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Durbin-Watson Statistic

Note

The Durbin-Watson (DW) statistic and corresponding sample autocorrelation are reported on the residual versus run graphs for ordinary least squares regression where they are based on the raw residuals, no matter the selected residual type. For Poisson regression and logistic regression, the values are always based on the deviance residuals. The statistic is not reported for mixed effect models such as split-plots.

  • A rough guide is \(1.5 < \text{ DW-statistic } < 2.5\).

One common trend that can occur in the residual run plot occurs when response values for runs at later times depend on the state of the system (measurement devices, operators, processes, etc.) at earlier times and, hence, are too similar or too different from response values at earlier times. Such serial dependence or autocorrelation can be caused by many things, including failure to properly reset factors or tare/calibrate measuring devices between runs. Like other violations of statistical independence, this can result in degraded confidence in reported p-values and other statistical tests.

In conjunction with the residual plots, the Durbin-Watson statistic (DW-statistic) [DW50, DW51] can be used as a rough guide to identifying such serial dependence [Fox16].

The Durbin-Watson statistic, \(DW\), for lag 1 autocorrelation is,

\[DW =\frac{\sum_{i=2}^n \left(e_i - e_{i-1}\right)^2}{\sum_{i=1}^n e_i^2}, 0 \leq d \leq 4~~\text{ (DW-statistic) }\]

where \(e_i, i=1,...,n\) are the raw residuals and \(n\) is the total number of residuals. Typically, \(n\) is equal to the number of runs in the design unless there is missing response data.

The corresponding lag 1 sample autocorrelation, \(r_1\), is,

\[r_1 = \frac{\sum_{i=2}^n e_i e_{i-1}}{\sum_{i=1}^n e_i^2}, -1 \leq r_1 \leq 1~~\text{ (Autocorrelation) }\]

This is a measure of the correlation between errors for responses recorded serially in time.

For large \(n\), the two statistics are related by \(DW \approx 2(1-r_1)\), so a rough rule of thumb for \(DW\) is that it should be close to \(2\) for no correlation (\(r_1 \approx 0\)) or at least approximately satisfy \(1.5 < DW < 2.5\), where the serial dependence is less likely to have a significant effect on statistical tests. However, the statistic can fall out of the range by chance, especially for small designs, so it should be used with caution [Oeh10].

References

DW50

J. Durbin and G. S. Watson. Testing for serial correlation in lease squares regression, i. Biometrika, 37(3-4):409–428, 1950.

DW51

J. Durbin and G. S. Watson. Testing for serial correlation in lease squares regression, ii. Biometrika, 38(1-2):159–179, 1951.

Fox16

John Fox. Applied Regression Analysis & Generalized Linear Models. Sage, 3rd edition, 2016.

Oeh10

Gary W. Oehlert. A first course in design and analysis of experiments. Gary W. Oehlert, 2010. ISBN 0-7167-3510-5.