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The Nonlinear Relationship
of Two Variables
Nonlinear Correlation Coefficients

Copyright © 1997-9 by Forrest W. Young.

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Three indices of correlation are used to investigate nonlinear relationships. The Spearman index is used for two ordinal variables. The Point-Biserial index is used when one variable is numeric and the other is binary. The Phi coefficient is used when both variables are binary.

The Spearman Correlation Coefficient
The Spearman Correlation Coefficient is designed to measure the degree of relation for two ordinal variables. It is designed to be used when:

• The original X and Y variables are ranks.
• The original X and Y variables are not ranks, but have been converted into ranks. It is especially useful when one of the variables is ordinal and the other is interval or ratio. Then both variables are changed into ranks.

To compute the Spearman Correlation Coefficient, you first convert your two variables into ranks, and then follow the procedure for Pearson correlation.

ViSta Applet: Transform variables into ranks by choosing the Transform menu's Ranks item. This produces a new dataset in which all ordinal or numeric variables have been converted into ranks. You now proceed as you would for the Pearson Correlation.

The commands to do this are:

(browse-data)
(ranks)
(browse-data)
(summarize-data :moments t :correlations t)
(correlations)
(browse-data)

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The Point-Biserial Correlation Coefficient
The Point-Biserial Correlation Coefficient is designed to measure the degree of relation for two variables when one variable is quantitative (interval or ratio) and the other variable is binary (has only two values --- also called dichotomous).

To compute the Point-Biserial Correlation Coefficient, you first convert your binary variable into 1's and 0's, and then follow the procedure for Pearson correlation.

Note: (July 9th, 1999) The new server has not yet been configured for these applets to run ViSta. They will, instead, download into your browser. This problem should be fixed soon.

ViSta Applet: Code your binary variable with 1's and 0's. Once you have done this you proceed as you would for the Pearson Correlation.

The commands to do this (given that you already have a binary variable, and that you want to select it and some other, non-binary variables) are:

(list-variables)
(select-variables '("GPA" "MathSAT" "VerbSAT" "GenderN"))
(summarize-data :moments t :correlations t)
(correlations)
(browse-data)

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The Phi Correlation Coefficient
The Phi Correlation Coefficient is designed to measure the degree of relation for two variables which are binary (each has only two values --- also called dichotomous).

To compute the Point-Biserial Correlation Coefficient, you first convert your two binary variable into 1's and 0's, and then follow the procedure for Pearson correlation.

ViSta Applet: Code both of your binary variables with 1's and 0's. Once you have done this you proceed as you would for the Pearson Correlation.

The commands to do this (given that you already have a datafile with at least two binary variables) are:

(browse-data)
(summarize-data :moments t :correlations t)
(correlations)
(browse-data)

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