Search results for: Multivariate control chart

Authors: Chunchom Pongchavalit

Abstract:

For the electrical metrics that describe photovoltaic cell performance are inherently multivariate in nature, use of a univariate, or one variable, statistical process control chart can have important limitations. Development of a comprehensive process control strategy is known to be significantly beneficial to reducing process variability that ultimately drives up the manufacturing cost photovoltaic cells. The multivariate moving average or MMA chart, is applied to the electrical metrics of photovoltaic cells to illustrate the improved sensitivity on process variability this method of control charting offers. The result show the ability of the MMA chart to expand to as any variables as needed, suggests an application with multiple photovoltaic electrical metrics being used in concert to determine the processes state of control.

Keywords: The multivariate moving average control chart, Photovoltaic processes control, Multivariate system.

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Authors: Francisco Aparisi, José Sanz

Abstract:

Multivariate quality control charts show some advantages to monitor several variables in comparison with the simultaneous use of univariate charts, nevertheless, there are some disadvantages. The main problem is how to interpret the out-ofcontrol signal of a multivariate chart. For example, in the case of control charts designed to monitor the mean vector, the chart signals showing that it must be accepted that there is a shift in the vector, but no indication is given about the variables that have produced this shift. The MEWMA quality control chart is a very powerful scheme to detect small shifts in the mean vector. There are no previous specific works about the interpretation of the out-of-control signal of this chart. In this paper neural networks are designed to interpret the out-of-control signal of the MEWMA chart, and the percentage of correct classifications is studied for different cases.

Keywords: Multivariate quality control, Artificial Intelligence, Neural Networks, Computer Applications

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Authors: O. Ikpotokin

Abstract:

In any production process, every product is aimed to attain a certain standard, but the presence of assignable cause of variability affects our process, thereby leading to low quality of product. The ability to identify and remove this type of variability reduces its overall effect, thereby improving the quality of the product. In case of a univariate control chart signal, it is easy to detect the problem and give a solution since it is related to a single quality characteristic. However, the problems involved in the use of multivariate control chart are the violation of multivariate normal assumption and the difficulty in identifying the quality characteristic(s) that resulted in the out of control signals. The purpose of this paper is to examine the use of non-parametric control chart (the bootstrap approach) for obtaining control limit to overcome the problem of multivariate distributional assumption and the p-value method for detecting out of control signals. Results from a performance study show that the proposed bootstrap method enables the setting of control limit that can enhance the detection of out of control signals when compared, while the p-value method also enhanced in identifying out of control variables.

Keywords: Bootstrap control limit, p-value method, out-of-control signals, p-value, quality characteristics.

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Authors: Myungraee Cha, Jun Seok Kim, Seung Hwan Park, Jun-Geol Baek

Abstract:

In manufacturing industries, development of measurement leads to increase the number of monitoring variables and eventually the importance of multivariate control comes to the fore. Statistical process control (SPC) is one of the most widely used as multivariate control chart. Nevertheless, SPC is restricted to apply in processes because its assumption of data as following specific distribution. Unfortunately, process data are composed by the mixture of several processes and it is hard to estimate as one certain distribution. To alternative conventional SPC, therefore, nonparametric control chart come into the picture because of the strength of nonparametric control chart, the absence of parameter estimation. SVDD based control chart is one of the nonparametric control charts having the advantage of flexible control boundary. However,basic concept of SVDD has been an oversight to the important of data characteristic, density distribution. Therefore, we proposed DW-SVDD (Density Weighted SVDD) to cover up the weakness of conventional SVDD. DW-SVDD makes a new attempt to consider dense of data as introducing the notion of density Weight. We extend as control chart using new proposed SVDD and a simulation study of various distributional data is conducted to demonstrate the improvement of performance.

Keywords: Density estimation, Multivariate control chart, Oneclass classification, Support vector data description (SVDD)

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Authors: Chia-Hau Liu, Tai-Yue Wang

Abstract:

Traditional multivariate control charts assume that measurement from manufacturing processes follows a multivariate normal distribution. However, this assumption may not hold or may be difficult to verify because not all the measurement from manufacturing processes are normal distributed in practice. This study develops a new multivariate control chart for monitoring the processes with non-normal data. We propose a mechanism based on integrating the one-class classification method and the adaptive technique. The adaptive technique is used to improve the sensitivity to small shift on one-class classification in statistical process control. In addition, this design provides an easy way to allocate the value of type I error so it is easier to be implemented. Finally, the simulation study and the real data from industry are used to demonstrate the effectiveness of the propose control charts.

Keywords: Multivariate control chart, statistical process control, one-class classification method.

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Authors: Joval P George, Dr. Zheng Chen, Philip Shaw

Abstract:

This paper deals with the application of Principal Component Analysis (PCA) and the Hotelling-s T2 Chart, using data collected from a drinking water treatment process. PCA is applied primarily for the dimensional reduction of the collected data. The Hotelling-s T2 control chart was used for the fault detection of the process. The data was taken from a United Utilities Multistage Water Treatment Works downloaded from an Integrated Program Management (IPM) dashboard system. The analysis of the results show that Multivariate Statistical Process Control (MSPC) techniques such as PCA, and control charts such as Hotelling-s T2, can be effectively applied for the early fault detection of continuous multivariable processes such as Drinking Water Treatment. The software package SIMCA-P was used to develop the MSPC models and Hotelling-s T2 Chart from the collected data.

Keywords: Principal component analysis, hotelling's t2 chart, multivariate statistical process control, drinking water treatment.

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Authors: Sheng-Shu Cheng, Fong-Jung Yu

Abstract:

C-control chart assumes that process nonconformities follow a Poisson distribution. In actuality, however, this Poisson distribution does not always occur. A process control for semiconductor based on a Poisson distribution always underestimates the true average amount of nonconformities and the process variance. Quality is described more accurately if a compound Poisson process is used for process control at this time. A cumulative sum (CUSUM) control chart is much better than a C control chart when a small shift will be detected. This study calculates one-sided CUSUM ARLs using a Markov chain approach to construct a CUSUM control chart with an underlying Poisson-Gamma compound distribution for the failure mechanism. Moreover, an actual data set from a wafer plant is used to demonstrate the operation of the proposed model. The results show that a CUSUM control chart realizes significantly better performance than EWMA.

Keywords: Nonconformities, Compound Poisson distribution, CUSUM control chart.

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Authors: Francisco Aparisi, Marco A. de Luna

Abstract:

In some real applications of Statistical Process Control it is necessary to design a control chart to not detect small process shifts, but keeping a good performance to detect moderate and large shifts in the quality. In this work we develop a new quality control chart, the synthetic T2 control chart, that can be designed to cope with this objective. A multi-objective optimization is carried out employing Genetic Algorithms, finding the Pareto-optimal front of non-dominated solutions for this optimization problem.

Keywords: Multi-objective optimization, Quality Control, SPC, Synthetic T2 control chart.

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Authors: N. Katemee, T. Mayureesawan

Abstract:

This paper developed the c-Chart based on a Zero- Inflated Poisson (ZIP) processes that approximated by a geometric distribution with parameter p. The p estimated that fit for ZIP distribution used in calculated the mean, median, and variance of geometric distribution for constructed the c-Chart by three difference methods. For cg-Chart, developed c-Chart by used the mean and variance of the geometric distribution constructed control limits. For cmg-Chart, the mean used for constructed the control limits. The cme- Chart, developed control limits of c-Chart from median and variance values of geometric distribution. The performance of charts considered from the Average Run Length and Average Coverage Probability. We found that for an in-control process, the cg-Chart is superior for low level of mean at all level of proportion zero. For an out-of-control process, the cmg-Chart and cme-Chart are the best for mean = 2, 3 and 4 at all level of parameter.

Keywords: average coverage probability, average run length, geometric distribution, zero-inflated poisson distribution

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Authors: S. Phanyaem

Abstract:

Exponentially weighted moving average control chart (EWMA) is a popular chart used for detecting shift in the mean of parameter of distributions in quality control. The objective of this paper is to compare the efficiency of control chart to detect an increases in the mean of a process. In particular, we compared the Maximum Exponentially Weighted Moving Average (MaxEWMA) and Maximum Generally Weighted Moving Average (MaxGWMA) control charts when the observations are Exponential distribution. The criteria for evaluate the performance of control chart is called, the Average Run Length (ARL). The result of comparison show that in the case of process is small sample size, the MaxEWMA control chart is more efficiency to detect shift in the process mean than MaxGWMA control chart. For the case of large sample size, the MaxEWMA control chart is more sensitive to detect small shift in the process mean than MaxGWMA control chart, and when the process is a large shift in mean, the MaxGWMA control chart is more sensitive to detect mean shift than MaxEWMA control chart.

Keywords: Maximum Exponentially Weighted Moving Average, Maximum General Weighted Moving Average, Average Run Length.

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Authors: Tai-Yue Wang, Hui-Min Chiang, Su-Ni Hsieh, Yu-Min Chiang

Abstract:

In many industries, control charts is one of the most frequently used tools for quality management. Hotelling-s T2 is used widely in multivariate control chart. However, it has little defect when detecting small or medium process shifts. The use of supplementary sensitizing rules can improve the performance of detection. This study applied sensitizing rules for Hotelling-s T2 control chart to improve the performance of detection. Support vector machines (SVM) classifier to identify the characteristic or group of characteristics that are responsible for the signal and to classify the magnitude of the mean shifts. The experimental results demonstrate that the support vector machines (SVM) classifier can effectively identify the characteristic or group of characteristics that caused the process mean shifts and the magnitude of the shifts.

Keywords: Hotelling's T2 control chart, Neural networks, Sensitizing rules, Support vector machines.

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Authors: S. Sukparungsee

Abstract:

In reality, the process observations are away from the assumption that are normal distributed. The observations could be skew distributions which should use an asymmetric chart rather than symmetric chart. Consequently, this research aim to study the robustness of the asymmetric Tukey’s control chart for skew and non-skew distributions as Lognormal and Laplace distributions. Furthermore, the performances in detecting of a change in parameter of asymmetric and symmetric Tukey’s control charts are compared by Average ARL (AARL). The results found that the asymmetric performs better than symmetric Tukey’s control chart for both cases of skew and non-skew process observation.

Keywords: Asymmetric control limit, average of average run length, Tukey’s control chart and skew distributions.

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Authors: Y. Areepong

Abstract:

The main goal of this paper is to study Statistical Process Control (SPC) with Exponentially Weighted Moving Average (EWMA) control chart when observations are serially-correlated. The characteristic of control chart is Average Run Length (ARL) which is the average number of samples taken before an action signal is given. Ideally, an acceptable ARL of in-control process should be enough large, so-called (ARL₀). Otherwise it should be small when the process is out-of-control, so-called Average of Delay Time (ARL₁) or a mean of true alarm. We find explicit formulas of ARL for EWMA control chart for Seasonal Autoregressive and Moving Average processes (SARMA) with Exponential white noise. The results of ARL obtained from explicit formula and Integral equation are in good agreement. In particular, this formulas for evaluating (ARL₀) and (ARL₁) be able to get a set of optimal parameters which depend on smoothing parameter (λ) and width of control limit (H) for designing EWMA chart with minimum of (ARL₁).

Keywords: Average Run Length1, Optimal parameters, Exponentially Weighted Moving Average (EWMA) control chart.

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Authors: W. Chattinnawat

Abstract:

Consider a mass production of HDD arms where hundreds of CNC machines are used to manufacturer the HDD arms. According to an overwhelming number of machines and models of arm, construction of separate control chart for monitoring each HDD arm model by each machine is not feasible. This research proposed a strategy to optimize the SPC management on shop floor. The procedure started from identifying the clusters of the machine with similar manufacturing performance using clustering technique. The three way control chart ( I - MR - R ) is then applied to each clustered group of machine. This proposed research has advantageous to the manufacturer in terms of not only better performance of the SPC but also the quality management paradigm.

Keywords: Three way control chart. I - MR - R , between/within variation, HDD arm.

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Authors: Y. Areepong

Abstract:

The objective of this paper is to present explicit analytical formulas for evaluating important characteristics of Double Moving Average control chart (DMA) for Poisson distribution. The most popular characteristics of a control chart are Average Run Length ( 0 ARL ) - the mean of observations that are taken before a system is signaled to be out-of control when it is actually still incontrol, and Average Delay time ( 1 ARL ) - mean delay of true alarm times. An important property required of 0 ARL is that it should be sufficiently large when the process is in-control to reduce a number of false alarms. On the other side, if the process is actually out-ofcontrol then 1 ARL should be as small as possible. In particular, the explicit analytical formulas for evaluating 0 ARL and 1 ARL be able to get a set of optimal parameters which depend on a width of the moving average ( w ) and width of control limit ( H ) for designing DMA chart with minimum of 1 ARL

Keywords: Optimal parameters, Average Run Length, Average Delay time, Double Moving Average chart.

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Authors: S. Sukparungsee

Abstract:

Generally, the traditional Shewhart p chart has been developed by for charting the binomial data. This chart has been developed using the normal approximation with condition as low defect level and the small to moderate sample size. In real applications, however, are away from these assumptions due to skewness in the exact distribution. In this paper, a modified Exponentially Weighted Moving Average (EWMA) control chat for detecting a change in binomial data by improving square root transformations, namely ISRT p EWMA control chart. The numerical results show that ISRT p EWMA chart is superior to ISRT p chart for small to moderate shifts, otherwise, the latter is better for large shifts.

Keywords: Number of defects, Exponentially Weighted Moving Average, Average Run Length, Square root transformations.

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Authors: Viliam Makis, Farnoosh Naderkhani, Leila Jafari

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In this paper, we present a model and an algorithm for the calculation of the optimal control limit, average cost, sample size, and the sampling interval for an optimal Bayesian chart to control the proportion of defective items produced using a semi-Markov decision process approach. Traditional p-chart has been widely used for controlling the proportion of defectives in various kinds of production processes for many years. It is well known that traditional non-Bayesian charts are not optimal, but very few optimal Bayesian control charts have been developed in the literature, mostly considering finite horizon. The objective of this paper is to develop a fast computational algorithm to obtain the optimal parameters of a Bayesian p-chart. The decision problem is formulated in the partially observable framework and the developed algorithm is illustrated by a numerical example.

Keywords: Bayesian control chart, semi-Markov decision process, quality control, partially observable process.

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Authors: Jun Seok Kim, Cheong-Sool Park, Jun-Geol Baek, Sung-Shick Kim

Abstract:

Control chart pattern recognition is one of the most important tools to identify the process state in statistical process control. The abnormal process state could be classified by the recognition of unnatural patterns that arise from assignable causes. In this study, a wavelet based neural network approach is proposed for the recognition of control chart patterns that have various characteristics. The procedure of proposed control chart pattern recognizer comprises three stages. First, multi-resolution wavelet analysis is used to generate time-shape and time-frequency coefficients that have detail information about the patterns. Second, distance based features are extracted by a bi-directional Kohonen network to make reduced and robust information. Third, a back-propagation network classifier is trained by these features. The accuracy of the proposed method is shown by the performance evaluation with numerical results.

Keywords: Control chart pattern recognition, Multi-resolution wavelet analysis, Bi-directional Kohonen network, Back-propagation network, Feature extraction.

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Authors: Farzaneh Ahmadzadeh

Abstract:

Quality control charts are very effective in detecting out of control signals but when a control chart signals an out of control condition of the process mean, searching for a special cause in the vicinity of the signal time would not always lead to prompt identification of the source(s) of the out of control condition as the change point in the process parameter(s) is usually different from the signal time. It is very important to manufacturer to determine at what point and which parameters in the past caused the signal. Early warning of process change would expedite the search for the special causes and enhance quality at lower cost. In this paper the quality variables under investigation are assumed to follow a multivariate normal distribution with known means and variance-covariance matrix and the process means after one step change remain at the new level until the special cause is being identified and removed, also it is supposed that only one variable could be changed at the same time. This research applies artificial neural network (ANN) to identify the time the change occurred and the parameter which caused the change or shift. The performance of the approach was assessed through a computer simulation experiment. The results show that neural network performs effectively and equally well for the whole shift magnitude which has been considered.

Keywords: Artificial neural network, change point estimation, monte carlo simulation, multivariate exponentially weighted movingaverage

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Authors: Syahanim Mohd Salleh, Zaihosnita Hood, Hairulliza Mohd Judi, Marini Abu Bakar

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CFlow is a flow chart software, it contains facilities to draw and evaluate a flow chart. A flow chart evaluation applies a simulation method to enable presentation of work flow in a flow chart solution. Flow chart simulation of CFlow is executed by manipulating the CFlow data file which is saved in a graphical vector format. These text-based data are organised by using a data classification technic based on a Library classification-scheme. This paper describes the file format for flow chart simulation software of CFlow.

Keywords: CFlow, flow chart, file format.

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Authors: R. Behmanesh, I. Rahimi

Abstract:

recurrent neural network (RNN) is an efficient tool for modeling production control process as well as modeling services. In this paper one RNN was combined with regression model and were employed in order to be checked whether the obtained data by the model in comparison with actual data, are valid for variable process control chart. Therefore, one maintenance process in workshop of Esfahan Oil Refining Co. (EORC) was taken for illustration of models. First, the regression was made for predicting the response time of process based upon determined factors, and then the error between actual and predicted response time as output and also the same factors as input were used in RNN. Finally, according to predicted data from combined model, it is scrutinized for test values in statistical process control whether forecasting efficiency is acceptable. Meanwhile, in training process of RNN, design of experiments was set so as to optimize the RNN.

Keywords: RNN, DOE, regression, control chart.

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Authors: K. Petcharat, Y. Areepong, S. Sukparungsri, G. Mititelu

Abstract:

These paper, we approximate the average run length (ARL) for CUSUM chart when observation are an exponential first order moving average sequence (EMA1). We used Gauss-Legendre numerical scheme for integral equations (IE) method for approximate ARL0 and ARL1, where ARL in control and out of control, respectively. We compared the results from IE method and exact solution such that the two methods perform good agreement.

Keywords: Cumulative Sum Chart, Moving Average Observation, Average Run Length, Numerical Approximations.

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Authors: Yi-Fan Lin, Peter P. Howley, Frank A. Tuyl

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Quality measurement and reporting systems are used in healthcare internationally. In Australia, the Australian Council on Healthcare Standards records and reports hundreds of clinical indicators (CIs) nationally across the healthcare system. These CIs are measures of performance in the clinical setting, and are used as a screening tool to help assess whether a standard of care is being met. Existing analysis and reporting of these CIs incorporate Bayesian methods to address sampling variation; however, such assessments are retrospective in nature, reporting upon the previous six or twelve months of data. The use of Bayesian methods within statistical process control for monitoring systems is an important pursuit to support more timely decision-making. Our research has developed and assessed a new graphical monitoring tool, similar to a control chart, based on the beta-binomial posterior predictive (BBPP) distribution to facilitate the real-time assessment of health care organizational performance via CIs. The BBPP charts have been compared with the traditional Bernoulli CUSUM (BC) chart by simulation. The more traditional “central” and “highest posterior density” (HPD) interval approaches were each considered to define the limits, and the multiple charts were compared via in-control and out-of-control average run lengths (ARLs), assuming that the parameter representing the underlying CI rate (proportion of cases with an event of interest) required estimation. Preliminary results have identified that the BBPP chart with HPD-based control limits provides better out-of-control run length performance than the central interval-based and BC charts. Further, the BC chart’s performance may be improved by using Bayesian parameter estimation of the underlying CI rate.

Keywords: Average run length, Bernoulli CUSUM chart, beta binomial posterior predictive distribution, clinical indicator, health care organization, highest posterior density interval.

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Authors: K. Atashgar

Abstract:

When an assignable cause(s) manifests itself to a multivariate process and the process shifts to an out-of-control condition, a root-cause analysis should be initiated by quality engineers to identify and eliminate the assignable cause(s) affected the process. A root-cause analysis in a multivariate process is more complex compared to a univariate process. In the case of a process involved several correlated variables an effective root-cause analysis can be only experienced when it is possible to identify the required knowledge including the out-of-control condition, the change point, and the variable(s) responsible to the out-of-control condition, all simultaneously. Although literature addresses different schemes to monitor multivariate processes, one can find few scientific reports focused on all the required knowledge. To the best of the author’s knowledge this is the first time that a multi task model based on artificial neural network (ANN) is reported to monitor all the required knowledge at the same time for a multivariate process with more than two correlated quality characteristics. The performance of the proposed scheme is evaluated numerically when different step shifts affect the mean vector. Average run length is used to investigate the performance of the proposed multi task model. The simulated results indicate the multi task scheme performs all the required knowledge effectively.

Keywords: Artificial neural network, Multivariate process, Statistical process control, Change point.

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Authors: Leila Jafari, Viliam Makis, Akram Khaleghei G.B.

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In this paper, we present a maintenance model of a two-unit series system with economic dependence. Unit#1 which is considered to be more expensive and more important, is subject to condition monitoring (CM) at equidistant, discrete time epochs and unit#2, which is not subject to CM has a general lifetime distribution. The multivariate observation vectors obtained through condition monitoring carry partial information about the hidden state of unit#1, which can be in a healthy or a warning state while operating. Only the failure state is assumed to be observable for both units. The objective is to find an optimal opportunistic maintenance policy minimizing the long-run expected average cost per unit time. The problem is formulated and solved in the partially observable semi-Markov decision process framework. An effective computational algorithm for finding the optimal policy and the minimum average cost is developed, illustrated by a numerical example.

Keywords: Condition-Based Maintenance, Semi-Markov Decision Process, Multivariate Bayesian Control Chart, Partially Observable System, Two-unit System.

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Authors: N. Pekin Alakoç, A. Apaydın

Abstract:

Quality control charts indicate out of control conditions if any nonrandom pattern of the points is observed or any point is plotted beyond the control limits. Nonrandom patterns of Shewhart control charts are tested with sensitizing rules. When the processes are defined with fuzzy set theory, traditional sensitizing rules are insufficient for defining all out of control conditions. This is due to the fact that fuzzy numbers increase the number of out of control conditions. The purpose of the study is to develop a set of fuzzy sensitizing rules, which increase the flexibility and sensitivity of fuzzy control charts. Fuzzy sensitizing rules simplify the identification of out of control situations that results in a decrease in the calculation time and number of evaluations in fuzzy control chart approach.

Keywords: Fuzzy set theory, Quality control charts, Run Rules, Unnatural patterns.

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Authors: Y. K. Chen, K. C. Chiou, C. Y. Chen

Abstract:

The cumulative conformance count (CCC) charts are widespread in process monitoring of high-yield manufacturing. Recently, it is found the use of variable sampling interval (VSI) scheme could further enhance the efficiency of the standard CCC charts. The average time to signal (ATS) a shift in defect rate has become traditional measure of efficiency of a chart with the VSI scheme. Determining the ATS is frequently a difficult and tedious task. A simple method based on a finite Markov Chain approach for modeling the ATS is developed. In addition, numerical results are given.

Keywords: Cumulative conformance count, variable sampling interval, Markov Chain, average time to signal, control chart.

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Authors: Asmaa M. Hussein, Amr Wassal, Ahmed Farouk Al-Sadek, A. F. Abd El-Rahman

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In this study, a multivariate analysis of potato spectroscopic data was presented to detect the presence of brown rot disease or not. Near-Infrared (NIR) spectroscopy (1,350-2,500 nm) combined with multivariate analysis was used as a rapid, non-destructive technique for the detection of brown rot disease in potatoes. Spectral measurements were performed in 565 samples, which were chosen randomly at the infection place in the potato slice. In this study, 254 infected and 311 uninfected (brown rot-free) samples were analyzed using different advanced statistical analysis techniques. The discrimination performance of different multivariate analysis techniques, including classification, pre-processing, and dimension reduction, were compared. Applying a random forest algorithm classifier with different pre-processing techniques to raw spectra had the best performance as the total classification accuracy of 98.7% was achieved in discriminating infected potatoes from control.

Keywords: Brown rot disease, NIR spectroscopy, potato, random forest.

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Authors: W. Wiyaratn, A. Watanapa

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The objective of this research is to study plant layout of iron manufacturing based on the systematic layout planning pattern theory (SLP) for increased productivity. In this case study, amount of equipments and tools in iron production are studied. The detailed study of the plant layout such as operation process chart, flow of material and activity relationship chart has been investigated. The new plant layout has been designed and compared with the present plant layout. The SLP method showed that new plant layout significantly decrease the distance of material flow from billet cutting process until keeping in ware house.

Keywords: Plant layout, Systematic Layout Planning, Flowanalysis, Activity relationship chart

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Authors: Satoshi Usami

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Among various HLM techniques, the Multivariate Hierarchical Linear Model (MHLM) is desirable to use, particularly when multivariate criterion variables are collected and the covariance structure has information valuable for data analysis. In order to reflect prior information or to obtain stable results when the sample size and the number of groups are not sufficiently large, the Bayes method has often been employed in hierarchical data analysis. In these cases, although the Markov Chain Monte Carlo (MCMC) method is a rather powerful tool for parameter estimation, Procedures regarding MCMC have not been formulated for MHLM. For this reason, this research presents concrete procedures for parameter estimation through the use of the Gibbs samplers. Lastly, several future topics for the use of MCMC approach for HLM is discussed.

Keywords: Gibbs sampler, Hierarchical Linear Model, Markov Chain Monte Carlo, Multivariate Hierarchical Linear Model

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