Monitoring multivariate simple linear profiles using robust estimators

Brief Abstract

This article focuses on estimation of multivariate simple linear profiles. While outliers may hamper the expected performance of the ordinary regression estimators, this study resorts to robust estimators as the remedy of the estimation problem in presence of contaminated observations. More specifically, three robust estimators M, S and MM are employed. Extensive simulation runs show that in the absence of outliers or for small amount of contamination, the robust methods perform as well as the classical least square method, while for medium and large amounts of contamination the proposed estimators perform considerably better than classical method.     

Adjusted-loss-function charts with variable sample sizes and sampling intervals

Recent research has shown that the control charts with adaptive features are quicker than the traditional static Shewhart charts in detecting process shifts. This article presents the design and implementation of a control chart based on Adjusted Loss Function (AL) with Variable Sample Sizes and Sampling Intervals (VSSI). This single chart (called the VSSI AL chart) is able to monitor the process shifts in mean and variance simultaneously. Our studies show that the VSSI AL chart is not only easier to design and implement than the VSSI X¯ & S (or X¯ & R) charts, but is also 10% more effective than the latter in detecting the process shifts from an overall viewpoint.     

The monitoring of simple linear regression profiles with two observations per sample

We evaluate and compare the performance of Phase II simple linear regression profile approaches when only two observations are used to establish each profile. We propose an EWMA control chart based on average squared deviations from the in-control line, to be used in conjunction with two EWMA control charts based on the slope and Y-intercept estimators, to monitor changes in the three regression model parameters, i.e., the slope, intercept and variance. Simulations establish that the performance of the proposed technique is generally better than that of other approaches in detecting parameter shifts.     

Monotonic change point estimation of generalized linear model-based regression profiles

In this article, a maximum likelihood estimator is derived in the generalized linear model-based regression profiles under monotonic change in Phase II. The performance of the proposed estimator is comprehensively investigated through some special cases, and compared with estimators under step change and drift. The results show that the proposed estimator has better performance in small and medium shifts under different increasing changes. Finally, the applicability of the proposed estimator is illustrated using a real case.     

Monitoring of autocorrelated general linear profiles

A collection of quality data represented by a functional relationship between response and explanatory variables is called a profile. In the literature, the errors of profiles are often assumed to be independent. However, quality data often exhibits time correlations in real applications. Therefore, in this paper, we investigate a general linear regression model with a between-profile autocorrelation. We propose a multivariate exponentially weighted moving average chart for monitoring shifts in the regression parameters, and an exponentially weighted moving average chart for monitoring shifts in the standard deviation. A simulation study reveals that our proposed schemes outperform competing existing schemes based on the average run length criterion. An example is used to illustrate the applicability of the proposed scheme.     

Monitoring the coefficient of variation using a variable sample size and sampling interval control chart

This article proposes a CV chart by using the variable sample size and sampling interval (VSSI) feature to improve the performance of the basic CV chart, for detecting small and moderate shifts in the CV. The proposed VSSI CV chart is designed by allowing the sample size and the sampling interval to vary. The VSSI CV chart's statistical performance is measured by using the average time to signal (ATS) and expected average time to signal (EATS) criteria and is compared with that of existing CV charts. The Markov chain approach is employed in the design of the chart.     

A direct procedure for monitoring the coefficient of variation using a variable sample size scheme

A variable sample size (VSS) scheme directly monitoring the coefficient of variation (CV), instead of monitoring the transformed statistics, is proposed. Optimal chart parameters are computed based on two criteria: (i) minimizing the out-of-control ARL (ARL₁) and (ii) minimizing the out-of-control ASS (ASS₁). Then the performances are compared between these two criteria. The advantages of the proposed chart over the VSS chart based on the transformed statistics in the existing literature are: the former (i) provides an easier alternative as no transformation is involved and (ii) requires less number of observations to detect a shift when ASS₁ is minimized.     

Improved best linear unbiased estimators for the simple linear regression model using double ranked set sampling schemes

ABSTRACT

In this paper, we consider the best linear unbiased estimators (BLUEs) based on double ranked set sampling (DRSS) and ordered DRSS (ODRSS) schemes for the simple linear regression model with replicated observations. We assume three symmetric distributions for the random error term, i.e., normal, Laplace and some scale contaminated normal distributions. The proposed BLUEs under DRSS (BLUEs-DRSS) and ODRSS (BLUEs-ODRSS) are compared with the BLUEs based on ordered simple random sampling (OSRS), ranked set sampling (RSS), and ordered RSS (ORSS) schemes. These estimators are compared in terms of relative efficiency (RE), RE of determinant (RED), and RE of trace (RET). It is found that the BLUEs-ODRSS are uniformly better than the BLUEs based on OSRS, RSS, ORSS, and DRSS schemes. We also compare the estimators based on imperfect RSS (IRSS) schemes. It is worth mentioning here that the BLUEs under ordered imperfect DRSS (OIDRSS) are better than their counterparts based on IRSS, ordered IRSS (OIRSS), and imperfect DRSS (IDRSS) methods. Moreover, for sensitivity analysis of the BLUEs, we calculate REs and REDs of the BLUEs under the assumption of normality when in fact the parent distribution follows a non normal symmetric distribution. It turns out that even under violation of normality assumptions, BLUEs of the intercept and the slope parameters are found to be unbiased with equal REs under each sampling scheme. It is also observed that the BLUEs under ODRSS are more efficient than the existing BLUEs.     

A New Adaptive Variable Sample Size Approach in EWMA Control Chart

Adaptive control charts have been developed for improving the capability of control charts in detecting small shifts. In this article, we propose a new exponential weighted moving average control chart with variable sample size, in which the sample size is determined as an integer linear function by EWMA statistic value. The performance of the proposed VSS EWMA control chart is compared with FSS EWMA as well as traditional VSS EWMA control charts. The results show the better performance of the proposed VSS strategy respect to the traditional one and fixed sample size.     

A Bayesian adaptive blinded sample size adjustment method for risk differences

Adaptive sample size adjustment (SSA) for clinical trials consists of examining early subsets of on trial data to adjust estimates of sample size requirements. Blinded SSA is often preferred over unblinded SSA because it obviates many logistical complications of the latter and generally introduces less bias. On the other hand, current blinded SSA methods for binary data offer little to no new information about the treatment effect, ignore uncertainties associated with the population treatment proportions, and/or depend on enhanced randomization schemes that risk partial unblinding. I propose an innovative blinded SSA method for use when the primary analysis is a non‐inferiority or superiority test regarding a risk difference. The method incorporates evidence about the treatment effect via the likelihood function of a mixture distribution. I compare the new method with an established one and with the fixed sample size study design, in terms of maximization of an expected utility function. The new method maximizes the expected utility better than do the comparators, under a range of assumptions. I illustrate the use of the proposed method with an example that incorporates a Bayesian hierarchical model. Lastly, I suggest topics for future study regarding the proposed methods. Copyright © 2015 John Wiley & Sons, Ltd.     

A change point method for Phase II monitoring of generalized linear profiles

In this article, we adopt the change point approach to monitor the generalized linear profiles in phase II Statistical process control (SPC). Generalized linear profiles include a large class of profiles defined in one framework. In contrast to the conventional change point approach, we adopt the Rao score test rather than the likelihood ratio test. Simulated results show that our approach has a good performance over any possible single step change in process parameters for two special cases of generalized linear profiles, namely Poisson and binomial profiles. Some diagnostic aids are also given and a real example is introduced to shed light on the merits of our approach in real applications.     

Three-level control charts with variable sample size,sampling interval,and control limits

Recently, three-level control chart has been developed for monitoring situations in which an appropriate quality measure uses three discrete levels to classify a product characteristic. In this paper, the variable parameters control charts for multinomial data are developed with a three-level classification scheme. We compare it with various adaptive charts to show that this modified scheme can improve the performance. In order to evaluate and compare the performance of this scheme, adjusted average time to signal is used as the performance measure. Results indicate that the proposed chart has improved performance and is relatively sensitive to small shifts.     

Monitoring prespecified changes in linear profiles using control charts with supplementary runs rules

In profile monitoring, some methods have been developed to detect the unspecified changes in the profiles. However, detecting changes away from the “normal” profile toward one of several prespecified “bad” profiles is one possible and challenging purpose. In this article, control charts with supplementary runs rules are developed to detect the prespecified changes in linear profiles. A control chart is first developed based on the Student's t-statistic in t test, and two runs rules are then supplemented to this chart, respectively. Simulation studies show that the proposed control schemes are effective and stable. Moreover, the control schemes are better than the existing alternative charts when the number of observations per sample profile is large. Finally, two illustrative examples indicate that our proposed schemes are effective and easy to be implemented.     

Economic-statistical design of simple linear profiles with variable sampling interval

Control charts are statistical tools to monitor a process or a product. However, some processes cannot be controlled by monitoring a characteristic; instead, they need to be monitored using profiles. Economic-statistical design of profile monitoring means determining the parameters of a profile monitoring scheme such that total costs are minimized while statistical measures maintain proper values. While varying sampling interval usually increases the effectiveness of profile monitoring, economic-statistical design of variable sampling interval (VSI) profile monitoring is investigated in this paper. An extended Lorenzen–Vance function is used for modeling total costs in VSI model where the average time to signal is employed for depicting the statistical measure of the obtained profile monitoring scheme. Two sampling intervals; number of set points and the parameters of control charts that are used in profile monitoring are the variables that are obtained thorough the economic-statistical model. A genetic algorithm is employed to optimize the model and an experimental design approach is used for tuning its parameters. Sensitivity analysis and numerical results indicate satisfactory performance for the proposed model.     

Phase II Monitoring of Geometric Profiles

Geometric profiles can be modeled effectively by large and small scale components. In several articles, a regression model with spatial autoregressive error term is combined with control charts to monitor geometric profiles. However, once a signal occurs, control charts would not be able to determine whether the shift is due to the large or small scale component. In this article, a combination of a multivariate and an omnibus control charts is used to monitor the large scale and small scale components to determine whether the shift is due to the large scale or small scale components.     

Bayesian sample size determination for the bounded linex loss function

In this paper, we discuss optimum sample size determination for a bounded linex loss function (blinex). The linex loss function is often used when losses are asymmetric, but it has the disadvantage that it can only be used if the moment-generating function of the posterior distribution exists. Blinex loss has the advantage that it can always be calculated. Also many authors have argued that a bounded loss function is to be preferred. We have obtained the optimum sample size for a number of distributions when the cost of sampling is linear. The form of the posterior risk function does not allow an analytical solution so simulation is necessary.     

The effect of Phase I sample size on the run length performance of control charts for autocorrelated data

Traditional control charts assume independence of observations obtained from the monitored process. However, if the observations are autocorrelated, these charts often do not perform as intended by the design requirements. Recently, several control charts have been proposed to deal with autocorrelated observations. The residual chart, modified Shewhart chart, EWMAST chart, and ARMA chart are such charts widely used for monitoring the occurrence of assignable causes in a process when the process exhibits inherent autocorrelation. Besides autocorrelation, one other issue is the unknown values of true process parameters to be used in the control chart design, which are often estimated from a reference sample of in-control observations. Performances of the above-mentioned control charts for autocorrelated processes are significantly affected by the sample size used in a Phase I study to estimate the control chart parameters. In this study, we investigate the effect of Phase I sample size on the run length performance of these four charts for monitoring the changes in the mean of an autocorrelated process, namely an AR(1) process. A discussion of the practical implications of the results and suggestions on the sample size requirements for effective process monitoring are provided.     

New control charts for simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles

Simultaneous monitoring of the mean vector and covariance matrix in multivariate processes allows practitioners to avoid the inflated false alarm rate that results from using two independent control charts. In this paper, we extend exponentially weighted moving average semicircle and generally weighted moving average semicircle control charts to monitor the mean vector and covariance matrix of multivariate multiple linear regression profiles in Phase II simultaneously. These new control charts are compared with the existing control charts in the literature in terms of the average run length criterion. Finally, a case is considered to show the application of the proposed charts.     

Adaptive blinded sample size adjustment for comparing two normal means—a mostly Bayesian approach

Adaptive sample size redetermination (SSR) for clinical trials consists of examining early subsets of on‐trial data to adjust prior estimates of statistical parameters and sample size requirements. Blinded SSR, in particular, while in use already, seems poised to proliferate even further because it obviates many logistical complications of unblinded methods and it generally introduces little or no statistical or operational bias. On the other hand, current blinded SSR methods offer little to no new information about the treatment effect (TE); the obvious resulting problem is that the TE estimate scientists might simply ‘plug in’ to the sample size formulae could be severely wrong. This paper proposes a blinded SSR method that formally synthesizes sample data with prior knowledge about the TE and the within‐treatment variance. It evaluates the method in terms of the type 1 error rate, the bias of the estimated TE, and the average deviation from the targeted power. The method is shown to reduce this average deviation, in comparison with another established method, over a range of situations. The paper illustrates the use of the proposed method with an example. Copyright © 2012 John Wiley & Sons, Ltd.