A Master Black Belt was perplexed by the software’s correlation and regression analysis output. The results were pure nonsense. In addition to regression coefficients that were negative when common sense told him they should be positive (and vice versa), some of the correlation coefficients were large, while the corresponding regression coefficient p-values were insignificant. What the heck was going on?

The problem with the Master Black Belt’s data was multicollinearity, a condition that’s encountered more often as Six Sigma practitioners move from engineering and manufacturing data to customer survey data.

Multicollinearity refers to linear intercorrelation among variables. Simply put, if nominally “different” measures actually quantify the same phenomenon to a significant degree–i.e., the variables are accorded different names and perhaps employ different numeric measurement scales but correlate highly with each other–they’re redundant.

Multicollinearity isn’t well understood by the applied statistics community. Re–searchers at Arizona State University found widespread misunderstanding of multicollinearity and other regression analysis concepts among graduate students who had taken, on average, five graduate and undergraduate statistics courses. I’ll try to explain multicollinearity by using a simple, everyday metaphor: a table.

Consider the photograph of a table shown in figure 1. This is a finely built table I purchased at Wal-Mart for only $99 (chairs were included). In my metaphor, the table is a model. The legs of the table represent x variables, or predictors. The top represents the y variable, the predicted variable. My intent is to build a table where the top is well supported by the legs, i.e., the predictions are well explained by the predictors.

The table legs are placed at 90° to each other. This angle provides maximum support for the top. In statistics, when two variables are uncorrelated with each other, they’re said to be orthogonal, or at 90° to one another. If you create a scatter diagram of two uncorrelated variables and draw best-fit lines, you get something that looks like figure 2: The lines are at 90° to one another.

Consider the two x variables to be the legs of the table, the circle to be the table top and the dots to be objects on the table’s surface. You can see that the x variables do a good job of “supporting” the y variable. If you add a few dots anywhere within the circle they’re unlikely to make the table tip over. In other words, the model is stable.

What if the table wasn’t built with the legs at 90°? Clearly, it wouldn’t be as stable. The statistical equivalent to this poorly designed table is shown in figure 3. The lines no longer cross each other at 90° because the two x’s are correlated with one another. There are areas within the circle (i.e., our y model and the equivalent of the table top) that aren’t well explained by the model. If we take additional samples and get y’s in either of these regions, the coefficients of the model will change drastically, i.e., the model, like the table, will be unstable. This is the multicollinearity problem in a nutshell.

Using our table metaphor, we can see that regression models are more stable when the x’s used to predict the y are uncorrelated with one another. If the x’s are correlated, there are a number of statistical techniques that can be used to address the problem. When many x’s are involved, I like to use principal components analysis, which replaces many correlated variables with a smaller number of uncorrelated factors. This not only overcomes the problem of multicollinearity, it usually also makes the model easier to understand and use.

A key feature of Six Sigma is the integrated use of design of experiments and robust design of product and process. Experiments are an important part of this; they serve both to find the most robust design settings and to verify that the designs are, in fact, robust.

As beneficial and productive as experiments can be, the process of conducting them has its drawbacks. The workplace — be it a factory, a retail establishment or an office — revolves around a routine. The routine is the “real work” that must be done to generate the sales that, in turn, produce the revenues which keep the enterprise in existence. Experimenting, by its very nature, means disrupting the routine. Important things are changed to determine what effect they have on various important metrics. Often, these effects are unpleasant; that’s why they weren’t changed in the first place. The routine was established to steer a comfortable course that avoids the disruption and waste that results from making changes.

Unless we change, however, we can never improve. Six Sigma generates as much improvement by changing things as it does by reducing variability. It’s part of the Six Sigma paradox, which states that to reduce variability in products and processes, we must encourage variability and experimentation.

It’s possible to conduct “virtual” experiments using existing data and artificial neural network (neural net) software.   Neural nets are a class of very powerful, general-purpose tools readily applied to prediction, classification and clustering. They have a proven track record in many data mining and decision-support applications. Neural nets have been applied across a broad range of industries, from predicting financial series to diagnosing medical conditions, from identifying fraudulent credit card transactions to predicting failure rates of engines.

Neural networks use a computer to model the neural connections in human brains. When used in well-defined domains, their ability to generalize and learn from data mimics our ability to learn from experience. However, there is a drawback. Unlike a well-planned and executed DOE, a neural network doesn’t provide an explicit mathematical model of the process. For the most part, neural networks must be approached as black boxes with mysterious internal workings, much like the human brain itself.

All companies record important information. Such data represents potential value to the Six Sigma team. It contains information that can be used to evaluate process performance. If the data include information on process settings, for example, they may be matched up to identify possible cause-and-effect relationships and point the direction for improvement. The activity of sifting through a database for useful information is known as data mining. The process works as follows:

1.Create a detailed inventory of data available throughout the organization. The information systems department may already have compiled this information.

2.Determine the variables that apply to the process being improved and for which historical data exist.

3.Using a subset of the data, train the neural net to recognize relationships between patterns in the independent variables and patterns in the dependent variables.

4.Validate and test the neural net’s predictive capacity with the remaining data.

5.Perform experimental designs as you would on a real process. However, instead of making changes to the actual process, make changes to the “virtual process” as represented by the neural net.

6.Once the sequential application of designed experiments has been completed using the neural net model, use the settings from the neural net as a starting point for conducting experiments on the actual process. If the experiment results confirm the results from the neural net, you can reduce sample sizes and move quickly along the path discovered by the virtual DOE  process.

The entire virtual experimentation process helps answer the question, “Where are we?” It’s important to recognize that neural net experiments aren’t the same as live experiments; they are only simulations. However, the cost of doing them is minimal compared with experiments in the real world and the process of identifying input and output variables. Moreover, the process of deciding at which levels to test these variables will bear fruit when the team moves on to the real thing. Virtual experiments allow a great deal more “what if” analysis, which may stimulate creative thinking from team members.

This article is based on an excerpt from The Complete Guide to Six Sigma, Quality Publishing LLC, scheduled for publication summer 1999. Copyright � 1999 by Quality America LLC. The article first appeared at www.qualityamerica.com . Reprinted by permission.