The distribution of the sample correlation coefficient with one variable fixed

Abstract

For th e usual st raight·line model, in whi c h the ind ependen t variable ta kes on a fixed, kno wn se t of va lues, it is shown th at th e sample co rre latio n coeffi cie nt is di stributed as Q with (n-2) degrees of freedo m and noncentralit y O=({3 /CT) V~(Xi -X)2. The Q variate has bee n defin ed and studi ed e lse· wh ere by Hogben et al. It is noted th at the square of th e correlation coeffi cient is di stributed as a nonce ntral beta variab le.Key Words: Anal ysis of variance, calibrali on, co rrelation coeffi cie nt, degrees of freedom, di stribution , fix ed vari abl e , nonce ntral beta variabl e, noncentrality, Q variate.

Correlation or Regression: Bias or Precision

“…The conditional distribution of r 2 given XI' X 2 , ••• , X N is known to be non-central beta (see, for example, Hogben, et al, 1964;Rao, 1965, p. 509;Hogben, 1968). We shall here outline the derivation as it applies to the special case which we wish to consider.…”

“…On introducing new variables, z = z r 2 = z/(w+z) and integrating out the z, we obtain Note that the distribution of r has been studied by Hogben et al (1964) and Hogben (1968).…”

Methods for detection of clusters of observations with an outlying correlation coefficient value

“…It is known that in the case of a bivariate normal, the distribution of the sample variances and the distribution of the Pearson correlation coefficient are not independent 7 8 , so that a test on sample variances and a test on correlations would not be independent. This is further compounded by the observation that in real situations, individual centers often include patients from a subpopulation that has different characteristics than the overall population across centers 5 , leading to large differences in variance across centers.…”

“…Hogben 8 describes the variability of the sample correlation coefficient of x and Y (denoted r xY ) in the simple linear regression model: Y i = α + βx i + ε i , i = 1, … n with i.i.d . normal errors with mean 0 and variance σ 2 .…”

Statistical model-based optimization for damage extent quantification

“…The majority of the work, however, focuses on the case where both vectors are stochastic [51][52][53]. A possible solution was investigated in [54], where the exact mean values of both vectors are required, which are not available in practice. In the present paper, the uncertainty quantification is achieved by the statistical delta method [55], which has been deployed previously as a first order linearization for evaluating uncertainties of modal parameters [38,56], modal indicators [57,58], or for physical parameters of mechanical systems [28], without a priori information on the solution.…”