%\VignetteIndexEntry{Attributes control charts (acc) in the rQCC package}
%\VignetteKeyword{rQCC}
%\VignetteKeyword{control chart}
%\VignetteKeyword{attribute chart}
%\VignetteKeyword{p chart}
%\VignetteKeyword{c chart}
%\VignetteKeyword{u chart}
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\title{Attributes control charts (\texttt{acc}) in the \texttt{rQCC} package}
%-------------------------------------------------------------------
\author{Chanseok Park\footnote{Applied Statistics Laboratory,
Department of Industrial Engineering, Pusan National University, Busan 46241, Korea.
His work was supported by the National Research Foundation of Korea(NRF) grant funded
by the Korea government(MSIT) (No.\ 2022R1A2C1091319).}
~{and}
Min Wang\footnote{Department of Management Science and Statistics,
The University of Texas at San Antonio, San Antonio, TX 78249, USA.}
}
\date{December 2022}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\begin{document}
\maketitle
\begin{abstract}
We provide a brief summary of attribute control charts
along with a description of how they are constructed
using the \texttt{acc} function in the R package \texttt{rQCC}. In addition, 
various traditional attribute control charts such as $p$, $np$, $c$, $u$, $g$, $h$, and $t$ can be easily
constructed using the \texttt{acc} function.
\end{abstract}

%--------------------------------------
\section{Introduction}
%--------------------------------------
In many practical applications, attribute control charts are widely used in
the scenario where the quality characteristics cannot be conveniently measured and/or represented.
Attribute (or count) data deal with quality characteristics, such as
color, types of design, defective or non-defective, conforming or nonconforming, etc.
In general, it is easier, cheaper, and faster to collect attribute data than quantitative data, 
whereas attribute data are less informative and precise.
For example, the fraction or percentage of nonconforming units does not provide any information
in which the measurements are centered and how dispersive they are.

Attribute control charts can also be constructed based on the idea of the traditional Shewhart control charts.
These are the control charts for the fraction of nonconforming units ($p$ chart),
for nonconformities ($c$ chart) and for nonconformities per unit ($u$ chart).
The traditional Shewhart-type control charts consist of the upper control limit (UCL),
the center line (CL) and the lower control limit (LCL). They have
the form of $\mathrm{CL}\pm g\cdot\mathrm{SE}$,
where the American Standard is based on $g=3$
with a target false alarm rate of 0.027\% and the British Standard is based on $g=3.09$
with a target false alarm rate of 0.020\%.
The UCL is given by $\mathrm{CL}+g\cdot\mathrm{SE}$ and the LCL is
$\mathrm{CL}-g\cdot\mathrm{SE}$.

In this note, we consider two kinds of attribute control charts for fraction nonconforming 
and for nonconformities (defects), which
are also called the $p$ and $u$ charts, respectively.
The Shewhart-type attribute control charts are constructed under the tacit assumption that
the normal approximation to the binomial distribution is acceptable. In what follows, we provide the construction of the $p$ and $u$ control charts
based on the traditional Shewhart-type control charts and implement these charts in the R package \texttt{rQCC}.
%======================================
\section{The $p$ and $np$ charts}
%======================================
The $p$ chart is usually used to monitor fraction nonconforming. 
Using this $p$ chart, it is also available to construct the control chart ($np$ chart) 
for the number of nonconforming
instead of fraction nonconforming. 

It is assumed that 
we have $m$ samples and that the $i$th sample have either the equal number of observations $n$ (balanced case)
or different number of observations $n_i$ (unbalanced case) for $i=1 ,2 ,\ldots, m$. For notational simplicity,  
let $X_{i}$ be the number of nonconforming units in the $i$th sample (subgroup) from a stable manufacturing process. 
We assume that $X_{i}$ are independent and identically distributed according to the binomial distribution
with the sample size $n$ or $n_i$ and the Bernoulli probability $p$.

%--------------------------------------
\subsection{The conventional $p$ chart}
%--------------------------------------
Let $\hat{p}_k = X_k/n_k$. We have $E(\hat{p}_k)=p$ and  $\mathrm{Var}(\hat{p}_k) = p(1-p)/n_k$.
To construct the control charts of the form $\mathrm{CL} \pm g\cdot\mathrm{SE}$,
we consider the relation
\[
\frac{\hat{p}_k - E(\hat{p}_k)}{\sqrt{\mathrm{Var}(\hat{p}_k)}} = \pm g,
\]
which showed that the control limits for the conventional Shewhart-type $p$ chart with $n_k$ are given by
%------------
\begin{equation} \label{EQ:pbar:01}
p \pm g \sqrt{ \frac{p(1-p)}{n_k} }.
\end{equation}
%------------
It is worth noting that the value of the parameter $p$ is unknown in practice. 
To tackle this issue, we combine all the subgroups and estimate $p$ by
\begin{equation} \label{EQ:pbar}
\bar{p} =\frac{\sum_{i=1}^m X_i}{ \sum_{i=1}^m n_i}.
\end{equation}
Replacing $p$ in (\ref{EQ:pbar:01}) with $\bar{p}$ provides the control limits of the $p$ chart 
%------------------
\begin{equation} \label{EQ:CIWald}
\bar{p} \pm g \sqrt{ \frac{\bar{p} (1- \bar{p})}{n_k} }.
\end{equation}
%%-----------------
Since the control limits should be between zero and one, the control limits for the $p$ chart
are actually given by
\begin{align*}
\mathrm{UCL} &= \min\left\{ \bar{p} + g \sqrt{\frac{\bar{p}(1-\bar{p})}{n_k}}, 1 \right\} ,  \\
\mathrm{CL}  &= \bar{p},    \\
\mathrm{LCL} &= \max\left\{ \bar{p} - g \sqrt{\frac{\bar{p}(1-\bar{p})}{n_k}}, 0 \right\} .
\end{align*}
For the case of the known $p$, use the known value instead of $\bar{p}$ in the above.

For example, one can obtain the control limits of the conventional $p$ chart using the \texttt{acc} function 
in the R package \texttt{rQCC}.

\bigskip\noindent\fbox{\parbox{\textwidth}{%
\begin{color}{red}
\texttt{> acc(x, n, type="p")}
\end{color}
}}
\smallskip

The conventional $np$ chart is easily obtained by multiplying $n$ and the control limits of the conventional $p$ chart.
It should be noted that this $np$ chart is appropriate only when samples are balanced. 
Thus, we have the control limits for the $np$ chart as follows.
\begin{align*}
\mathrm{UCL} &= \min\left\{ n\bar{p} + g \sqrt{ n\bar{p}(1-\bar{p})} , n \right\} ,  \\
\mathrm{CL}  &= n\bar{p},    \\
\mathrm{LCL} &= \max\left\{ n\bar{p} - g \sqrt{ n\bar{p}(1-\bar{p})}, 0 \right\} .
\end{align*}
One can obtain the control limits of the conventional $np$ chart 
by setting \texttt{type="np"} in the \texttt{acc} function.


%---------------------------------------
\subsection{The $p$ chart based on the Wilson confidence interval}
%---------------------------------------
In the statistics literature, Equation (\ref{EQ:CIWald}) is the most commonly-used interval for the Bernoulli parameter $p$ and 
is also related to the Wald confidence interval for $p$; see \cite{Agresti/Coull:1998}.
However, it is well-known that the Wald confidence limits are unsatisfactory for small $n$.
See the references in \cite{Agresti/Coull:1998,Blyth/Still:1983,Ghosh:1979}.
Wilson~\cite{Wilson:1927} obtained an improved confidence interval by using the idea of directly inverting a test statistic.

Park~\cite{Park:2013b} introduced an improvement of the $p$ chart based on the approach of Wilson~\cite{Wilson:1927}.
When $p$ is known, the control limits with the sample size $n_k$ are given by
%---------------
\[
\frac{{p} + g^2/(2N)}{1+g^2/N} \pm
\frac{g}{1+g^2/N} \sqrt{ \frac{{p}(1-{p})}{n_k} + \frac{g^2}{4n_k N} },
\]
%---------------
where  $N=\sum_{i=1}^{m}n_k$.
When $p$ is unknown, the control limits are then given by 
%---------------
%% \begin{equation} \label{EQ:WilsonCC1}
\[
\frac{\bar{p} + g^2/(2N)}{1+g^2/N}
\pm \frac{g}{1+g^2/N}
\sqrt{ \frac{\bar{p}(1-\bar{p})}{n_k} + \frac{g^2}{4n_k N} },
\]
%% \end{equation}
%---------------
where $\bar{p}$ is given in (\ref{EQ:pbar}).
%%-----------------
Since the control limits should be between zero and one, the control limits are actually given by
\begin{align*}
\mathrm{UCL} &= \min\left\{ \frac{\bar{p} + g^2/(2N)}{1+g^2/N}
               +\frac{g}{1+g^2/N} \sqrt{\frac{\bar{p}(1-\bar{p})}{n_k}+\frac{g^2}{4n_k N}}, 1\right\} ,\\
\mathrm{CL}  &= \frac{\bar{p} + g^2/(2N)}{1+g^2/N} ,     \\
\mathrm{LCL} &= \max\left\{ \frac{\bar{p} + g^2/(2N)}{1+g^2/N}
               - \frac{g}{1+g^2/N} \sqrt{\frac{\bar{p}(1-\bar{p})}{n_k}+\frac{g^2}{4n_k N}}, 0\right\} .
\end{align*}
As afore-mentioned, for the case of the known $p$,
we can use the known value instead of $\bar{p}$ in the above.

For example, one can obtain the control limits of the $p$ chart 
using the \texttt{acc} function in the R package \texttt{rQCC}. 

\bigskip\noindent\fbox{\parbox{\textwidth}{%
\begin{color}{red}
\texttt{> acc(x, n, type="p", pEstimator="Wilson")}
\end{color}
}}
\smallskip

The $np$ chart based on the Wilson confidence interval is also obtained 
by multiplying $n$ and the control limits of the $p$ chart based on the Wilson confidence interval. 
This chart is also appropriate only when samples are balanced. 
Thus, we have the following control limits.
\begin{align*}
\mathrm{UCL} &= \min\left\{ \frac{n\bar{p} + g^2/(2m)}{1+g^2/(nm)}
               +\frac{g}{1+g^2/(nm)} \sqrt{n\bar{p}(1-\bar{p}) + {g^2}/{(4m)}}, n\right\} ,\\
\mathrm{CL}  &= \frac{n\bar{p} + g^2/(2m)}{1+g^2/(nm)} ,     \\
\mathrm{LCL} &= \max\left\{ \frac{n\bar{p} + g^2/(2m)}{1+g^2/(nm)}
               - \frac{g}{1+g^2/(nm)} \sqrt{n\bar{p}(1-\bar{p}) + {g^2}/{(4m)}}, 0\right\} .
\end{align*}
One can obtain the control limits of the $np$ chart based on the Wilson confidence interval 
with \texttt{type="np"} as follows. 

\bigskip\noindent\fbox{\parbox{\textwidth}{%
\begin{color}{red}
\texttt{> acc(x, n, type="np", pEstimator="Wilson")}
\end{color}
}}
\smallskip

%======================================
\section{The $u$ and $c$ charts}
%======================================
The $u$ chart is the control chart for monitoring nonconformities (defects) that occur in each unit.
Let $X_i$ be the number of defects with the $n_i$ inspection units.
We assume that the number of defects in a \emph{single} unit is distributed as the Poisson
distribution with the mean $\lambda$.
Then $X_i$ are independent and identically distributed according to the Poisson distribution 
with the mean $\lambda n_i$, such that 
$E(X_i) = n_i\lambda$ and $\mathrm{Var}(X_i) = n_i\lambda$.

Let $U_k = X_k / n_k$. We have
$E(U_k) = \lambda$ and $\mathrm{Var}(U_k) = \lambda/n_k$.
To construct the  $\mathrm{CL} \pm g\cdot\mathrm{SE}$ control limits,
we consider the relation
\[
\frac{U_k - E(U_k)}{\sqrt{\mathrm{Var}(U_k)}} = \pm g.
\]
The control limits for the Shewhart-type $u$ chart with the $n_k$ inspection units are given by
%------------
\[
\lambda \pm g \sqrt{ \frac{\lambda}{n_k} } .
\]
%------------
It is worth noting that if $n_k=1$, this chart can be regarded as the $c$ chart.

When $\lambda$ is unknown, it can be estimated by
\[
\bar{\lambda} = \frac{\sum_{i=1}^{m}X_i}{\sum_{i=1}^{m}n_i}.
\]
Since the control limits are nonnegative, we consider the following control limits
\begin{align*}
\mathrm{UCL} &= \bar{\lambda} + g\sqrt{\frac{\bar{\lambda}}{n_k}} ,  \\
\mathrm{CL}  &= \bar{\lambda},  \\
\mathrm{LCL} &= \max\left\{ \bar{\lambda} - g\sqrt{\frac{\bar{\lambda}}{n_k}}, 0 \right\} .
\end{align*}
Again, for the case of the known $\lambda$,
we just use the value of $\lambda$ instead of $\bar{\lambda}$ in the above.

For example, one can obtain the control limits of the $u$ chart 
using the \texttt{acc} function as follows. 

\bigskip\noindent\fbox{\parbox{\textwidth}{%
\begin{color}{red}
\texttt{> acc(x, n, type="u")}
\end{color}
}}
\smallskip

The control limits of the $c$ chart are also easily obtained as follows.

\bigskip\noindent\fbox{\parbox{\textwidth}{%
\begin{color}{red}
\texttt{> acc(x, type="c")}
\end{color}
}}
\smallskip

%======================================
\section{The $g$ and $h$ charts}
%======================================
The geometric distribution can also be used for counting event data. 
It is widely used for monitoring the number of conforming cases between the two consecutive
appearances of nonconformities such as defects and infections.

Based on this, Kaminsky et al.\ \cite{Kaminsky/etc:1992} proposed $g$ and $h$ charts. 
The probability mass function for the geometric distribution with location $a$ is given by 
\begin{equation} \label{EQ:pmfofGeo}
f(y) = P(Y=y) = p(1-p)^{y-a},
\end{equation}
where $y=a, a+1, \ldots$ and 
$a$ is the known minimum possible number of events (usually $a=0,1$).
The mean and variance of $Y$ are given by 
\[
\mu = E(Y) = \frac{1-p}{p}+a \quad\textrm{and}\quad 
\sigma^2 = \mathrm{Var}(Y) = \frac{1-p}{p^2}.
\]

We construct the control chart for the total number of events ($g$ chart)
and the average number of events (the $h$ chart) per subgroup with $\mathrm{CL} \pm g\cdot\mathrm{SE}$ control limits.

The $g$ chart (total number of events) with the sample size $n_k$
has the following control limits which are actually
the function of $\mu$ and $\sigma^2$. 
For more details, see Kaminsky et al.\ \cite{Kaminsky/etc:1992} and Park and Wang~\cite{Park/Wang:2023}.
\begin{alignat}{2}
\mathrm{UCL} &= n_k{\mu}+g\sqrt{n_k \sigma^2} && = n_k\left(\frac{1-p}{p}+a\right) +g\sqrt{\frac{n_k(1-p)}{p^2}},\notag\\
\mathrm{CL}  &= n_k{\mu}                      && = n_k\left(\frac{1-p}{p}+a\right), \label{EQ:CLgchart}   \\
\mathrm{LCL} &= n_k{\mu}-g\sqrt{n_k \sigma^2} && = n_k\left(\frac{1-p}{p}+a\right) -g\sqrt{\frac{n_k(1-p)}{p^2}}. \notag 
\end{alignat}
The smallest possible value of the total number of events is $n_k a$.
Thus, the $\mathrm{LCL}$ cannot be smaller than $n_k a$.
If $\mathrm{LCL} < n_k a$ in the above limit, we set up $\mathrm{LCL} = n_k a$.

The $h$ chart (average number of events) with the sample size $n_k$
has the following control limits.
\begin{alignat}{2}
\mathrm{UCL} &= {\mu} + g \sqrt{\frac{\sigma^2}{n_k}} && = \frac{1-p}{p}+a  + g\sqrt{\frac{1-p}{n_k p^2}}, \notag \\
\mathrm{CL}  &= {\mu}                                 && = \frac{1-p}{p}+a,   \label{EQ:CLhchart}  \\
\mathrm{LCL} &= {\mu} - g \sqrt{\frac{\sigma^2}{n_k}} && = \frac{1-p}{p}+a  - g\sqrt{\frac{1-p}{n_k p^2}}. \notag
\end{alignat}
The smallest possible value of the average number of events is $a$.
Thus, we set up $\mathrm{LCL} = a$ if $\mathrm{LCL} < a$ in the above limit.

The value of the process parameter $p$ is unknown in general so that we estimate $p$.
We assume that we obtain $m$ samples and denote the size of each sample by $n_i$ for $i=1,2,\ldots,m$.
Suppose that $X_{ij}$ be the number of independent Bernoulli trials (cases)
until the first nonconforming case in the $i$th sample for $i=1,2,\ldots,m$ and $j=1,2,\ldots,n_i$.
Then $X_{ij}$'s are independent and identically-distributed geometric random variables with location shift $a$ and $p$.
We let $\bar{\bar{X}} = \sum_{i=1}^{m}\sum_{j=1}^{n_i} X_{ij}/N$ with $N=\sum_{i=1}^{m} n_i$.
We can estimate $p$ using the maximum likelihood (ML) and  
minimum variance unbiased (MVU) estimators~\cite{Park/Wang:2023} which are given by 
\[
\hat{p}_{\mathrm{ML}}  = \frac{1}{\bar{\bar{X}}-a+1}  \quad\textrm{and}\quad 
\hat{p}_{\mathrm{MVU}} = \frac{1-1/N}{\bar{\bar{X}}-a+1-1/N},
\]
respectively.
It should be noted that Benneyan~\cite{Benneyan:2001b} proposed 
\[
\hat{p}_{\mathrm{B}} = \frac{1-1/N}{\bar{\bar{X}}-a+1}
\]
as the MVU estimator, but Park and Wang~\cite{Park/Wang:2023} showed that 
$\hat{p}_{\mathrm{B}}$ is not a correct MVU estimator. 

The ML estimator has the invariance property.
Thus, by substituting $\hat{p}_{\mathrm{ML}}$  into (\ref{EQ:CLgchart}) and (\ref{EQ:CLhchart}), 
we can construct the $g$ and $h$ charts based on the ML method as follows.  
%-----------
\begin{itemize}
\item $g$ chart
\begin{align*}
\mathrm{UCL} &= n_k\bar{\bar{X}} + g \sqrt{n_k (\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1)},  \\
\mathrm{CL}  &= n_k\bar{\bar{X}},   \\
\mathrm{LCL} &= n_k\bar{\bar{X}} - g \sqrt{n_k (\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1)}.
\end{align*}
\item $h$ chart:
\begin{align*}
\mathrm{UCL} &= \bar{\bar{X}} + g\sqrt{\frac{(\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1)}{n_k}},  \\
\mathrm{CL}  &= \bar{\bar{X}},   \\
\mathrm{LCL} &= \bar{\bar{X}} - g\sqrt{\frac{(\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1)}{n_k}}.
\end{align*}
\end{itemize}
%-----------

When we construct the charts based on the MVU estimator, $\hat{p}_{\mathrm{MVU}}$, 
a care should be taken because the MVU estimator has no invariance property like the ML estimator.
Park and Wang~\cite{Park/Wang:2023} showed that the MVU estimators of $\mu$ and $\sigma^2$ are given by 
\[
\hat{\mu}_{\mathrm{mvu}} = \bar{\bar{X}}
\quad\textrm{and}\quad 
\hat{\sigma}^2_{\mathrm{mvu}} = \frac{N}{N+1}  (\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1),
\]
and they construct the  $g$ and $h$ charts based on $\hat{\mu}_{\mathrm{mvu}}$ and $\hat{\sigma}^2_{\mathrm{mvu}}$ 
as follows.
%-----------
\begin{itemize}
\item $g$ chart:
\begin{align*}
\mathrm{UCL} &= n_k\bar{\bar{X}} + g \sqrt{\frac{n_k N}{N+1} (\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1)},  \\
\mathrm{CL}  &= n_k\bar{\bar{X}},   \\
\mathrm{LCL} &= n_k\bar{\bar{X}} - g \sqrt{\frac{n_k N}{N+1} (\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1)}.
\end{align*}
\item $h$ chart:
\begin{align*}
\mathrm{UCL} &= \bar{\bar{X}} + g\sqrt{\frac{N}{N+1}\frac{(\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1)}{n_k}},  \\
\mathrm{CL}  &= \bar{\bar{X}},   \\
\mathrm{LCL} &= \bar{\bar{X}} - g\sqrt{\frac{N}{N+1}\frac{(\bar{\bar{X}}-a) (\bar{\bar{X}}-a+1)}{n_k}}.
\end{align*}
\end{itemize}
%-----------

The control limits of the $g$ and $h$ charts are also easily obtained as follows.

\bigskip\noindent\fbox{\parbox{\textwidth}{%
\begin{color}{red}
\texttt{> acc(x, type="g")}

\texttt{> acc(x, type="h")}
\end{color}
}}
\smallskip


%======================================
\section{The $t$ chart}
%======================================
The $t$ chart is the control chart for monitoring the stability of a process
\cite{Santiago/Smith:2013} with the assumption that the time between
events can be modeled with exponential or Weibull distribution.
Since the underlying distributions are not symmetric, 
the use of the probability-limit control chart is suggested 
instead of the traditional Shewhart-type three-sigma limits.

The cumulative distribution function (cdf) of the exponential distribution is given by 
\[
F(x) = 1 - e^{-x/\theta}.
\]
Then the ML estimator is obtained by 
$\hat{\theta}=\bar{X}=(1/n)\sum_{i=1}^n X_i$ with a sample $\{X_1, X_2, \allowbreak \ldots,\allowbreak X_n\}$.
Then the LCL, CL and UCL are calculated equating
$F(x)$ to $\alpha/2$, $1/2$, and $1-\alpha/2$, respectively.
Solving these for $x$, we have the exponential $t$ chart given by
\begin{align*}
\textrm{LCL} &= -\log(1-\alpha/2)\cdot\hat{\theta},  \\
\textrm{CL}  &= -\log(1/2)\cdot\hat{\theta},         \\
\textrm{UCL} &= -\log(\alpha/2)\cdot\hat{\theta},
\end{align*}
where $\alpha/2$ is generally given by $\alpha/2 = \Phi(-g)$.
Here $\Phi(\cdot)$ is the cdf of the standard normal distribution.

The cdf of the Weibull distribution is given by
\[
F(x) = 1 - \exp\left\{-\left(\frac{x}{\theta}\right)^\beta\right\},
\]
where $\theta > 0$ and $\beta >0$ represent the scale and shape parameters, respectively.
Let $\hat{\beta}$ and $\hat{\theta}$ be the ML estimators of the Weibull parameters.
Then, similar to the exponential $t$ chart, the Weibull $t$ chart is given by 
\begin{align*}
\textrm{LCL} &= \left\{-\log(1-\alpha/2)\right\}^{1/\hat{\beta}} \cdot\hat{\theta},  \\
\textrm{CL}  &= \left\{-\log(1/2)\right\}^{1/\hat{\beta}}  \cdot\hat{\theta},         \\
\textrm{UCL} &= \left\{-\log(\alpha/2)\right\}^{1/\hat{\beta}}  \cdot\hat{\theta}.
\end{align*}
The control limits of the exponential and Weibull $t$ charts are obtained as follows.

\bigskip\noindent\fbox{\parbox{\textwidth}{%
\begin{color}{red}
\texttt{> acc(x, type="t")} \qquad \texttt{\# Exponential t chart}

\texttt{> acc(x, type="t", tModel="W")} \qquad \texttt{\# Weibull t chart}
\end{color}
}}
\smallskip


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