Nonsense associations in Markov random fields with pairwise dependence

Summaryidentified the issue of ′nonsense correlations’ in time series data, where dependence within each of two random vectors causes overdispersion, i.e., variance inflation, for measures of dependence between the two. Since then much has been written about nonsense correlations, but nearly all of it confined to the time series literature. In this paper we provide the first, to our knowledge, rigorous study of this phenomenon for other forms of (positive) dependence, specifically for Markov random fields on lattices and graphs. We consider binary and continuous random vectors and three different measures of association: correlation, covariance and the ordinary least-squares coefficient from projecting one random vector onto the other. In some settings we find variance inflation consistent with Yule’s nonsense correlation. Surprisingly, we also find variance deflation in some settings, and in others the variance is unchanged under dependence. Perhaps most notably, we find general conditions under which ordinary least-squares inference that ignores dependence is valid despite positive dependence in the regression errors, contradicting the presentation of ordinary least squares in countless textbooks and courses.