Reliability Coefficients in a Correlation Matrix

Given s fallible tests t₁,t₂, …t_s, the problem is to express their intercorrelations in terms of the average correlations between a varying number of parallel forms contained within each test. A new correlation determinant Δ′ is derived containing d_ii instead of unity as an element on the principal diagonal, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} $$d_{ii} = [1 + (m_i - 1)\bar r_{ii} ]/m_i ,$$ in which m_i is the number of parallel forms comprising the i'th test and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} $$\bar r_{ii} $$ is the average intercorrelation of the m_i (m_i − 1) / 2 parallel forms. As m_i → ∞, d_ii approaches the correlation “corrected for attenuation.” These results make explicit the assumptions, as to intrisic accuracy of all measures, which are implicit in the usual multiple and partial correlation analysis. These results also make possible a simple procedure for estimating the effect on various partial correlation measures of improving the accuracy of part or all of the measures by including additional parallel forms.