We propose control chart methods for process monitoring when the quality of a process or product is characterized by a linear function. In the historical analysis of Phase I data, we recommend methods including the use of a bivariate T^{2}chart to check for stability of the regression coefficients in conjunction with a univariate Shewhart chart to check for stability of the variation about the regression line. We recommend the use of three univariate control charts in Phase II. These three charts are used to monitor the Y-intercept, the slope, and the variance of the deviations about the regression line, respectively. A simulation study shows that this type of Phase II method can detect sustained shifts in the parameters better than competing methods in terms of average run length performance. We also relate the monitoring of linear profiles to the control charting of regression-adjusted variables and other methods.Key Words: Calibration, Exponentially Weighted Moving Average Control Chart, Multivariate T

^{2}Control Charts, Statistical Process Control.

*By* **KEUNPYO KIM, MAHMOUD A. MAHMOUD, and WILLIAM H. WOODALL,
Virginia Tech, Blacksburg, VA 24061-0439**

**Introduction**

IN many practical situations, the quality of a process or product is characterized by a relationship (or profile) between two or more variables instead of by the distribution of a single quality characteristic. Lawless et al. (1999) gave examples in automotive engineering. Kang and Albin (2000) presented two examples of situations for which product profiles are of interest. The first example involved aspartame (an artificial sweetener), which is characterized by the amount that dissolves per liter of water at different levels of temperature. In this case, there is a desired functional relationship between the amount of dissolved aspartame and the temperature. The other example is a semiconductor manufacturing application involving calibration of a mass .ow controller in which the performance of the process is characterized by a linear function. Mestek et al. (1994) and Stover and Brill (1998) gave similar calibration applications.

Kang and Albin (2000) proposed Phase I and Phase II control chart methods for monitoring a process in which the quality of a product is characterized by a linear relationship. Stover and Brill (1998) also considered the Phase I problem, while Brill (2001) considered possible extensions to relationships more general than a linear one.

In this paper, we propose statistical process
control charts for monitoring in Phase II a process or product that is
characterized by a linear profile. Kang and Albin (2000) proposed two
control chart strategies to monitor such a process. One approach involves
a multivariate *T*^{2} chart. The other uses statistics based
on the successive samples of deviations from the in-control line, in a
combination of an exponentially weighted moving average (EWMA) chart to
monitor the average deviation and a range (*R*-) chart to monitor
the variation of the deviations. Both approaches are described in the
next section. Our method is more similar to their second approach. Instead
of using the deviations from the in-control line, however, we code the
independent variable so that the average value is zero, and use the estimated
regression coefficients from each sample, i.e., the estimates of the *Y*-intercept
and slope, to construct two univariate EWMA charts. Also, we study two
different one-sided EWMA charts for monitoring a process standard deviation
as replacements for their Rchart. One of these charts was developed by
Crowder and Hamilton (1992). In practice, a two-sided chart might be more
appropriate in many applications to detect decreases, as well as increases,
in variability about the regression line.

The statistical performance of the combined use of the three EWMA charts is compared to that of the methods of Kang and Albin (2000) later in our paper. We use simulation to show that our proposed method has better overall performance than the competing methods. Later in the paper we make recommendations for the Phase I analysis.

The monitoring of linear profiles is very closely
related to the control charting of regression-adjusted variables, as proposed
by Mandel (1969), Zhang (1992), Hawkins (1991, 1993), Wade and Woodall
(1993), and Hauck et al. (1999). In these approaches, a regression model
is often used to account for the effect of an input quality variable *X*
on the output quality variable *Y* when monitoring a particular stage
of a manufacturing process. One can adjust the output variable, however,
based on any number of upstream process or quality variables in what Hawkins
(1993) referred to as a "cascade process." In a cascade process, variables
have a natural ordering, and if any variable undergoes a parameter shift,
it may then a.ect some or all of the variables following it but none preceding
it. The use of regression adjustment of a single quality variable based
on a simple linear regression model is very similar to the linear profile
situation, except that the Phase I data usually consist of a single set
of bivariate data points. In Phase II, one observes a sequence of deviations
from the predicted values of *Y* based on the .tted Phase I regression
model. In the regression-adjusted applications, however, the *X*-variable
is usually considered to be a random variable, and does not take fixed
values as typically assumed in the linear pro.le monitoring application.

The problem of modeling and monitoring process or product quality using a function has been approached with other methods. Walker and Wright (2002) used additive models to represent the curves of interest in the monitoring of density profiles of particleboard. Jin and Shi (2001) used wavelets to monitor "waveform signals" for diagnosis of process faults. The use of linear functions as responses in designed experiments has also been studied recently. See, for example, Miller (2002) and Nair et al. (2002).

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