Statistics for Environmental Engineers

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An Example of Autocorrelated Errors

The laboratory data presented for the case study were created to illustrate the consequences of autocorrelation on regression. The true model of the experiment is n = 20 + 0.5x. The data structure is shown in Table 41.1. If there were no autocorrelation, the observed values would be as shown in Figure 41.2. These are the third column in Table 41.1, which is computed as yi + 20 + 0.5xi + a, where the ai are independent values drawn randomly from a normal distribution with mean zero and variance of one (the at’s actually selected have a variance of 1.00 and a mean of -0.28).

In the flawed experiment, hidden factors in the experiment were assumed to introduce autocorrelation. The data were computed assuming that the experiment generated errors having first-order autocorrelation with p = 0.8. The last three columns in Table 41.1 show how independent random errors are converted to correlated errors. The function producing the flawed data is:

yt = n + ei = 20 + 0.5 xi + 0.8ei-1 + ai

If the data were produced by the above model, but we were unaware of the autocorrelation and fit the simpler model n = во + во x, the estimates of во and в1 will reflect this misspecification of the model. Perhaps more serious is the fact that t-tests and F-tests on the regression results will be wrong, so we may be misled as to the significance or precision of estimated values. Fitting the data produced from the autocorrelation model of the process gives yi = 21.0 + 0.12xi. The 95% confidence interval of the slope is [-0.12 to 0.35] and the t-ratio for the slope is 1.1. Both of these results indicate the slope is not significantly different from zero. Although the result is reported as statistically insignificant, it is wrong because the true slope is 0.5.

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