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In certain cases, the quality of a process or a product can be effectively characterized by two or more multiple linear regression profiles in which response variables are correlated. This structure can be modeled as multivariate multiple linear regression profiles. When linear profiles are monitored separately, then correlation between response variables is ignored and misleading results could be expected. To overcome this problem, the use of methods that consider the multivariate structure between response variables is inevitable. In this paper, we propose four methods to monitor this structure in Phase II. The performance of the methods is compared through simulation studies in terms of the average run length criterion. Furthermore, a method based on likelihood ratio approach is developed to determine the location of shifts and a numerical simulation is used to evaluate the performance of the proposed method. Finally, the use of the methods is illustrated by a numerical example.

In some statistical process control applications, the quality of a process or product can be characterized by a relationship between a response variable and one explanatory variable, which is referred to as profile. We give an example here of a profile that can be described using a polynomial model. This example comes from the automotive industry, where one of the most important quality characteristics of an automobile engine is the relationship between the torque produced by an engine and the engine speed in revolutions per minute. We find for this data set that a second-order polynomial works well. In addition, we show that there is autocorrelation within each profile, thus an ordinary least-square method that ignores the autocorrelation is inappropriate. We propose a linear mixed model method as an alternative approach. After the reduction of the data to a series of parameter estimates, we then conduct a step-by-step Phase I analysis of the polynomial profiles monitoring using a T 2 -based procedure to check the stability of the process and whether or not there are outlying profiles. The remaining profiles are used to form the estimated mean vector and variance-covariance matrix to be used in Phase II studies. Finally, a brief discussion is presented to show how one can use these parameters in Phase II.

In some statistical process control applications, quality of a process or product is characterized by a relationship between two or more variables which is referred to as profile. In many practical situations, a profile can be modeled as a polynomial regression. In this article, three methods are developed for monitoring polynomial profiles in Phase I. Their performance is evaluated using power criterion. Furthermore, a method based on likelihood ratio test is developed to identify the location of shifts. Numerical simulation is used to evaluate the performance of the developed method.

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