Durbin-Watson test

For similar reasons, the statement If the Durbin-Watson test demonstrates a correlation, then the relationship between the two assays is not linear is not exactly correct, either. Under some circumstances, a linear correlation can also give rise to a statistically significant value of DW. In fact, for any statistic, it is always possible to construct a data set that gives a high-probability value for the statistic, yet the data clearly and obviously fail to meet the pertinent criteria (again, Anscombe is a good example of this for a few common statistics). So what should we do Well, different statistics show different sensitivities to particular departures from the ideal, and this is where DW comes in. 

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. 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

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. 

The residuals e, = (y - yt) should also fit a normal distribution, i.e. they should not correlate with xt. An easy way to check this is to plot et versus x . If the residuals do not scatter randomly around zero, the linear model may not be adequate for the data. An indication of a wrongly specified model may be the occurrence of auto-correlated residuals, which can be checked by the DURBIN-WATSON test [MAGER, 1982],... 

Stage 1. The MeOH/H20/NaCl data are subjected to the correlation procedure described previously which gives values of the Wilson energy constants (Zi and Z2) and a new set of data for temperature and vapor composition that are internally consistent (see Table I). The small values of the standard deviation and the bias indicate good quality data in the salt effect field. For the analysis of serial correlation among the residuals we use the Durbin-Watson test (9). A run of positive or negative signs in the series of residuals is some indication that the model... 

Finally the original data are shown on Figures 5 and 6 together with the confidence regions. Now we see that 42% of the pressure differences lie within the confidence levels while 66% of the vapor composition differences are within the levels. Included in Table III are the standard deviations, bias, and results of the Durbin-Watson test. Comparison of the two sets of results indicates appreciably larger values for standard deviation and bias for the experimental results whereas for the D-test the... 

Whenever researchers perform a regression analysis using data collected over time, they should conduct the Durbin-Watson Test. Most statistical software packages have it as a standard subroutine and it can be chosen for inclusion in... 

More often than not, serial correlation will involve positive correlation, where each e, value is directly correlated to the e, i value. In this case, the Durbin-Watson test is a one-sided test, and the population serial correlation component—under the alternative hypothesis— is P>0. The Durbin-Watson formula for 1 lag is... 

If P > 0, then e, = e, i + dt. The Durbin-Watson test can be evaluated using the six-step procedure ... 

Step 3 Write out the Durbin-Watson test statistic for 1 lag... 

Although the pattern displayed by the residuals may be due to lack of linear fit (as described in Chapter 2), before any linearizing transformation, the researcher should perform the Durbin-Watson test. Let us do that, using the six-step procedure. 

Regression Analysis with the Durbin-Watson Test, Example 3.1... 

The Durbin-Watson test is very popular, so when discussing time series correlation, most researchers who employ statistics are likely to know it. It is also the most common one found in statistical software. 

Nevertheless, one additional step must be included to determine the final Durbin-Watson test value Dw- Dw is computed as Dw =4 Dw Step 4 Decision mle ... 

Two-Tail Durbin-Watson Test Procedure Step 1 State the hypothesis. 

Note that Se, i e, is not the same numerator term used in the Durbin—Watson test. Here, the e, is and e,s are multiphed but, in the Durbin-Watson test, they are subtracted and squared. 

Step 3 Evaluate the transformed regression equation by using the Durbin-Watson test to determine if it is still significantly serially correlated. If the test shows no serial correlation, the procedure stops. If not, the residuals from the fitted equation are used to repeat the entire process again, and the new regression that results is tested using the Durbin-Watson test, and so on. 

Step 3 We again test for serial correlation using the Durbin-Watson test procedure. Because this is the second computation of the regression equation, we lost one value to the lag adjustment, so = 17. For every iteration, the lag adjustment reduces n by 1. 

We can now regress y on xj, which produces a regression equation nearly through the origin or bo = 0 (Table 3.17). However, the Durbin-Watson test... 

Let us compute the Durbin-Watson test for a lag of 1, using the six-step procedure. 

This test is best employed when the error terms are not highly serially correlated, either by assuring this with the Durbin-Watson test or after the serial correlation has been corrected. It is best used when the sample size is large, assuring normality of the data. 

TABLE E (continued) Durbin-Watson Test Bounds Level of Significance a = 0.01 ... 

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