Revisiting the Normalization of the Moran's I Index: A Correlation‐Based Approach With Inference

Abstract

Summary Moran's index, the canonical measure of spatial autocorrelation, may take values outside the interval when arbitrary spatial weight matrices are used. This lack of boundedness complicates interpretation and may lead to misleading conclusions. We systematically evaluate seven approaches aimed at addressing this issue. We first examine conventional rescaling methods based on extreme eigenvalues and show their limitations. We then demonstrate that standard row‐normalizations of the weight matrix do not guarantee the desired bounds. Next, we propose three theoretically sound normalization methods derived from the Cauchy‐Schwarz inequality. These approaches (i) interpret the index as a genuine correlation coefficient, (ii) measure the correlation between values and neighbourhood means and (iii) incorporate weighted spatial operators. All three methods provably constrain the index to while maintaining a clear spatial interpretation of a correlation coefficient. An empirical example using the meuse dataset confirms the theoretical results and shows that the traditional Moran index may fail to attain its theoretical upper or lower bound even when observations and neighbourhood means are perfectly correlated. We therefore recommend using a Pearson correlation coefficient between observed values and neighbourhood averages. Finally, we show how to test the hypothesis of no spatial autocorrelation using this new index.