Correlation: Confidence intervals...


Calculates confidence intervals for the correlation coefficients from the selected Correlation object(s) and saves these intervals in a new TableOfReal object.
Settings

Confidence level

the confidence level you want for the confidence intervals.

Number of tests

determines the Bonferroni correction for the significance level. If the default value (zero) is chosen, it will be set equal to the number of correlations involved (a matrix of dimension n has n·(n1)/2 correlations).

Approximation

defines the approximation that will be used to calculate the confidence intervals. It is either Fisher's z transformation or Ruben's transformation. According to Boomsma (1977), Ruben's approximation is more accurate than Fisher's.
Algorithm
We obtain intervals by the largesample conservative multiple tests with Bonferroni inequality and the Fisher or Ruben transformation. We put the upper values of the confidence intervals in the upper triangular part of the matrix and the lower values of the confidence intervals in lower triangular part of the resulting TableOfReal object.
In Fisher's approximation, for each element r_{ij} of the correlation matrix the confidence interval is:
[ tanh (z_{ij}  z_{α′} / √(N  3)) , tanh (z_{ij} + z_{α′} / √(N  3)) ], 
where z_{ij} is the Fisher ztransform of the correlation r_{ij}:
z_{ij} = 1/2 ln ((1 + r_{ij}) / (1  r_{ij})), 
z_{α′} the Bonferroni corrected zvalue z_{α/(2·numberOfTests)},
and N the number of observations that the correlation matrix is based on.
In Ruben's approximation the confidence interval for element r_{ij} is:
[ x_{1} / √(1  x_{1}^{2}), x_{2} / √(1  x_{2}^{2}) ] 
in which x_{1} and x_{2} are the smallest and the largest root from
a x^{2} + b x + c = 0, with 
b =  2 r′ √((2N  3)(2N  5)) 
c = (2N  5  z_{α′}^{2}) r′^{2}  2z_{α′}^{2}, and 
r′ = r_{ij} / √(1  r_{ij}^{2}), 
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© djmw, April 7, 2004