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Multivariate Control Charts: T2 and Generalized Variance

Multivariate analysis techniques may be useful in statistical process control (SPC) whenever there is more than one process variable. Multivariate control charting is usually helpful when the effect of multiple parameters is not independent or when some parameters are correlated. This article focuses on parameters that correlate when the Pearson correlation coefficient is greater than 0.1. (Ranging from -1 to +1, the Pearson correlation coefficient is a measure of the degree of the linear relationship between two variables.) Applying univariate control charts is possible but inefficient – and can lead to erroneous conclusions. Multivariate methods that jointly monitor these variables is an alternative approach.

Here is a simple hypothetical two-variable example to illustrate how an erroneous conclusion may be made. Figures 1 and 2 are a simple two-variable example that illustrates the potential for an analysis error.

Figure 1: X-bar and R Charts of X1

Figure 1: X-bar and R Charts of X1

Figure 2: X-bar and R Charts of X2

Figure 2: X-bar and R Charts of X2

The control charts for X1 and X2 are stable and in control. None of the Western Electric rules (rules for detecting out-of-control conditions on control charts) have been violated. It seems as though all is well; however, there is more to the story. Plotting X1 versus X2 shows a point separated from the rest (Figure 3). The joint control region is the ellipse formed by the normal data.

Figure 3: Marginal Plot of Mean X2 Versus Mean X1

Figure 3: Marginal Plot of Mean X2 Versus Mean X1

(Note: If the process is not creating defects in the region where X1 and X2 are not typically operating [see red circled area of Figure 3 above], then there is no point to alerting the process. This approach only works if a defect is made.)

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That observation (data point circled in red) appears unusual compared to the others when examined simultaneously. Two others might be questionable. What is going on? The probability of an out-out-control point in a univariate chart (X1 or X2 exceeding their respective 3 sigma limits) is 0.0027, 1 in 370 (Z = 3, P(d) = 0.00135 per tail, so 2 tailed = 0.0027). The risk of making a wrong decision increases rapidly as the number of variables increase. The probability of simultaneously detecting an out of control point in 2D space using univariate charts is the straight multiplication of the probability of detection in the individual charts. The probability of X1 and X2 simultaneously exceeding their respective 3 sigma limits is (0.0027) x (0.0027) = 0.00000729.

The alpha risk – the risk of not detecting a special cause – is:

α‘ = 1 – (1- α)p
where p = the number of factors
α = the risk of error (0.0027)
α‘ = the adjusted risk

The alpha risk increases linearly as the number of factors increase. So, in a multivariate situation the likelihood of missing a special cause increases as the number of factors increase.

Figure 4: Risk of Missing a Special Cause

Figure 4: Risk of Missing a Special Cause

The overall goal of process control is to produce products that are uniform around a target. Walter A. Shewhart, the father of control charts, described two types of mistakes that can be made in the effort to achieve uniformity. They are:

  1. Attributing an outcome to special cause of variation when it comes from common causes of variation.
  2. Attributing an outcome to common causes of variation when it comes from a special cause.
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The goal is to make these mistakes infrequently. Here, however, the risk of these mistakes increased.

In this example, the process is not dominated by a single factor, but by more than one important related factors. Multivariate control chart techniques can be applied to monitor our process. By using a multivariate model with a control ellipse, as in this example, the out-of-control point can be detected.

Figure 5: Out-of-control and Joint Control Points

Figure 5: Out-of-control and Joint Control Points

When correlation exists among quantitative variables on a single experimental unit, a single control chart based on Hotelling’s T2 statistic (testing hypotheses on multiple measures) will provide an alpha level (no more risk inflation) and greater sensitivity to out-of-control points than multiple X-bar and R charts.

Why are these multivariate control charts not used more frequently? The associated calculations are laborious. We need to calculate the subgroup means for each variable, X1 and X2Xn, calculate the subgroup variances and the subgroup covariances. Then calculate the mean of the subgroup means, the mean of subgroup variances and the mean of the covariances. Next, designate the sample covariance matrix and the mean vector. Then calculate T2 and the control limits. Finally T2 can be compared to the control limits to determine if individual points are out of control. These calculations are time-consuming and confusing; however, conceptually it is simple. For this two-factor example, the calculations establish an ellipse (Figure 6). The T2 statistic measures the distance from the centerline of the ellipse to each point. T2 is plotted on a Shewhart type of chart. A T2 chart is the multivariate analogy of the X-bar chart. Control limits are based on the F-distribution for subgroups.

Figure 6: Calculating Hotelling’s T2 Statistic

Figure 6: Calculating Hotelling’s T2 Statistic

The generalized variance chart is analogous to the R-chart but with more intensive calculations.

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equations

Fortunately, software is able to perform these calculations with just a few clicks of a mouse. For this example, two points would be flagged as special cause as well as a variation-driven stability concern (Figure 7).

Figure 7: T2 and Generalized Variance Chart of X1, X2

Figure 7: T2 and Generalized Variance Chart of X1, X2

Is X1 or X2 – or both – driving the out-of-control points? An automatic decomposition output is shown here.

Test Results for T2 Chart of X1, X2
PointVariablep-value
Greater than UCL3X10.0001
5X10.0209
X20.0018

Test Results for Generalized Variance Chart of X1, X2

TEST. One point beyond control limits.
Test failed at points:  1

This output indicates that point 3 is driven by X1 and point 5 is driven by both X1 and X2; therefore, both must be investigated even though the univariate charts indicated both variables were stable and in control (refer back to Figures 1 and 2).

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After finding the issue with points 3 and 5, corrections can be made and control can be monitored with the T2 and general variance chart.

That simple two-variable example was an easy one for visualizing the joint control space. Let’s look at a complex injection-molding process with many related variables such as fill time, recovery time, transfer pressure, transfer position, nozzle temperature and cycle time (Figure 8).

Figure 8: Injection-molding Process

Figure 8: Injection-molding Process

There are correlations between these variables that indicate they are not independent.

Correlations Among Variables
Cycle TimeFill TimeRecovery TimeTransfer PositionTransfer Press
Fill Time0.090
Recovery Time0.2990.083
Transfer Position-0.023-0.115-0.426
Transfer Press-0.0770.110-0.257-0.200
Nozzle Temperature-0.025-0.059-0.0240.065-0.072
Cell contents: Pearson correlation

All of these variables exhibit special cause variation in the individual control charts. Where should improvement efforts be focused and prioritized?

Figure 9: Individual Control Charts

Figure 9: Individual Control Charts

T2 and generalized variance charts can be created (Figure 10). There are still many out-of-control points.

Figure 10: T2-Generalized Variance Chart of All Variables – Before

Figure 10: T2-Generalized Variance Chart of All Variables – Before

Then look to the decomposition and create a Pareto of the number of times each variable contributed to out-of-control points (Figure 11).

Figure 11: Pareto Chart of Significant Variables – Before

Figure 11: Pareto Chart of Significant Variables – Before

This indicates that the factors that drive recovery time should be looked at. It is time to consult the process experts and allow improvements to be made. The multivariate chart constructed after changes were implemented shows improvement.

Figure 12: T2-Generalized Variance Chart of All Variables – After

Figure 12: T2-Generalized Variance Chart of All Variables – After

Figure 13: Pareto Chart of Significant Variables – After

Figure 13: Pareto Chart of Significant Variables – After

The process of looking to the decomposition and creating a Pareto chart must be recreated now. The practitioner must investigate what about these points is driving the variables (transfer pressure, cycle time, fill time, recovery time and transfer pressure) to an abnormal region of the control space. It is time to return to the experts to continue the process improvement work.

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When multiple variables are related, individual univariate control charts can be misleading and at best are inefficient. Hotelling’s T2 and generalized variance control charts are useful for continuous improvement and process monitoring. The availability of reliable software takes the math “magic” out of these control charts. Although the example used here is manufacturing related, the tool is not limited to production issues.

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Comments 6

  1. Andy Urquhart

    What an excellent article. Motorola Semiconductor Products Sector in Austin started using Hotelling’s T-squared in 1986 and then broke all yield records month on month. Integrated into Motorola’s Multi-Vari Chart software developed by Mike Wolfe, we were able to switch between univariate and multivariate charts easily so we could readily identify sources of variation in the process.

    Andy Urquhart
    Motorola Six Sigma Champion, Austin

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  2. Togy

    Excellent article.

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  3. Aishah

    I need to know how you perform the automatic decomposition in order to identify which process variables contribute to the out-of-control state.

    Thank you for your kind sharing.

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  4. jaime rebolledo

    Great reading… Can you suggest me a software to perform a decomposition of the most related variable that caused the out of control points.

    Best
    Jaime Reboledo

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