Some of my colleagues have recently been pushing multivariate charts. These charts combine several different process outcome values into a single number that’s placed on a chart. Unfortunately, this theoretically elegant approach breaks down in practice for reasons that will be explained here.

We shall use the example of Adhesive 442 to illustrate the ideas behind multivariate charts. Each batch of Adhesive 442 is characterized by two key properties, pH and viscosity. Twenty consecutive batches had the values shown in Figure 1.

Figure 1: Adhesive 442 data

The average pH for Adhesive 442 is 7.9825. The average moving range for the pH values is 0.221. Based upon this value, the sigma(X) value for pH is 0.196. The XmR chart for the pH values is shown in Figure 2.

Figure 2: XmR chart for Adhesive 442 pH values

Even though the average pH is near the target value of 8.00, Western Electric detection Rule Two calls our attention to two long runs near the limits. These suggest that this process is not being operated consistently.

The average viscosity for Adhesive 442 is 5.020. The average moving range for viscosity is 0.421. Based upon this value, the sigma(X) value for viscosity is 0.373. The XmR chart for the viscosity values is given in Figure 3.

Figure 3: XmR chart for Adhesive 442 viscosity values

The average viscosity is near the target value of 5.00, but once again there are long runs near the limits. So this chart also shows that this process is not being operated consistently.

The parallelism seen on these two X charts reinforces the message of each. This process changed between batches 10 and 11, and again between batches 18 and 19. Whatever happened at these points in time affected both pH and viscosity. By finding and controlling the assignable cause of these changes, we can improve both the consistency of the process and the consistency of the future product stream. Thus, as always, the separate process behavior charts are sufficient to tell you when and how to take action to improve the process.

Simple multivariate charts

If we wish to compute a combined measure of how close each batch is to the collective target value, we’ll need to combine the various outcomes for each batch. This requires that each variable be transformed to a common scale; this is done by standardization. Each variable will have its own measurement units. But estimates of the standard deviation tell us how many of these measurement units correspond to a standard unit of dispersion. So when we subtract the target value and divide by sigma(X), we are converting our variables to a common scale so that they can be combined correctly. These standardized values tell us how many standard deviations lie between that observation and its target value:

For the pH values, the target value is a pH of 8.00 units, and sigma(X) for the pH values was found to be 0.196 pH units per standard deviation. The standardized pH values, Z1, are listed in Figure 4. For the viscosity values, the target value is a viscosity of 5.00 units; sigma(X) for the viscosity values was found to be 0.373 units per standard deviation. The standardized viscosity values, Z2, are listed in Figure 4.

Figure 4: Standardized pH and viscosity values

When we plot the standardized values for each batch as points in the coordinate plane we get Figure 5, where the horizontal axis shows the standardized pH values and the vertical axis shows the standardized viscosity values. Here, each batch is represented by a single point, and the distance from that point to the origin represents how far off-target that batch is.

The red lines forming the square represent the standardized limits centered on the target values. While Figure 5 shows how these two variables are related to each other, it hides the time-order sequence shown by the XmR charts.

Figure 5: Batch outcomes as points on the plane

In Figure 6, to quantify the distance by which each batch misses the target value, we compute the two-dimensional Euclidean distance of each point from the origin, D₂, using the standardized values in Figure 4.

Figure 6: Euclidean distances to target

Theory tells us that when a process is operated consistently, the two-dimensional Euclidean distances, D₂, should have an average of 1.25 standard deviations and a three-sigma upper limit of 3.22 standard deviations. These two values are shown in Figure 7 as circles.

Figure 7: Limits for standardized Euclidean distances

Notice that five batches which managed to fit just inside the square are outside the outer circle. In addition, only one batch managed to fall below the expected average distance shown by the inner circle. So, once again, we have evidence that these batches are inconsistent. This process essentially produced two different products and shipped them both under the same label.

The square and circle in Figure 7 show the difference between using separate charts and using a multivariate chart. The separate charts are like using rectangular coordinates, while the multivariate chart is like using polar coordinates. Unfortunately, we have trouble drawing Figure 7 when working with more than two variables. To get around this problem, we usually plot the radial distances for the batches in sequence as in Figure 8.

Figure 8: A simple multivariate chart for Adhesive 442

Figure 8 reveals that batches 6, 8, 15, 16, and 17 are too far from the origin for the deviations to have been due to routine variation. In addition, Figure 8 shows batches 5, 7, 13, 19, and 20 to be close to the limit. So, in all, 10 of the 20 batches appear to be substantially off-target.

However, unlike Figure 7 and figures 2 and 3, Figure 8 doesn’t tell you which batches are on the low side and which are on the high side. Moreover, even though Figure 8 uses the time order, it doesn’t show the transition from low to high after Batch 10, and the transition from high to low after Batch 18 that’s shown by figures 2 and 3.

More than two variables

The Euclidean distance for p variables will be the square root of the sum of the squares of the p standardized values:

So, to get a simple multivariate chart that works for more than two variables, we plot the Euclidean distances for each batch in their time order as in Figure 8.

Where do the limits come from? We use theoretical limits here, rather than empirical limits, because we’re no longer working with the original data.

Both theory and practice have shown that when data from a predictable process are standardized and combined into Euclidean distances, they can be modeled by the square root of a chi-square distribution (known as a chi-distribution). So when we use the parameters of a chi-distribution to create limits, we’re comparing the transformed data to limits that assume the process was operated predictably. The mean and standard deviation of a chi-distribution with p degrees of freedom are:

Figure 9 lists the means and three-sigma upper limits for p = 2 to p = 25.

Figure 9: Central lines and upper limits for Euclidean distances

Reasonable approximations to these values may be found by using the square root of p as the central line, and the square root of the 99.5th percentile of a chi-square distribution with p d.f. as the upper limit.

An unconscious shift

There’s a subtle change in emphasis between the process behavior charts and the simple multivariate chart. This shift is an unavoidable side effect of combining variables. Figures 2 and 3 were designed to show the process behavior over time. They allow you to detect the presence of unknown causes that have appreciable impact on process and product. Once detected, these causes can then be controlled to reduce variation and improve on-target operation.

In Figures 7 and 8, the emphasis shifts to describing each batch of product. This shift from process to product is profound. It takes you back to looking for bad stuff while it buries the information needed to improve both process and product!

Dr. W. Edwards Deming observed that the Western approach to production could be summarized as “Burn the toast, then scrape it.” Scraping burned toast doesn’t improve the overall quality. Process monitors seek to steer the process to a minimum level of burned toast, but steering doesn’t eliminate the variation that creates the burned toast. Process behavior charts provide a way to learn how to quit burning the toast.

So, when you think of a process behavior chart as merely another type of process monitor, you’re unlikely to be aware of this shift in emphasis. Nevertheless, it will still severely limit what you may accomplish with your data analysis.

But wait, there’s more!

The simple multivariate chart of Figure 8 is not what your software is likely to give you. Your software is probably going to subject your data to a more complex transformation known as Hotelling’s T-square. Read on to see what some of my colleagues want you to use.

The Euclidean distance essentially ignores the possibility that the p process outcomes may be correlated. Hotelling’s T-square statistic attempts to use the correlation structure by changing the circles in Figure 7 into ellipses. This is done by estimating the variance-covariance matrix for the p process outcomes, and using this matrix to “standardize” the deviations from target.

The central line and upper limit for a T-square chart depend upon an F-distribution with p and k-p degrees of freedom. For our example, p = 2 and k = 20.

Without getting lost in the complexities of finding and inverting the variance-covariance matrix, the T-square value for a batch is the square of a distance to the origin as measured with a rubber ruler. For the Adhesive 442 data, the T-square values are shown in Figure 10 and plotted in Figure 11.

Figure 10: T-square values for Adhesive 442 batches

Figure 11: T-square chart for Adhesive 442 batches

And the wonderful, mathematically sophisticated, T-squared transformation has made all the signals disappear! Figure 8 showed 10 batches to be far from the target. Here, these 10 batches are hugging the central line. Figure 8 and Figure 11 have nothing in common except the fact that they were computed from the same data. Since they don’t tell the same story, one of them has to be lying. To see what’s happening, we return to our two-dimensional plot.

Figure 12: Ellipses created by the T-square transformation

The inner ellipse in Figure 12 shows those points with a T-square score of 1.6. Since most of the points are near this inner ellipse, they have similar T-square values, and so they hug the central line on Figure 11. The outer ellipse shows points with a T-square score of 11.0. Batch Two is labeled with the number 2. It’s the only point in the northwest quadrant, and being close to the outer ellipse, it has a T-square value of 10.8. Its elliptical distance to the origin is much greater than the elliptical distances of all the other points.

To understand the rubber ruler being used here, compare Batch Two and Batch 14. Batch Two is 0.25 low on pH and 0.48 high on viscosity (D₂ = 1.81, T-square = 10.8). Batch 14 is 0.30 high on pH and 0.58 high on viscosity (D₂ = 2.18, T-square = 0.69). So, despite similar deviations from their targets, these two batches get wildly different T-square values. Batch 2 stands out on the T-square chart because it doesn’t fit in with the pattern assumed by the ellipses in Figure 12.

The ellipses are created by the correlation between our two variables, but this correlation was actually created by the inconsistent operation shown in figures 2 and 3. Without the shifts in level seen there, the correlation would be much smaller and the ellipses would be more circular. This brings us to an inherent fallacy in the T-square transformation: It’s impossible to compute meaningful correlations when the underlying process is operated inconsistently.

When the process is operated inconsistently, the variance-covariance matrix will get distorted and the standardization will be wrong. As a result, signals will be missed, and perfectly ordinary batches will suddenly appear to be extreme. And this problem is unavoidable because correlations have to be computed globally.

So, for our example, where both XmR charts show evidence of inconsistent operation, and where the simple multivariate chart reveals 10 batches that are off-target by more than 2.85 standard deviations, the T-square chart shows all but one batch to be hugging the central line! This makes the T-square chart nothing more than a complex way to lie with statistics.

Summary

When we combine outcomes into a summary metric for each batch, the emphasis shifts from process characterization and improvement to detecting bad batches.

The Euclidean distance and T-square values are based on radial distances. As such, they don’t reveal whether a batch is above or below the target values, or which of the p variables are responsible for a large deviation. This is why multivariate charts can never be more than a process monitor. Yet when they distort the distances they can be much less—they can be complete nonsense. In any case, they can never begin to tell you how to improve your process.

The primary assumption of classical statistics is that the data are observations on independent and identically distributed random variables. This is equivalent to “assume the process is predictable so that the data are homogeneous.” When this assumption is not true, the T-square transformation will be incorrect. It will distort the results and bury the true signals as it did with our example, and it can create false signals as well.

As always, the best analysis is the simplest analysis that provides the needed insight. All of the useful information about this process was revealed by the two process behavior charts. Everything else revealed, after much effort, by the multivariate charts was either redundant, irrelevant, or completely wrong.

Process behavior charts were created for process improvement. Those who think of them as merely another process monitor technique sell them short and will fail to realize their full potential to improve quality, productivity, and competitive position.

Donald J. Wheeler’s complete “Understanding SPC” seminar may be streamed for free; for details, see spcpress.com.