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Correlation Methods and Statistics


Correlation and Covariance Matrices

You can generate a correlation or covariance matrix from numeric data columns, and have the choice of storing the computation results in an-autogenerated worksheet, or display the results in a table format whose values can be color coded.

Using Fisher's z Transformation (zr)

This option is provided to allow transforming a skewed sampling distribution into a normalized format.

  • The theoretical sampling distribution of the correlation coefficient can be approximated by the normal distribution when the value of a population correlation ρ = 0, but as the value of r deviates from zero, the sampling distribution becomes increasingly skewed. Fisher's z transformation transforms a skewed sampling distribution into a normalized format.

    • The relationship between Pearson's product-moment correlation coefficient and the Fisher-Transformed values are shown in the right-hand side image.

    The image below shows the Fisher-transformed values of the correlation matrix displayed above.

Pearson Product-Moment Correlation Coefficient (Pearson's r)

Pearson r is a measure of correlation between two variables. The associated tests evaluate whether the correlation coefficient for the underlying population is different from zero, i.e., whether there is a monotonous relationship between the two variables. Pearson's r varies from -1 to +1, with 0 indicating no relationship and 1 indicating perfect relationship.

The test reports from Aabel include:

  • the critical one-tailed and two-tailed r(0.01) and r(0.05), Fisher z transformation, as well as t statistic and the associated p values.

Spearman's Rank-Order Correlation Coefficient (Spearman's ρ)

Spearman's ρ is the non-parametric analog of the Pearson product-moment correlation coefficient. The results or the former and latter are closely similar, as the Spearman correlation is calculated in a very similar manner as Pearson, except that Spearman first ranks the data.

Example:

Kendall's Rank Correlation Coefficient (Kendall's τ)

Like Spearman's ρ, Kendall's τ is a non-parametric method of correlation between two variables, but has an advantage over Spearman's ρ: Kendall's τ also indicates the difference between the probability that the observed data are in the same order for the two variables vs. the probability that the observed data are in different orders for the two variables.

  • The method requires two numeric variables.
  • The test results include the estimated Kendall's τ the critical one-tailed and two-tailed τ(0.05) and τ(0.01), t statistic, Z statistic and the corresponding p values (two-tailed and one-tailed).

    • The right-hand side image shows the Kendall's &tau report from the same data used or the Spearman's ρ above.