1 ARL Table for X-bar Chart

To create an ARL table for an X-bar chart, we do the following. Consider as an example a process with a mean of 10, standard deviation of 1, and subgroup size of 9.

1.1 Define the ARL function

ARL<-function(mu_0,mu_1,sigma_0,n){
  UCL<-mu_0+3*sigma_0/sqrt(n)
  LCL<-mu_0-3*sigma_0/sqrt(n)
  beta<-pnorm(q=UCL,mean=mu_1,sd=sigma_0/sqrt(n))-pnorm(q=LCL,mean=mu_1,sd=sigma_0/sqrt(n))
  1/(1-beta)
}

1.2 Table of ARLs

Define a table of ARLs for an in-control mean of \(\mu_o=10\), \(\sigma_o=1\), and \(n=9\).

vals<-c(9.2,9.4,9.6,9.8,10,10.2,10.4,10.6,10.8)
df<-data.frame(vals,ARL(10,vals,1,9))
colnames(df)<-c("mu","ARL")
#use kable() to make the table output "pretty"
kable(df,caption = "ARL Table for X-Bar Chart with mu_o=10",align=c("c","c"),digits=2) %>% kable_styling(full_width=T,"striped") %>% row_spec(5, bold=TRUE)
ARL Table for X-Bar Chart with mu_o=10
mu ARL
9.2 3.65
9.4 8.69
9.6 27.82
9.8 119.67
10.0 370.40
10.2 119.67
10.4 27.82
10.6 8.69
10.8 3.65

1.3 Plot of ARLs

This is a plot the ARLs from \(\mu_1=9\) to \(\mu_1=11\).

vals<-seq(9,11,.01)
plot(vals,ARL(mu_0=10,vals,sigma_0=1,n=9),
     type="l",main="ARL Plot for X-Bar Chart with n=9 and mu_0=10",
     xlab="mu_1",
     ylab="Avg Number of Subgroups to Signal Out-of-Control")
abline(v=10,lty=2)

1.4 Generate \(\bar{X}\) chart

Generate some sample data from a Normal(10,1) distribution. Arrange into \(m=20\) subgroups of size \(n=5\). Create \(\bar{X}\) chart.

testdat<-rnorm(100,10,1) #generate 100 random variates from a Normal(10,1) distribution
testmatrix<-matrix(testdat,20,5) #put into a matrix representing m=20 subgroups of size n=5
qcc(testmatrix, type = "xbar", title = "X-bar Chart")

## List of 11
##  $ call      : language qcc(data = testmatrix, type = "xbar", title = "X-bar Chart")
##  $ type      : chr "xbar"
##  $ data.name : chr "testmatrix"
##  $ data      : num [1:20, 1:5] 10.58 10.97 9.27 9.5 10.06 ...
##   ..- attr(*, "dimnames")=List of 2
##  $ statistics: Named num [1:20] 10.91 9.93 10.03 10.19 9.79 ...
##   ..- attr(*, "names")= chr [1:20] "1" "2" "3" "4" ...
##  $ sizes     : int [1:20] 5 5 5 5 5 5 5 5 5 5 ...
##  $ center    : num 9.92
##  $ std.dev   : num 1.11
##  $ nsigmas   : num 3
##  $ limits    : num [1, 1:2] 8.44 11.41
##   ..- attr(*, "dimnames")=List of 2
##  $ violations:List of 2
##  - attr(*, "class")= chr "qcc"

2 ARL Table for np-chart

Generate and ARL table for an np-chart

2.1 Define ARL Function for the Binomial

ARL<-function(p_0,p_1,n){
  UCL<-n*p_0+3*sqrt(n*p_0*(1-p_0))
  LCL<-n*p_0-3*sqrt(n*p_0*(1-p_0))
  if(LCL<0) beta=pbinom(UCL,n,p_1) else beta=pbinom(UCL,n,p_1)-pbinom(LCL,n,p_1)
  1/(1-beta)
}

2.2 Table of ARLs for np-chart

Define a table of ARLs for subgroup sizes of \(n=100\) and an in-control proportion of defects being \(p_0=0.02\).

vals<-c(0.005,0.01,0.015,0.02,0.025,0.03,0.035,0.04)
df<-data.frame(vals,ARL(.02,vals,100))
colnames(df)<-c("p_0","ARL")
#use kable() to make the table output "pretty"
kable(df,caption = "ARL Table for np-chart with $p_o=.02$.  Note that chart cannot detect downward shifts since the LCL=0, which correspondings entries are greyed out.  The in-control ARL is in red.",align=c("c","c"),digits=3) %>% kable_styling(full_width=T,"striped") %>% row_spec(4, col="red") %>% row_spec(1:3,col="grey")
ARL Table for np-chart with \(p_o=.02\). Note that chart cannot detect downward shifts since the LCL=0, which correspondings entries are greyed out. The in-control ARL is in red.
p_0 ARL
0.005 1200586.272
0.010 14067.930
0.015 1233.277
0.020 246.181
0.025 77.056
0.030 32.023
0.035 16.166
0.040 9.399

2.3 Plot of ARLs for np-chart

This is a plot the ARLs from \(p_0=.02\) to \(p_1=.05\).

vals<-seq(.02,.05,.001)
plot(vals,ARL(p_0=.02,vals,n=100),
     type="l",main="ARL Plot for np-chart with n=100 and p_0=.02",
     xlab="p_1",
     ylab="Avg Number of Subgroups to Signal Out-of-Control")

3 ARL Table for a C-Chart

Generate and ARL table for a c-chart

3.1 Define ARL Function for the Poisson

ARL<-function(lambda_0,lambda_1){
  UCL<-lambda_0+3*sqrt(lambda_0)
  LCL<-lambda_0-3*sqrt(lambda_0)
  if(LCL<0) beta=ppois(UCL,lambda_1) else beta=ppois(UCL,lambda_1)-ppois(LCL,lambda_1)
  1/(1-beta)
}

3.2 Table of ARLs for np-chart

Define a table of ARLs with an in-control rate of defect being \(\lambda_0=4\).

vals<-c(2,2.5,3,3.5,4,4.5,5,5.5,6)
df<-data.frame(vals,ARL(4,vals))
colnames(df)<-c("lambda_0","ARL")
#use kable() to make the table output "pretty"
kable(df,caption = "ARL Table for c-chart with lambda_0=4.  Note that chart cannot detect downward shifts since the LCL=0, which correspondings entries are greyed out.  The in-control ARL is in red.",align=c("c","c"),digits=3) %>% kable_styling(full_width=T,"striped") %>% row_spec(5, col="red") %>% row_spec(1:4,col="grey")
ARL Table for c-chart with lambda_0=4. Note that chart cannot detect downward shifts since the LCL=0, which correspondings entries are greyed out. The in-control ARL is in red.
lambda_0 ARL
2.0 120362.662
2.5 16226.678
3.0 3420.710
3.5 980.975
4.0 352.142
4.5 149.955
5.0 73.018
5.5 39.602
6.0 23.463

3.3 Plot of ARLs for c-chart

This is a plot the ARLs from \(\lambda_0=4\) to \(\lambda_1=6\).

vals<-seq(4,6,.01)
plot(vals,ARL(4,vals),
     type="l",main="ARL Plot for c-chart with lambda_0=4",
     xlab="lambda_1",
     ylab="Avg Number of Subgroups to Signal Out-of-Control")