/// <summary>
/// Performs a Spearman rank-order test of association between the two variables.
/// </summary>
/// <returns>The result of the test.</returns>
/// <remarks>
/// <para>The Spearman rank-order test of association is a non-parametric test for association between
/// two variables. The test statistic rho is the correlation coefficient of the <em>rank</em> of
/// each entry in the sample. It is thus invariant over monotonic reparameterizations of the data,
/// and will, for example, detect a quadratic or exponential association just as well as a linear
/// association.</para>
/// <para>The Spearman rank-order test requires O(N log N) operations.</para>
/// </remarks>
/// <exception cref="InsufficientDataException">There are fewer than three data points.</exception>
/// <seealso cref="PearsonRTest"/>
/// <seealso cref="KendallTauTest"/>
/// <seealso href="http://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient"/>
public TestResult SpearmanRhoTest()
{
if (Count < 3)
{
throw new InsufficientDataException();
}
// analytic expressions for the mean and variance of ranks
double M = (Count - 1) / 2.0;
double V = (Count + 1) * (Count - 1) / 12.0;
// compute the covariance of ranks
int[] rx = xData.GetRanks();
int[] ry = yData.GetRanks();
double C = 0.0;
for (int i = 0; i < Count; i++)
{
C += (rx[i] - M) * (ry[i] - M);
}
C = C / Count;
// compute rho
double rho = C / V;
// for small enough sample, use the exact distribution
Distribution rhoDistribution;
if (Count <= 10)
{
// for small enough sample, use the exact distribution
// it would be nice to do this for at least slightly higher n, but computation time grows dramatically
// would like to ensure return in less than 100ms; current timings n=10 35ms, n=11 72ms, n=12 190ms
rhoDistribution = new DiscreteAsContinuousDistribution(new SpearmanExactDistribution(Count), Interval.FromEndpoints(-1.0, 1.0));
}
else
{
// for larger samples, use the normal approximation
// would like to fit support and C_4 too; look into logit-normal
// i was not happy with Edgeworth expansion, which can fit C_4 but screws up tails badly, even giving negative probabilities
rhoDistribution = new NormalDistribution(0.0, 1.0 / Math.Sqrt(Count - 1));
}
return(new TestResult(rho, rhoDistribution));
}
