Statistics for Environmental Engineers

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Can the dramatic difference in the outcome of the first and second experiments possibly be due to the presence of autocorrelation in the experimental data? It is both possible and likely, in view of the lack of randomization in the order of running the tests.

The Consequences of Autocorrelation on Regression

An important part of doing regression is obtaining a valid statement about the precision of the estimates. Unfortunately, autocorrelation acts to destroy our ability to make such statements. If the error terms are positively autocorrelated, the usual confidence intervals and tests using t and F distributions are no longer strictly applicable because the variance estimates are distorted (Neter et al., 1983).

Why Autocorrelation Distorts the Variance Estimates

Suppose that the system generating the data has the true underlying relation n = в + fax, where x could be any independent variable, including time as in a times series of data. We observe n values: y1 = n + e1,…, yi-2 = n + ei-2, yi-1 = n + ei-i, yt = П + ei, —, yn = П + en. The usual assumption is that the residuals (e) are independent, meaning that the value of ei is not related to ei-1, ei-2, etc. Let us examine what happens when this is not true.

Suppose that the residuals (ei), instead of being random and independent, are correlated in a simple way that is described by ei = p ei-1 + ai, in which the errors (at) are independent and normally distributed with constant variance a2. The strength of the autocorrelation is indicated by the autocorrelation coefficient (p), which ranges from -1 to +1. If p = 0, the ei are independent. If p is positive, successive values of ei are similar to each other and:

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