Reliability Coefficients in a Correlation Matrix

Given s fallible tests t 1, t 2, … 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 12ptminimalamsmathwasysymamsfontsamssymbamsbsymathrsfsupgreek-69ptdocumentd₈₈ = 1 + (mᵢ - 1) r₈₈ /mᵢ, document in which m i is the number of parallel forms comprising the i 'th test and 12ptminimalamsmathwasysymamsfontsamssymbamsbsymathrsfsupgreek-69ptdocument r₈₈ document 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.