Durbin-Watson Statistic

We continue in out next chapter with a discussion of using the Durbin-Watson Statistic for testing for nonlinearity. 

Linearity in Calibration Act III Scene II - A Discussion of the Durbin-Watson Statistic, a Step in the Right Direction... 

As we left off in Chapter 63, we had proposed a definition of linearity. Now let us 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 have talked about the Durbin-Watson statistics in previous chapters, 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 indeed 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. 

In both the linear and the nonlinear cases the total variation of the residuals is the sum of the random error, plus the departure from linearity. When the data is linear, the variance due to the departure from nonlinearity is effectively zero. For a nonlinear set of data, since the X-difference between adjacent data points is small, the nonlinearity of the function makes minimal contribution to the total difference between adjacent residuals and most of that difference contributing to the successive differences in the numerator of the DW calculation is due to the random noise of the data. The denominator term, on the other hand, is dependent almost entirely on the systematic variation due to the curvature, and for nonlinear data this is much larger than the random noise contribution. Therefore the denominator variance of the residuals is much larger than the numerator variance when nonlinearity is present, and the Durbin-Watson statistic reflects this by assuming a value less than 2. 

The key to calculating the Durbin-Watson statistic is that prior to performing the calculation, the data must be put into a suitable order. The Durbin-Watson statistic is then sensitive to serial correlations of the ordered data. While the serial correlation is often thought of in connection with time series, that is only one of its applications. 

Draper and Smith [1] discuss the application of DW to the analysis of residuals from a calibration their discussion is based on the fundamental work of Durbin, et al in the references listed at the beginning of this chapter. While we cannot reproduce their entire discussion here, at the heart of it is the fact that there are many kinds of serial correlation, including linear, quadratic and higher order. As Draper and Smith show (on p. 64), the linear correlation between the residuals from the calibration data and the predicted values from that calibration model is zero. Therefore if the sample data is ordered according to the analyte values predicted from the calibration model, a statistically significant value of the Durbin-Watson statistic for the residuals in indicative of high-order serial correlation, that is nonlinearity. 

Draper and Smith point out that you need a minimum of fifteen samples in order to get meaningful results from the calculation of the Durbin-Watson statistic [1], Since the... 

Anscombe data set contains only eleven readings, statistically meaningful statements cannot be made, nevertheless it is interesting to see the results of the Durbin-Watson statistic applied to the nonlinear set of Anscombe data the value of the statistic is 1.5073. For comparison, the Durbin-Watson statistic for the data set representing normal good data is 2.4816. 

But for ordinary data, we would not expect such a sequence to happen. This is the reason most statistics work as general indicators of data performance the special cases that cause them to fail are themselves low-probability occurrences. In this case the problem is not whether or not the data are nonlinear, the problem is that they are nonrandom. This is a perfect example of the data failing to meet a criterion other than the one you are concerned with. Therefore the Durbin-Watson test fails, as would any statistical test fail for such data they are simply not amenable to meaningful statistical calculations. Nevertheless, a blind computation of the Durbin-Watson statistic would give an apparently satisfactory value. But this is a warning that other characteristics of the data can cause it to appear to meet the criteria. 

We also note that as a practical matter, meaningful calculation of the Durbin-Watson Statistic requires many samples worth of data. We noted above that for fewer than... 

Three tests of this characteristic were discussed in the previous chapters the FDA/ICH recommendation of linear regression with a report of various regression statistics, visual inspection of a plot of test results versus the actual concentrations, and use of the Durbin-Watson Statistic. Since we previously analyzed these tests we will not further discuss them here, but will summarize them in Table 65-1, along with other tests for nonlinearity that we explain and discuss in this chapter. 

Durbin-Watson statistic Works Objective Is statistically testable Can be computerized Has fatal flaw Requires large number of samples Low statistical power... 

This test thus shares several characteristics with the Durbin-Watson test. It is based on well-known and rigorously sound statistics. It is amenable to automated computerized calculation, and suitable for automatic operation in an automated process situation. It does not have the fatal flaw of the Durbin-Watson Statistic. 

On the other hand, it also shares some of the disadvantages of the Durbin-Watson Statistic. It is also based on a comparison of variances, so that it is of low statistical power. It requires many more samples and readings than the Durbin-Watson statistic does, since each sample must be measured many times. In general, it is not applicable... 

This is the basis for our new test of linearity. It has all the advantages we described it gives an unambiguous determination of whether any nonlinearity is affecting the relationship between the test results and the analyte concentration. It provides a means of distinguishing between different types of nonlinearity, if they are present, since only those that have statistically significant coefficients are active. It also is more sensitive than any other statistical linearity test including the Durbin-Watson statistic. The tables... 

Durbin-Watson Statistic, a Step in the Right Direction 427... 

D. N. Rutledge and A. S. Barros, Durbin-Watson statistic as a morphological estimator of information content, Anal. Chim. Acta, 454(2), 2002, 277-295. 

It is commonly asserted that the Durbin-Watson statistic is only appropriate for testing for first order autoregressive disturbances. What combination of the coefficients of the model is estimated by the Durbin-Watson statistic in each of the following cases AR(1), AR(2), MA(1) In each case, assume that the regression model does not contain a lagged dependent variable. Comment on the impact on your results of relaxing this assumption. 

Another useful figure of merit is the Durbin-Watson (/-statistic ... 

---