--- title: "Quality Control" author: "The ASTA team" output: pdf_document: fig_caption: no highlight: tango number_section: yes toc: yes slidy_presentation: fig_caption: no highlight: tango html_document: toc: yes --- ```{r include=FALSE} library(qcc) library(mosaic) ``` ## Outline * **Quality control** * **Continuous process variable** * **Binomial process variable** * **Poisson process variable** # Quality control ## Quality control chart Control charts are used to routinely monitor quality. Two major types: * **univariate control**: a graphical display (chart) of one quality characteristic * **multivariate control**: a graphical display of a statistic that summarizes or represents more than one quality characteristic The control chart shows * the value of the quality characteristic versus the sample number or versus time * a **center line** (CL) that represents the mean value for the in-control process * an **upper control limit** (UCL) and a **lower control limit** (LCL) The control limits are chosen so that almost all of the data points will fall within these limits **as long as the process remains in-control**. ## Example ```{r message=FALSE} library(qcc) data(pistonrings) head(pistonrings,3) ``` Piston rings for an automotive engine are produced by a forging process. The inside diameter of the rings manufactured by the process is measured on 25 samples(`sample=1,2,..,25`), each of size 5, for the control phase I (`trial=TRUE`), when preliminary samples from a process being considered 'in-control' are used to construct control charts. Then, further 15 samples, again each of size 5, are obtained for phase II (`trial=FALSE`). Reference: Montgomery, D.C. (1991) Introduction to Statistical Quality Control, 2nd ed, New York, John Wiley & Sons, pp. 206-213 ## Example ```{r echo=FALSE} phaseI <- pistonrings$diameter[1:125] phaseII <- pistonrings$diameter[126:200] h <- qcc(phaseI, std.dev = "SD", type = "xbar.one", newdata = phaseII, title = "qq chart: pistonrings") ``` We shall treat different methods for determining LCL,CL and UCL. In that respect, it is crucial that we have * **phase I data**, where the process is in-control. * These data are used to determine LCL,CL and UCL. ## The simple six sigma model Assume that measurements * is a sample, i.e\ they are independent * they have a normal distribution * we know the mean $\mu_0$ and standard deviation $\sigma_0$. In this case we dont need phase I data. * CL=$\mu_0$. * LCL=$\mu_0-k\sigma_0$. * UCL=$\mu_0+k\sigma_0$. The only parameter to determine is $k$. We dont want to give a lot of false warnings, and a popular choise is * k=3, known as the 3*sigma rule. * The probability of a measurement outside the control limits is then 0.27%, when the proces is in-control. This means that the span of allowable variation is $6\sigma_0$. The concept ``Six Sigma'' has become a mantra in many industrial communities. ## Average Run Length (ARL) Let $\mathtt{pOut}$ denote the probability that a measurement is outside the control limits. On average this means that we need $1/\mathtt{pOut}$ observations before we get an outlier. This is known as the *the Average Run Length*: $$AVL=\frac{1}{\mathtt{pOut}}$$ An in-control process with the 3*sigma rule has AVL ```{r} round(1/(2*pdist("norm", -3, plot = FALSE))) ``` An in-control process with AVL=500 has k*sigma rule, where k equals ```{r} -qdist("norm", (1/2)*(1/500), plot = FALSE) ``` ## Types of quality control charts. Depending on the type of control variable, there are various types of control charts. +-------------+--------------+---------------+-------------+ | chart | distribution | statistic | example | +=============+==============+===============+=============+ | xbar | normal | mean | means of a continuous process variable | +-------------+--------------+---------------+-------------+ | S | normal | standard | standard deviations of a \ | | | | deviation | continuous process variable | +-------------+--------------+---------------+-------------+ | R | normal | range | ranges of a continuous\ | | | | | process variable | +-------------+--------------+---------------+-------------+ | p | binomial | proportion | percentage of faulty items | +-------------+--------------+---------------+-------------+ | c | poisson | count | number of faulty items\ | | | | | during a workday | +-------------+--------------+---------------+-------------+ #Continuous process variable ## Continuous process variable Phase I data: * $m$ samples with $n$ measurements in each sample. * For sample $i=1,2,\ldots m$ calculate mean $\bar{x}_i$ and standard deviation $s_i$. * Calculate $$\bar{x}=\frac{1}{m}\sum_{i=1}^m \bar{x}_i \quad\text{and}\quad \bar{s}=\frac{1}{m}\sum_{i=1}^m s_i$$ When the sample is normal, it can be shown that $\bar{s}$ is a biased estimate of the true standard deviation $\sigma$: * $E(\bar{s})=c_4(n)\sigma$ * $c_4(n)$ is tabulated in textbooks and available in the `qcc` package. Unbiased estimate of $\sigma$: $$\hat{\sigma}_1=\frac{\bar{s}}{c_4(n)}$$ Furthermore $\bar{s}$ has estimated standard error $$se(\bar{s})=\bar{s}\frac{\sqrt{1-c_4(n)^2}}{c_4(n)}$$ ## xbar chart $$ \begin{aligned} \text{UCL:}\quad &\bar{x}+3\frac{\hat{\sigma}_1}{\sqrt{n}}\\ \text{CL:}\quad &\bar{x}\\ \text{LCL:}\quad &\bar{x}-3\frac{\hat{\sigma}_1}{\sqrt{n}} \end{aligned} $$ This corresponds to * The probability of a measurement outside the control limits is 0.27%. If we want to change this probability, we need another z-score. E.g if we want to lower this probability to 0.1%, then $3$ should be substituted by $3.29$. ## Example ```{r eval=FALSE} phaseI <- matrix(pistonrings$diameter[1:125] , nrow=25, byrow=TRUE) phaseII <- matrix(pistonrings$diameter[126:200], nrow=25, byrow=TRUE) h <- qcc(phaseI, type = "xbar", std.dev = "UWAVE-SD", newdata = phaseII, title = "xbar chart: pistonrings") ``` * `phaseI` is a matrix with $m=25$ rows, where each row is a sample of size $n=5$. * Similarly `phaseII` has 15 samples. The function `qcc` calculates the necessary statistics and optionally makes a plot. * `phaseI` and `type=` are the only arguments required. * We want that the limits are based on the unweighted average of standard deviations - `UWAVE-SD`. This is not the default. * We also want to evaluate the phase II data: `newdata=phaseII`. * Optionally, we can specify the title on the plot. ## Example ```{r echo=FALSE} phaseI <- matrix(pistonrings$diameter[1:125] , nrow = 25,byrow=TRUE) phaseII <- matrix(pistonrings$diameter[126:200], nrow = 15,byrow=TRUE) h <- qcc(phaseI, std.dev = "UWAVE-SD", type = "xbar", newdata = phaseII, title = "xbar chart: pistonrings") ``` Besides limits we are also told whether the process is above/below CL for 7 or more consecutive samples (yellow dots). `run.length=7` is default, but may be changed. If we e.g. want this to happen with probability 0.2%, then we specify `run.length=10`. ## S chart: Monitoring variability In most situations, it is crucial to monitor the process mean. But it may also be a problem if the variability in "quality" gets too high. In that respect, it is relevant to monitor the standard deviation, which is done by the S-chart: $$ \begin{aligned} \text{UCL:}\quad&\bar{s}+3se(\bar{s})\\ \text{CL:}\quad &\bar{s}\\ \text{LCL:}\quad &\bar{s}-3se(\bar{s})\\ &se(\bar{s})=\bar{s}\frac{\sqrt{1-c_4(n)^2}}{c_4(n)} \end{aligned} $$ Where $3$ may be substituted by some other z-score depending on the required confidence level. ```{r eval=FALSE} h <- qcc(phaseI,type="S", newdata=phaseII, title="S chart: pistonrings") ``` ## S chart example ```{r echo=FALSE} h <- qcc(phaseI,type="S", newdata=phaseII, title="S chart: pistonrings") ``` Remark that the plot does not allow values below zero. Quite sensible when we are talking about standard deviations. ## R chart: Range statistics If the sample size is relatively small ($n\leq 10$), it is custom to use the range $R$ instead of the standard deviation. The range of a sample is simply the difference between the largest and smallest observation. When the sample is normal, it can be shown that: * $E(\bar{R})=d_2(n)\sigma$, where $\bar{R}$ is the average of the $m$ sample ranges. * $d_2(n)$ is tabulated in textbooks and available in the `qcc` package. Unbiased estimate of $\sigma$: $$\hat{\sigma}_2=\frac{\bar{R}}{d_2(n)}$$ Furthermore $\bar{R}$ has estimated standard error $$se(\bar{R})=\bar{R}\frac{d_3(n)}{d_2(n)}$$ $d_3(n)$ is tabulated in textbooks and available in the `qcc` package. ## Charts based on R xbar chart based on $\bar{R}$: $$ \begin{aligned} \text{UCL:}\quad&\bar{x}+3\frac{\hat{\sigma}_2}{\sqrt{n}}\\ \text{CL:}\quad &\bar{x}\\ \text{LCL:}\quad &\bar{x}-3\frac{\hat{\sigma}_2}{\sqrt{n}} \end{aligned} $$ This is actually the default in the `qcc` package. R chart to monitor variability: $$ \begin{aligned} \text{UCL:}\quad & \bar{R}+3se(\bar{R})\\ \text{CL:}\quad & \bar{R}\\ \text{LCL:}\quad & \bar{R}-3se(\bar{R})\\ \end{aligned} $$ ## R chart example ```{r} h <- qcc(phaseI, type="R", newdata=phaseII, title="R chart: pistonrings") ``` # Binomial process variable ## Binomial variation Let us suppose that the production process operates in a stable manner such that * the probability that an item is defect is $p$. * successive items produced are independent In a random sample of n items, the number D of defective items follows a binomial distribution with parameters $n$ and $p$. Unbiased estimate of $p$: $$\hat{p}=\frac{D}{n}$$ which has standard error $$se(\hat{p})=\sqrt{\frac{p(1-p)}{n}}$$ ## p chart Data from phase I: * $m$ samples with estimated proportions $\hat{p}_i,\ \ i=1,\ldots,m$ * $\bar{p}$ is the average of the estimated proportions. p chart: $$ \begin{aligned} \text{UCL:}\quad & \bar{p}+3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}\\ \text{CL:}\quad & \bar{p}\\ \text{LCL:}\quad & \bar{p}-3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}\\ \end{aligned} $$ ## Example ```{r message=FALSE} data(orangejuice) head(orangejuice, 3) ``` Production of orange juice cans. * The data were collected as 30 samples of 50 cans. * The number of defective cans `D` were observed. * After the first 30 samples, a machine adjustment was made. * Then further 24 samples were taken from the process. ```{r message=FALSE,eval=FALSE} with(orangejuice, qcc(D[trial], sizes=size[trial], type="p", newdata=D[!trial], newsizes=size[!trial])) ``` ## Example ```{r echo=FALSE} h <- with(orangejuice, qcc(D[trial], sizes=size[trial], type="p", newdata=D[!trial], newsizes=size[!trial])) ``` The machine adjustment after sample 30 has had an obvious effect. The chart should be recalibrated. # Poisson process variable ## Poisson variation Let us suppose that the production process operates in a stable manner such that * defective items are produced at a constant rate The number D of defective items over a time interval of some fixed length follows a poisson distribution with mean value $c$. Unbiased estimate of $c$: $$\hat{c}=D$$ which has standard error $$se(\hat{c})=\sqrt{c}$$ ## c chart Data from phase I: * $m$ sampling periods with mean estimates $\hat{c}_i,\ \ i=1,\ldots,m$ * $\bar{c}$ is the average of the estimated means. c chart: $$ \begin{aligned} \text{UCL:}\quad & \bar{c}+3\sqrt{\bar{c}}\\ \text{CL:}\quad & \bar{c}\\ \text{LCL:}\quad & \bar{c}-3\sqrt{\bar{c}}\\ \end{aligned} $$