## 2 Research Methodology  
### Proposed Methodology  
The study follows a methodology that involves a series of steps to calculate an approximate PCI for multivariate nonnormal data, as illustrated in Table 1. The methodology starts by transforming collected data (\(X_{i}\)) to normal distribution using root transformation method. The root transformation approach looks for a correct root (\(r\)) of right-skewed nonnormal data in which, when the data are brought up to the power \(r\left(X^{r}\right)\), the skewness in the converted data distribution would be basically small. The bisection technique is used to calculate \(r\)'s value in accordance with the idea that a function switches its skewness sign as it goes through zero. The bisection approach can half interval between zero and one in successive iterations and finally identifies the root by evaluating the function in the midst of an interval and substituting whichever limit has the same sign of skewness.  
Additionally, the technique is applied to produce the transformed specification limits (\(USL_{y}\)) by the same transformation equation, where \(Y_{i}\) represents the transformed data and p is the number of QCs. In order to increase the similarity between all multivariate variables, the data of each variable are presented as a fraction of their specification limits. The transformed variables are then divided by their corresponding specification limits to  
\begin{table}
\begin{tabular}{c c c} \hline  
form new relative variables and specifications. Also, specification limits are divided by themselves. Consequently, each variable or QC will have the same specification limits.  
\[r_{i}=\frac{Y_{i}}{USL_{y}} \tag{1}\]  
where \(Y_{i}\) is the transformed normal data of variable \(i\), and \(USL_{y}\) is its upper specification limit. This will reveal the same specification limits for all variables, so then it could be treated as single variable. The relative variables are then averaged to one variable \(\overline{r}\).  
\[\overline{r}=\frac{r_{2}+r_{1}+\ldots+r_{p}}{p} \tag{2}\]  
where \(p\) is the number of multivariate variables, and \(\overline{r}\) is the average of each vector of the relative multivariate data.  
The standard deviation of the multivariate data is used to generate a single variable with a large sample size. This is called pooled standard deviation, \(S_{P}\) of \(Y_{i}\), which is used to generate a large sample from a normal distribution with the mean of \(\overline{r}\) and \(S_{P}\). Generating the large sample size is performed to estimate the percent of nonconforming (PNC) data (fraction of data out of specification limit). PNC is then used:  
\[S_{P}=\sqrt{\frac{(n_{1}-1)*S^{2}_{1}+(n_{2}-1)*S^{2}_{2}+\ldots+(n_{p}-1)*S^{2 }_{p}}{n_{1}+n_{2}+\ldots+n_{p}-2}} \tag{3}\]  
where \(p\) is the number of multivariate variables, \(n_{p}\) is the sample size of variable \(p\), and \(S^{2}_{p}\) is the variance of variable \(p\). Then, based on the relative specification limits, the PNC of the generated sample is estimated to be used to estimate the MPCI. Finally, the following formula is used to estimate the process PCI for two sides limits:  
\[C_{p}=\frac{\varphi^{-1}(0.5+0.5(1-PNC))}{3} \tag{4}\]  
where \(C_{p}\) is the process capability index, \(\varphi^{-1}\) is the inverse of the normal distribution, and PNC is the percent of nonconforming in the simulated data. Meanwhile, the one-sided process capability index is estimated by the following formula:  
\[C_{p}=\frac{\varphi^{-1}(1-PNC)}{3}.\]  
There are two main assumptions for the proposed methodology to be valid:  
* First, the data should be normally distributed; thus, the proposed method consists of checking the normality assumption and transforming the nonnormal data into normal using the explained root transformation technique.
* Second, the proposed multivariate process capability index is specific for correlated data. Consequently, unrelated multivariate variables could be individually investigated using a univariate capability index.