As we left off in our last column, we had proposed a definition of linearity. Now let's start by delving into the ins and outs of the Durbin-Watson statistic (1-6) and looking at how to use it to test for nonlinearity.

In fact, we've talked about the Durbin-Watson statistic previously in our columns, although a long time ago and under a different name. Quite a while ago we published a column titled "Alternative Ways to Calculate Standard Deviation" (7). One of the alternative ways described was the calculation by Successive Differences. As we shall see, that calculation is very closely related to the Durbin-Watson statistic. More recently we described this statistic (more directly named) in a sidebar to an article in the American Pharmaceutical Review (8).

To relate the Durbin-Watson statistic to our current concerns, we go back to the basics of statistical analysis and remind ourselves how statisticians think about statistics. Here we get into the deep thickets of statistical theory, and meaning and philosophy. We will try to keep it as simple as possible, though.

How DB Works Let us start with two of the formulas for standard deviation presented in our earlier column (7). One of the formulas is the "ordinary" formula for standard deviation:

The other formula is the formula for calculating standard deviation by Successive Differences:

Now we ask ourselves the question: "If we calculate the standard deviation for a set of data (or errors) from these two formulas, will they give us the same answer?" And the answer to that question is that they will, if (that's a very big "if") the data and the errors have the characteristics that statisticians consider "good" statistical properties: random, independent (uncorrelated), constant variance, and in this case, a Normal distribution, and for errors, a mean (μ) of zero, as well. For a set of data that meets all these criteria, we can expect the two computations to produce the same answer (within the limits of what sometimes loosely is called "statistical variability").

So under conditions where we expect the same answer from both computations, we expect the ratio of the computations to equal 1 (unity). Basically, this is a general description of how statisticians think about problems: First, compare the results of two computations of what is nominally the same quantity when all conditions meet the specified assumptions. Then if the comparison fails, this constitutes evidence that something about the data is not conforming to the expected characteristic (that is, is not random, is correlated, is heteroscedastic, is not Normal, and so forth). The Durbin-Watson statistic is that type of computation, stripped to its barest essentials. Dividing equation 2 by equation 1 above, canceling similar terms, noting that the mean error is zero and ignoring the constant factor (2) we arrive at: