PA.

Kaminsky, F. C., J. C. Benneyan and R. D. Davis (1992). Statistical Control Charts Based on a Geometric Distribution. Journal of Quality Technology, 24, 63-69.

Montgomery, D. C. (2013). Statistical Quality Control: An Modern Introduction, 7th edition. John Wiley & Sons, New York, NY.

Park, C. (2013). An Improved p Chart Based on the Wilson Interval. Journal of Statistics and Management Systems, 16(2-03), 201-221.
tools:::Rd_expr_doi("10.1080/09720510.2013.777576")

Park, C. and M. Wang (2022). A study on the g and h control charts. Communications in Statistics - Theory and Methods, To Appear.
tools:::Rd_expr_doi("10.1080/03610926.2022.2044492")

Examples

Run this code

# ==============================
# Example 1a: p chart (balanced)
# ------------------------------
# Refer to Table 31 of ASTM (2010).
x = c(1, 3, 0, 7, 2, 0, 1, 0, 8, 5, 2, 0, 1, 0, 3)
n = 400

# The conventional p chart with the balanced samples.
# Print LCL, CL, UCL.
result = acc(x, n)
print(result)

# Summary of a control chart.
summary(result)

# Plot of a control chart.
plot(result, cex.text=0.8)
text(15, 0.5, labels="p chart (with balanced sample)" )

# The p chart based on the Wilson confidence interval.
acc(x, n, pEstimator="Wilson")


# ===============================
# Example 1b: np chart (balanced)
# -------------------------------
# The data are the same as in Example 1a.
# The conventional np chart with the balanced samples.
# Print LCL, CL, UCL.
result = acc(x, n, type="np")
print(result)
summary(result)
plot(result, cex.text=0.8)
text(15, 25, labels="np chart" )

# The np chart based on the Wilson confidence interval.
acc(x, n, type="np", pEstimator="Wilson")


# ================================
# Example 2a: p chart (unbalanced)
# --------------------------------
# Refer to Table 32 of ASTM (2010).
x = c( 9, 7, 3, 9,13,14,14,10,12,14, 6,12, 7,11, 5,
       4, 2, 4, 7, 9, 7,12, 8, 5,18, 7, 8, 8,15, 3, 5)
n = c( 580, 550, 580, 640, 880, 880, 640, 550, 580, 880,
       800, 800, 580, 580, 550, 330, 330, 640, 580, 550,
       510, 640, 300, 330, 880, 880, 800, 580, 880, 880, 330)

# The conventional p chart with the unbalanced samples.
# Print LCL, CL, UCL.
result = acc(x, n, nk=880)
print(result)

# Summary of a control chart.
summary(result)

# Plot of a control chart.
plot(result, cex.text=0.8)
text(15, 0.2, labels="p chart (with unbalanced sample)" )


# ================================
# Example 2b: p chart (unbalanced)
# --------------------------------
# Refer to Table 7.4 of Montgomery (2013).
x = c(12, 8, 6, 9, 10, 12, 11, 16, 10, 6, 20, 15, 9, 8, 6, 8, 10, 7, 5, 8, 5, 8, 10, 6, 9)
n = c(100,80,80,100,110,110,100,100,90,90,110,120,120,120,110,80,80,80,90,100,100,100,100,90,90)

# The conventional p chart with the unbalanced samples.
# Print LCL, CL, UCL.
# If nk is missing, the average sample size is used.
result = acc(x, n)
print(result)

# Summary of a control chart.
summary(result)

# Plot of a control chart.
# Refer to Figure 7.8 of Montgomery (2013).
plot(result, cex.text=0.8)
text(15, 0.2, labels="p chart (with unbalanced sample)" )


# ================================
# Example 2c: p chart (unbalanced)
#             p is known 
# --------------------------------
# Refer to Table 41 of ASTM (2010).
x = c(2, 2, 1, 1, 5, 2, 0, 3, 0, 15, 7, 2, 5, 2, 0, 3, 0, 4, 8, 4)
n = c(600,1300,2000,2500,1550,2000,1550,780,260,2000,1550,950,950,950,35,330,200,600,1300,780)

# The fraction nonconforming is known as 0.0020
# The control limits at the size nk=600.
# If nk (sample size for Phase II) is unknown, the average of subsample sizes is used.
result = acc(x, n, parameter=0.002, nk=600)
summary(result)


# ===============================
# Example 3a: u chart (balanced)
# -------------------------------
# Refer to Table 33 of ASTM (2010).
x = c(17, 14, 6, 23,  5,  7, 10, 19, 29, 18, 25, 5,
       8, 11, 18, 13, 22, 6, 23, 22,  9, 15, 20, 6, 24)
n = 10

# The u chart with the balanced samples.
# Print LCL, CL, UCL.
result = acc(x, n, type="u")
print(result)

# Summary of a control chart
summary(result)

# Plot of a control chart
plot(result, cex.text=0.8)
text(13, 3, labels="u chart" )


# ================================
# Example 3b: u chart (unbalanced)
# --------------------------------
# Refer to Table 34 of ASTM (2010).
x = c(72, 38, 76, 35, 62, 81, 97, 78, 103, 56,
      47, 55, 49, 62, 71, 47, 41, 52, 128, 84)
n = c(20, 20, 40, 25, 25, 25, 40, 40, 40, 40,
      25, 25, 25, 25, 25, 20, 20, 20, 40, 40)

# The u chart with the unbalanced samples.
# Print LCL, CL, UCL.
result = acc(x, n, type="u", nk=20)
print(result)

# Summary of a control chart
summary(result)

# Plot of a control chart
plot(result, cex.text=0.8)
text(12, 3.5, labels="u chart (with unbalanced sample)" )


# ===============================
# Example 4: c chart 
# -------------------------------
# Refer to Table 35 of ASTM (2010).
x = c(0, 1, 1, 0, 2, 1, 3, 4, 5, 3, 0, 1, 1, 1, 2, 4, 0, 1, 1, 0,
      6, 4, 3, 2, 0, 0, 9,10, 8, 8, 6,14, 0, 1, 2, 4, 5, 7, 1, 3,
      3, 2, 0, 1, 5, 3, 4, 3, 5, 4, 2, 0, 1, 2, 5, 9, 4, 2, 5, 3)

# Print LCL, CL, UCL.
result = acc(x, type="c")
print(result)

# Summary of a control chart
summary(result)

# Plot of a control chart
plot(result, cex.text=0.8)
text(40, 14, labels="c chart" )


# ===============================
# Example 5: g and h charts
# -------------------------------
# Refer to Kaminsky et al. (1992).
tmp = c(
11,  2,  8,  2, 4,   1,  1, 11,  2, 1,   1,  7,  1,  1, 9, 
 5,  1,  3,  6, 5,  13,  2,  3,  3, 4,   3,  2,  6,  1, 5,  
 2,  2,  8,  3, 1,   1,  3,  4,  6, 5,   2,  8,  1,  1, 4,  
13, 10, 15,  5, 2,   3,  6,  1,  5, 8,   9,  1, 18,  3, 1,  
 3,  7, 14,  3, 1,   7,  7,  1,  8, 1,   4,  1,  6,  1, 1, 
 1, 14,  2,  3, 7,  19,  9,  7,  1, 8,   5,  1,  1,  6, 1,  
 9,  5,  6,  2, 2,   8, 15,  2,  3, 3,   4,  7, 11,  4, 6,  
 7,  5,  1, 14, 8,   3,  3,  5, 21,10,  11,  1,  6,  1, 2,  
 4,  1,  2, 11, 5,   3,  5,  4, 10, 3,   1,  4,  7,  3, 2, 
 3,  5,  4,  2, 3,   5,  1,  4, 11,17,   1, 13, 13,  2, 1)  
data = matrix(tmp, byrow=TRUE, ncol=5)

# g chart with ML method.
# Print LCL, CL, UCL.
result = acc(data, type="g", location=1)
print(result)

# Summary of a control chart
summary(result)

plot(result, cex.text=0.8)

# h chart with MVU method.
acc(data, type="h", location=1, gEstimator="MVU")


# ===============================
# Example 6: g and h charts (unbalanced data)
# -------------------------------
x1 = c(11, 2,  8,  2, 4)
x2 = c(1,  1, 11,  2, 1)
x3 = c(1,  7,  1)
x4 = c(5,  1,  3,  6, 5)
x5 = c(13, 2,  3,  3)
x6 = c(3,  2,  6,  1, 5)
x7 = c(2,  2,  8,  3, 1)
x8 = c(1,  3,  4,  6, 5)
x9 = c(2,  8,  1,  1, 4)
data = list(x1, x2, x3, x4, x5, x6, x7, x8, x9)

result = acc(data, type="g", location=1, gEstimator="MVU", nk=5)
summary(result)
plot(result)


# ===============================
# Example 7: t charts 
# -------------------------------
x = c(0.35, 0.92, 0.59, 4.28, 0.21, 0.79, 1.75, 0.07, 3.3, 
1.7, 0.33, 0.97, 0.96, 2.23, 0.88, 0.37, 1.3, 0.4, 0.19, 1.59)

# Exponential t chart
result = acc(x, type="t", tModel="E")
summary(result)

plot(result, cex.text=0.8)
text(10, 6, labels="Exponential t chart" )


# Weibull t chart
result = acc(x, type="t", tModel="W")
summary(result)

plot(result, cex.text=0.8)
text(10, 6, labels="Weibull t chart" )

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