...of a quality characteristic. However, in some situations, the quality of a process is better characterized and summarized by a relationship between the response variable and one or more explanatory variables. Studies focusing on simple linear-regression profiles have received particular attention. In recent years, various methods to monitor linear profiles have been proposed in the literature. Kang and Albin (2000) proposed two control charts for the phase II monitoring of linear profiles. One of them is a multivariate [T.sup.2] chart and the other is a combination of an Exponentially Weighted Moving Average (EWMA) chart and a range (R) chart. In Kim et al. (2003), a method based on the combination of three EWMA charts was proposed to detect shifts in either the intercept, the slope, or standard deviation.

Simulation studies showed that the three EWMA charts performed better in detecting sustained shifts in the parameters than the methods in Kang and Albin (2000) in terms of the Average Run Length (ARL). Moreover, they appear to be much simpler to interpret. Mahmoud and Woodall (2004) studied a phase I method for monitoring linear profiles. Mahmoud et al. (2005) proposed a change-point method, based on the likelihood ratio statistics, to detect sustained changes in a linear profile data set in phase I. They concluded that to detect both kinds of changes, sustained and randomly occurring unsustained shifts, one could use the change-point method in conjunction with the methods proposed by Mahmoud and Woodall (2004). A discussion of the problems associated with the monitoring of linear profiles is given in Woodall et al. (2004).

In a phase II analysis, the process parameters are usually assumed to be known. This is true for all the control charts that monitor linear profiles as mentioned above. However, the process parameters, the intercept, the slope and the standard deviation are usually unknown in the early stages of process improvement, and they are usually estimated using m In-Control (IC) historical samples of size n (or by the phase I study). Some authors have recommended using 20 to 30 samples of size four or five to estimate the process parameters for traditional control charts (see Montgomery (2004) or Ryan (1989)). Quesenberry (1993) and Jones et al. (2001, 2004), among others, have investigated the effect of the estimated parameters on the performance of traditional control charts. A recent literature review paper by Jensen et al. (2005) provides a thorough discussion on the effects that parameter estimation has on control chart performance. They concluded that when the number of reference samples is small, control charts with estimated parameters produce a large bias in the IC ARL, and reduce the sensitivity of the chart in detecting the process changes as measured by the Out-of-Control (OC) ARL. Moreover, after short runs, the false alarm probabilities from the charts increase dramatically. In fact, to attain a performance similar to a chart with known parameters, 20 or 30 samples are insufficient. For example, for the traditional EWMA chart with [lambda] = 0.2, 300 samples of five observations are needed to achieve the desired level of IC performance (Jones et al., 2001). However, in most cases, it may not be feasible to wait for the accumulation of enough subgroups, because the users usually want to monitor and adjust the process in the start-up stages. Hence, many authors have studied design procedures for traditional control charts with estimated parameters (e.g., Hillier (1967, 1969), Yang and Hillier (1970), Nedumaran and Pignatiello (2001) and Jones (2002)). In response, self-starting methods have been developed that update the parameter estimates with new observations and simultaneously check for the OC conditions (Hawkins, 1987; Quesenberry, 1991, 1995; Hawkins and Olwell, 1998; Sullivan and Jones, 2002). These methods have in fact been developed for situations where the number of available samples is too small to obtain a control chart performance comparable to that obtained with the true parameters.

As we know, the problem of detecting a step shift in a parameter of the process is similar to sequential change-point detection. A commonly used change-point model is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [tau] is known as the change point. Sequential change-point detection is addressed in Pollak and Siegmund (1991), Siegmund and Venkatraman (1995), Gombay (2000) and in an excellent review paper by Lai (2001), which presents a summary of these methods as well as a class of sequential detection rules. Pignatiello and Simpson (2002) proposed a control chart based on the likelihood ratio approach for on-line detection and show that it has a robust performance when the magnitude of the shifts varies.

Recently, based on the change-point model, Hawkins et al. (2003) and Hawkins and Zamba (2005a) have proposed two control charts to detect shifts in the mean and variance, respectively, when the true parameters of the process are unknown. Hawkins and Zamba (2005b) proposed an attractive alternative to the traditional charting method. They suggest a single chart to detect a change in the mean and/or variance based on the likelihood ratio test for unknown parameters. They showed that this change-point formulation not only had the desired run length behavior but was also comparable to the best of the traditional formulations for detecting step changes in parameters. The success of their method inspires us to employ a change-point detection approach in control charts. Since their method can monitor multiple process parameters at the same time, we expect it also to be effective for monitoring linear profiles with three parameters that need to be controlled simultaneously.

Since linear profiles are often modeled by a regression model, the change-point problem associated with regression models is also relevant and has been studied before (see Quandt (1958), Holbert (1982), Hawkins (1989), Kim and Siegmund (1989), Kim (1994) and Chen (1998)). It should be noted that the sample sizes are fixed in these papers which may not be appropriate for a phase II analysis. The change-point model of Mahmoud et al. (2005) was designed to monitor linear profiles but primarily focused on the phase I analysis, although the authors claimed that their method might be generalized to phase II settings.

In this paper, a control chart based on a likelihood ratio statistic is proposed for monitoring linear profiles when the true process parameters are unknown. The benefit of such a control chart is that it may eliminate the need to collect a large number of reference samples to estimate the process parameters before the control scheme begins (although it is still advisable to collect a few preliminary or historical samples): We demonstrate the effectiveness of our proposed approach through Monte Carlo simulations.

2. Control chart for linear profiles

In this section, we first present the model for linear profiles under consideration and a brief description of the change-point model. We then discuss our proposed control chart, its design, and some diagnostic aids.

2.1. The change-point model for linear profiles

Denote by {([x.sub.i], [y.sub.ij]), i = 1, 2,..., n} the jth random sample collected over time. For an IC process, the relationship between the response and explanatory variables is assumed to be:

[y.sub.ij] = [A.sub.0] + [A.sub.1][x.sub.i] + [[epsilon].sub.ij] i = 1, 2,..., n, (2)

where the [[epsilon].sub.ij]/[sigma] are independent, identically distributed (i.i.d) standard Normal random variables, and the explanatory variable X takes on n values that are fixed. This is usually the case in practical applications and is consistent with the assumptions in Kang and Albin (2000), Kim et al. (2003) and Mahmoud and Woodall (2004).

When the parameters [A.sub.0], [A.sub.1] and [[sigma].sup.2] are unknown, a widely used method is to estimate them using historical data. Suppose...

NOTE: All illustrations and photos have been removed from this article.



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