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PearsonResult.ConfidenceInterval

The confidence interval for the correlation, through the Fisher z transform.

public (double Low, double High) ConfidenceInterval(double level = 0.95)

Parameters — level is the confidence level, strictly between 0 and 1.

Returns — (double Low, double High): the interval around the correlation the test measured. A one-sided test's interval is half-open, and the far bound is the correlation's own limit, -1 or 1, rather than an infinity — a correlation cannot leave that range, so there is no larger number for the bound to be.

Exceptions — ArgumentOutOfRangeException when level is NaN or outside (0, 1).

Example — the same eight panels, at two levels.

using Lodestar.Stats;

double[] sugar = [4.0, 6.5, 5.0, 9.0, 7.5, 3.0, 8.0, 6.0];
double[] sweetness = [3.1, 5.9, 4.0, 7.4, 6.8, 2.2, 6.1, 5.2];

PearsonResult result = Pearson.Test(sugar, sweetness);
(double low95, double high95) = result.ConfidenceInterval();
(double low99, double high99) = result.ConfidenceInterval(0.99);

double narrow = Math.Round(low95, 4);   // => 0.8783
double wide = Math.Round(low99, 4);     // => 0.7979

Remarks — the interval is not symmetric about the coefficient, and that is the transform doing its job rather than an artefact. r is bounded and its sampling distribution is skewed near the ends; atanh carries it onto a scale where the distribution is close to normal, the interval is built there, and tanh carries the two bounds back. A correlation of 0.9778 with an upper bound of 0.9961 has far less room above it than below, because there is far less room above it.

Below four pairs the interval is the whole range. The standard error is 1 / sqrt(n - 3), which has nothing to say at three pairs or fewer, so (-1, 1) is returned rather than a NaN — the same answer scipy gives there, and an honest one: the data constrains nothing.

using Lodestar.Stats;

(double low, double high) = Pearson.Test([1.0, 2.0, 3.0], [1.0, 3.0, 2.0])
    .ConfidenceInterval();

double lower = low;    // => -1
double upper = high;   // => 1

A one-sided interval is half-open, and its fixed bound survives a NaN coefficient. A constant sample has no correlation, so the computed bound is NaN — but the other bound was never computed from the data, and -1 or 1 is still true of it.

Applies to — net10.0, netstandard2.0.