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·(n-1)/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 large-sample 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 rij of the correlation matrix the confidence interval is:

[ tanh (zij - zα′ / √(N - 3)) , tanh (zij + zα′ / √(N - 3)) ],

where zij is the Fisher z-transform of the correlation rij:

zij = 1/2 ln ((1 + rij) / (1 - rij)),

zα′ the Bonferroni corrected z-value zα/(2·numberOfTests),

α = 1 - confidenceLevel,

and N the number of observations that the correlation matrix is based on.

In Ruben's approximation the confidence interval for element rij is:

[ x1 / √(1 - x12), x2 / √(1 - x22) ]

in which x1 and x2 are the smallest and the largest root from

a x2 + b x + c = 0, with

a = 2N - 3 - zα′2

b = - 2 r′ √((2N - 3)(2N - 5))

c = (2N - 5 - zα′2) r′2 - 2zα′2, and

r′ = rij / √(1 - rij2),

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