Variance Estimation, the Effective Sample Size, and the Bootstrap

An important statistical consequence of positive spatial autocorrelation is that it in£ates the variance of the test statistic, and this may in turn result in an in£ated Type I error rate in hypothesis tests (Section 3.3). Let us review and summarize the discussion in that section. Suppose we have obtained a sample {Y1, Y2, …, Yn} from a normally distributed, spatially autocorrelated population, and we are testing the null hypothesis H0: μ = 0 against the alternative Ha: μ ≠ 0 (Equation 3.6). Recall from Equation 3.9 that the variance σ2 of the population from which the Yi are drawn can be estimated by the sample variance s2, which satis–es s n Y Yi 1 1 = − −( ) = ∑ . (10.1) The t statistic (Equation 3.11) with μ set to 0 becomes, t = Y-/s{Y-}, where s2{Y-} = s2/n and s Y s Y{ } { }= 2 . We use this statistic to carry out the test. Reproducing Equation 3.12, var cov , ,Y n n { } = + { } ≠ ∑σ2 2 (10.2) which indicates that if cov{Yi, Yj} > 0 (i.e., if the values are autocorrelated), then s2{Y –} underestimates the true variance of the mean (see Section 3.5.2 for a more complete discussion). Thus, the denominator of the t statistic is smaller than it should be, so that the value of t is arti–cially in£ated, resulting in an increased probability of rejecting the null hypothesis (i.e., an increased Type I error rate).