1998 Fall Technical Conference Abstracts

Ajmani, "Robust Multivariate Control Charts"

Control charts are one of the most powerful tools for monitoring industrial processes. Developed in the 1920s by Walter A. Shewhart, they have gained wide acceptance in industry particularly in the manufacturing sector. Univariate control charts are used to monitor processes that manufacture products with a single quality characteristic of interest. In many instances, manufactured items may have two or more quality characteristics that jointly determine the usefulness or quality of the product. In most cases these quality variables are correlated. Multivariate control charts are needed to monitor such processes.

The performance of the multivariate control charts that are currently being used in industry and that are being cited in the literature have been studied under the assumption that the underlying distribution of the process is multivariate normal. In reality this assumption rarely holds. Our simulation studies have indicated that the normal theory multivariate control charts perform poorly when deviations from multivariate normality occur. We have proposed three robust multivariate Shewhart type charts that are based on the affine invariant multivariate one-sample sign and sign-rank tests that were developed by Hettmansperger et al. (1994), Randles (1989), and Peters and Randles (1991). Our simulation studies have indicated that the proposed charts perform better than the normal theory charts when deviations from multivariate normality occur.

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A variety of control charts are used for SPC including, Shewhart, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA). The Shewhart chart can detect large changes quickly, but is slow in detecting small shifts in the mean, whereas CUSUM and EWMA charts are better at detecting a small mean shift, but may be slow in detecting a large shift, and require tuning of their filter parameters. For multivariate SPC, similar control charts are used for the T2 value and residual of the latent variables determined by principal component analysis (PCA) or partial least squares (PLS), or for the T2 of the original variables. To detect large and small shifts quickly, multiple charts may be used together.

This paper will present a general framework for SPC based on representing the measurements at multiple scales by projection on a family of wavelets. This framework unifies several existing SPC techniques such as Shewhart, CUSUM, and EWMA. These existing methods will be special cases of the framework since they differ only in the scale at which they represent the measurements. Shewhart charts operate at the finest scale, CUSUM charts operate at the scale of all the measurements, which is the coarsest scale, while EWMA charts operate at a scale determined by the value of the tuning parameters. The resulting multiscale SPC (MSSPC) method will detect deterministic changes as large wavelet coefficients that violate detection limits determined separately at each scale. If the measurements are autocorrelated, the detection limits will change according to the power spectrum of the normal measurements, and the wavelet coefficients will be approximately decorrelated. Consequently, MSSPC will be able to monitor both uncorrelated and autocorrelated measurements without additional processing. The state of the process will be confirmed by reconstructing the signal based on the large wavelet coefficients at each scale, and applying a detection limit determined from the scales at which the event was detected. Thus, MSSPC will automatically adjust the detection limits for each measurement, and extract the signal feature representing the abnormal operation. This approach will be extended to multivariate SPC by using MSPCA and MSPLS.

Since MSSPC automatically selects the best filtered signal or residual at any scale for every measurement, it performs significantly better than existing methods for both, uncorrelated and autocorrelated measurements. Furthermore, simultaneous extraction of the signal feature relevant to abnormal operation improves the task of fault diagnosis, and integrates data preprocessing, process monitoring and fault diagnosis. The superior performance of MSSPC will be demonstrated by a run-length analysis and application to industrial problems. To the best of our knowledge, this work represents the first application of wavelets to SPC.

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Borror, "Optimal Design Strategies for Experiments Involving Noise Variables"

This paper discusses evaluating various designed experiments that involve both controllable and uncontrollable (noise) factors. It is desired to simultaneously minimize the influence of noise factors on the products and processes and determine the levels of the controllable factors that will optimize the overall response or outcome. A method is introduced that determines a model describing both the mean of the process and the variability of the process. The method is evaluated using a form of the variance dispersion graphs. Some examples of the industrial use of these designs are given.

This talk presents a short demonstration of response surface methodology (RSM). It then discusses the reasons for RSM's development. Topics include: the role of screening designs, steepest ascent methods, approximating models, transformations, canonical analysis, and ridge analysis. This talk emphasizes that RSM is a method of sequential investigation. It discusses the difference between the mathematical paradigm of theorem and proof and the scientific paradigm for the acquisition of knowledge.

1. Short demonstration of RSM.
2. Reasons for its development.
3. The role of screening designs, steepest ascent methods, approximating models, transformations, canonical analysis and ridge analysis.
4. RSM as a method of sequential investigation.
5. The difference between paradigm of theorem and proof and the scientific paradigm for the acquisition of knowledge.

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The latent variable multivariate regression model occurs frequently in the field of chemometrics. The subject has been addressed to date using a technique called partial least squares (PLS) regression. Latent variable spaces estimated using this method often play critical roles in such areas as process monitoring and designed experiments. They have become a key tool in the quest to take the large amount of process and experimental data available today and turn it into relevant, timely, information on the systems under investigation.

However, to date the statistical model for the data has not been dealt with in any detail. The statistical community has virtually ignored the latent variable multivariate regression model, usually assuming that the standard multivariate regression model holds. This can give misleading results, and may seriously limit the information that one can obtain from data that come from a latent variable model. Although the model can be used to obtain predictions of the response variables in a manner similar to standard multivariate regression, prediction is often not the goal in chemometrics applications. These applications often require estimates of the structure of the data. In this talk we clearly define the statistical model for latent variable multivariate regression and discuss its implications for data analysis. Four real datasets are used to illustrate the common occurrence of such data, two from chemistry and two from chemical engineering. We discuss some of the advantages of the latent variable model over the standard or reduced rank model.

It is hoped that this work will encourage more work on the statistical basis for latent variable regression modeling. Several key areas where more research is needed are presented.

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The background for this paper was an application involving a quality assurance laboratory in a high technology manufacturing company. The laboratory was responsible for assuring a stable measuring system for such characteristics as concentration of critical organic compounds. Laboratory management wanted to use control charts to provide graphical evidence of measurement system ability.

The standard deviation, a common index of precision, was hard to apply for this laboratory since the precision varied as a function of the chemical concentration. Over time the company adopted the coefficient of variation as a way to better compare and contrast the measuring system performance across chemicals and their respective concentrations. Since the laboratory had standardized on the use of the coefficient of variation, management wished to set up control charts based on this statistic.

Factors for control limits as they apply to the coefficient of variation are not readily available owing to the overall intractability of the sampling distribution. This paper shows how a simple simulation can be embedded in a spreadsheet to derive approximate control limits over a range of sample sizes. The straightforward for the simulation should encourage practitioners to apply it for similar applications. The intractability of a particular sampling distribution should seem less of a barrier to setting up specially tailored control charts.

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In many manufacturing environments such as the nuclear weapons complex, emphasis has shifted from the regular production and delivery of large orders to infrequent small orders. However, the challenge to maintain the same high quality standards while building much smaller lot sizes remains. To meet this challenge, specific areas need more attention, including fast and on-target process start-up, low-volume statistical process monitoring and control, relating key product characteristics to important process parameters via validated mathematical/statistical models, and estimating reliability given few actual performance tests of the product.

In this paper we address the issue of low-volume statistical process monitoring and control. We investigate an adaptive filtering approach to process monitoring with a relatively short times series of autocorrelated data. The emphasis is on estimation and minimization of mean squared error rather than the traditional hypothesis testing and run length analyses associated with process control charting. We develop an adaptive filtering technique that assumes initial process parameters are not known, and updates the process parameters as more data become available. Using simulation techniques, we study the data requirements (the length of a time series of autocorrelated data) necessary to adequately estimate process parameters. We show that far fewer data values are needed than is typically recommended for process control applications. And we demonstrate the techniques with a case study from the nuclear weapons manufacturing complex.

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To optimize the control factors of a product or process, often multiple quality characteristics must be considered. For some consumer products, in particular food products, some of the quality characteristics are evaluated by sensory panels. The data produced by these panels is in the form of matrices of paired comparisons of the treatments. The contribution here is a method of incorporating sensory panel data into the analysis of a multiresponse optimization experiment.

We apply the Hierarchical Method for Multiresponse Experiments to a case study. The problem is the optimization of 26 sensory characteristics of a military ration product. The panel consisted of 8 subjects with 4 replications each.

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The new philosophy of quality management characterized by a clear focus on processes and a commitment to continuous quality improvement is only now beginning to impact the nuclear energy industry. The main reasons for this delay were the strong governmental regulations and the much less global competition in that industry. Also the apparent need for always more control activities, because of the high risks involved, makes the adoption of a modern quality philosophy difficult.

Efforts currently under way at PreussenElektra, an important electrical utility in northern Germany, to convince their suppliers of nuclear fuel elements to use the new quality concepts, will be described. The role and responsibilities of governmental control authorities will be discussed. Examples of statistical tools used to demonstrate that error prevention instead of error correction will increase quality and at the same time reduce costs will be shown. The concepts and keys elements of the Guidelines "Quality System for Fuel Elements", developed especially for that purpose, will be exposed. The talk will address the issues facing statisticians working in an environment resistant to change and the problem of communication statistician-engineer-governmental representative.

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Traditional experimental design (DOE) lives in a data-poor environment in which at most a few dozen experimental runs are possible. However, in automated in vitro microplate bioassays, thousands of runs are quite feasible, although under complex randomization restrictions. Such capabilities require new thinking about how to do DOE in a data-rich environment. This case study illustrates these challenges.

The context was this. High throughput screening (HTS) assays are used to test thousands of compounds for biological activity. These tests are carried out in 96 or 384 well plates in which small(µl = 10-6 liters) quantities of reagents and test compounds are (robotically) dispensed, mixed, incubated, and read. The experimental goal was to investigate to what degree a common setup could be used for ten such related assays. Eight continuous and one 6-level categorical factors were varied. However, one continuous factor was incubation time, which had to be done on a plate by plate basis, so split-plotting was inherent. Several 96 well plates could be done per day, allowing several thousand runs to be done in the one month allotted time.

The analysis also presented challenges. With traditional polynomial models, the large sample size would produce many unimportant but statistically "significant" effects; higher level interactions due to the categorical factor were likely; and it was important to present results graphically so that the experimenters could clearly interpret them. We found that binary regression tree methods met these needs well. However, the purpose of the presentation is not to provide definitive answers but to stimulate discussion, so audience participation will be welcomed.

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Two-level factorial experiments with zero or parsimonious center point replication are efficient components of response surface methods and are applied widely. Experiments simulating production justify assuming effects sparsity and have been treated in many articles.

But efficient experiments in research and development are essentially saturated. And most of the coefficients will have moderate absolute values, a few will be large, and a few will be small. They will approximate a straight line on a half-normal plot. Such experiments were treated by Holms (1980) assuming neither factorial nor center point replication. The empirically optimized procedure used Cochran’s distribution.

The present work includes the option of center point replication. That option suggests that the F-distribution as well as Cochran’s distribution should be used. The results include empirically optimized combined F- and Cochran’s distribution procedures, for an unfavorable distribution of population coefficients, for each of the options of zero through six center points.

In preparation for a major initiative starting in 1997 to improve safety, I was asked by Sr. Mgmt. to analyze historical safety data from over 400 reporting Units. The objective was to present data from each Unit to each Unit in a simple, unambiguous manner and to show whether or not "significant" (real) change is occurring. Very quickly I found that many Units had only a few injuries that could be used to define "Mean Time Between Injuries" in the "before 1/1/1997" period and could expect even fewer as 1997 unfolded. I also found that the literature addressed change detection in the Poisson parameter, but did so using large sample theory (known parameter during baseline). I found no papers showing small sample theory for Poisson parameter change detection.

I have developed a reasonably complete system for detecting change in the Poisson parameter, a system easily understood and correctly interpreted by people with no statistical background. In the process I have derived the small sample distribution of a variety of useful statistics that can deal with the issue of change detection.The fundamental concept is to use a ratio, current data divided by past MTBI. This approach completely removes the Poisson parameter from the problem. The distribution is entirely a function of known sample sizes. The pdf and DF are reasonably easy to use and the DF has a closed form solution.

The ratio statistic has a strong asymmetry, so must be transformed into the Normal before laymen will feel comfortable with a control chart. Optimal power transformations depend upon sample size. The recommended power transform currently in vogue is to use 1/3.5 or 1/3.6 as the exponent. This is shown to be (1) correct for asymptotic situations (large n), but (2) very incorrect for small samples. A table with the optimal power transform is given as a function of sample size. Change detection is done with a CUSUM.

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True replication in a designed experiment permits calculation of a pure error mean square that is used to check model lack-of-fit. In 2-level factorial and fractional factorial designs, replication is usually done for center points, although other replication patterns may be useful. The number of replicate points are often small (hence the term, "almost unreplicated factorial") and the usual analysis to determine important effects typically ignores these points. For instance, half-normal and normal probability plots display only the effect estimates, without any graphical representation to account for replicate observations. A valuable contribution of Lenth (1989) provides a numerical method for selecting important effects from unreplicated factorials. As the purpose of Lenth's paper was to provide a method for analyzing unreplicated factorials, he made no provision to take advantage of replication.

We propose supplementing Lenth's method by combining his estimate of error variance from small factorial effects with the pure error variance from replicate observations or (possibly) other estimates of error variance. We demonstrate that there can be a substantial increase in power to detect effects by incorporating this additional information. In fractional factorials, this increase in power is particularly valuable. We also propose augmenting half-normal or full normal probability plots of effects by adding points that represent error effects estimated from the replicate observations and (possibly) other estimates of the error variance. As these points are truly error, they provide a set of points on the plots to judge the potential importance of all effects. The number of extra points added equals the number of degrees of freedom associated with the within sum of squares for the replicate observations.

These new methods supplement the valuable methods of Lenth and probability plotting to select effects in factorial and fractional factorial studies. Specific advantages of the new methods include:

Two-level factorial and fractional factorial experiments are frequently used with success in industrial applications. However, it is also common that process changes indicated by such experiments fail to produce the expected benefits when one seeks to confirm the results. One explanation for such disappointments is that the effects of factors under study actually vary over time, as a result of interactions between these process factors and temporal process changes such as raw material lots, environmental conditions, etc. Under such conditions, short-run experiments can be misleading, and experiments spread out over time are essential to identify accurately the long-term average effect of process changes. Although manufacturing process experiments spread out over time require a greater commitment of resources, they are more likely to produce useful results. We propose 2^k factorial and 2^k-p fractional factorial designs in incomplete blocks for applications where factors interact with blocks. When the individual blocks are resolution IV fractions, testing of main effects is straightforward. For situations where many smaller blocks are utilized, partial confounding schemes are considered. We illustrate the analysis of such experiments using both specialized plots and significance tests.

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The fabrication of integrated circuits (ICs) is accomplished through a vast sequence of processing steps. Moreover, the silicon wafers on which the ICs are produced move through the process in lots of size 24 or more. Although some processing steps are applied to individual wafers, for other steps several wafers (or even several lots) are processed simultaneously as a group. To facilitate experimentation with such a multistage batch process, "split lot" experimental designs are attractive, since they allow the experimental wafers to be split into sub-lots for processing. The designs are obtained by using different sets of factorial effects to define the composition of the sub-lots at each step. Specific examples are given with up to nine processing steps. A split lot design balances the way in which the wafers are repartitioned at each stage in the experiment. Taguchi (1987) refers to such experiments as multi-way split unit designs. Two-way split-unit experiments arise naturally in agriculture, where some factors are assigned to rows and other factors to columns in a field. The term "strip plot," which originated in this agricultural context, remains in common usage when the experiment involves only two processing steps (see e.g. Miller, 1997). Although semiconductor fabrication motivated our interest in these designs, their applicability includes any industry with batch processing of discrete units.

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The Wavelet Transform (WT) has recently attracted interest due to its ability to provide simultaneously time/ spatial and frequency information. This is an advantage in comparison to the Fourier transform that gives only frequency information. The ability of the WT to compress the information in a few coefficients is widely exploited for de-noising signals by thresholding. The basic assumption is that small coefficients bear little information and can therefore be removed or shrunken. De-noising improves the signal-to-noise ratio and hence often the precision of a calibration. Generally, a signal consists of a background, chemical information (e.g. spectral bands) and noise, which are mainly represented by low, medium and high frequencies, respectively. The WT can approximately separate these components by representing them at different frequency levels in the wavelet domain.

Considering only the high frequency levels, it is possible to analyze the structure of the noise. Non-stationarity, heteroscedasticty and correlation in the signal can be detected and/or estimated, due to the fact that time/spatial information is available.

Regression utilizes mainly medium frequency wavelet coefficients. The time/spatial information allows to distinguish between local features (e.g.: IR absorption bands) and facilitates a parsimonious and interpretable calibration. The property of vanishing moments makes calibration results based on wavelet coefficients more robust with respect to a varying background, in the case it can locally approximated by a polynomial.

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Quality management ideas have had a substantial impact on the non-academic operations of many colleges and universities, but very little impact on the academic dise. Why? There are "cultural barriers" on both sides of the academic/industry divide, always clearer to those on the opposite side. Yet there are many specific ways in which quality ideas could quickly improve academic processes. Statisticians should be leaders in encouraging change.