On a Statistic Arising in Testing Correlation

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Abstract
This paper is devoted to the study of a certain statistic, u, defined on samples from a bivariate population with variances σ11, σ22 and correlation ρ. Let the parameter corresponding to u be υ. Under binormality assumptions the following is demonstrated. (i) If σ11 = σ22, then the distribution of u can be obtained rapidly from the F distribution. Statistical inferences about ρ = υ may be based on F. (ii) In the general case, allowing for σ11 ≠ σ22, a certain quantity involving u, r (sample correlation between the variables) and υ follows a t distribution. Statistical inferences about υ may be based on t. (iii) In the general case a quantity t′ may be constructed which involves only the statistic u and only the parameter υ. If treated like a t distributed magnitude, t′ admits conservative statistical inferences. (iv) The F distributed quantity mentioned in (i) is equivalent to a certain t distributed quantity as follows from an appropriate transformation of the variable. (v) Three test statistics are given which can be utilized in making statistical inferences about ρ = υ in the case σ11 = σ22. A comparison of expected lengths of confidence intervals for ρ obtained from the three test statistics is made. (vi) The use of the formulas derived is illustrated by means of an application to coefficient alpha.