Comparison of correlation coeffìcients when conditional independence constraints between variables are present

Partial correlations are the natural interaction terms to be associated with the edges of the independence graph of a multivariate normal distribution. Two edges of the graph can thus be compared by comparing the corresponding partial correlations. The comparison of "dependent" correlations is a well known problem in statistics but, when the variables satisfy some conditional independence relations, the maximum likelihood estimates of partial correlations are different from the sample partial correlations, so that classical results no longer bold. In this paper we analyze to what extent the classical test statistics can be applied. We show that maximum likelihood estimates of partial correlations are more efficient than sample partial correlations. Furthermore we show that the conditional independence structure of the model can be used to turn the comparison of "dependent" correlations into the comparison of "independent" parameters