Chi-plots (Fisher and Switzer 1983, 2001) provide a method to diagnose multivariate non-independence among Y variables.
chi.plot(Y1, Y2, ...)Returns a chi-plot.
A Y variable of interest. Must be quantitative vector.
A second Y variable of interest. Must also be a quantitative vector.
Additional arguments from plot.
Ken Aho and Tom Taverner (Tom provided modified original code to eliminate looping)
The method relies on calculating all possible pairwise differences within y\(_1\) and within y\(_2\). Let pairwise differences associated with the first observation in y\(_1\) that are greater than zero be transformed to ones and all other differences be zeros. Take the sum of the transformed values, and let this sum divided by (1 - n) be be the first element in the 1 x n vector z. Find the rest of the elements (2,..,n) in z using the same process.
Perform the same transformation for the pairwise differences associated with the first observation in y\(_2\). Let pairwise differences associated with the first observation in y\(_2\) that are greater than zero be transformed to ones and all other differences be zeros. Take the sum of the transformed values, and let this sum divided by (1 - n) be be the first element in the 1 x n vector g. Find the rest of the elements (2,..,n) in g using the same process.
Let pairwise differences associated with the first observation in y\(_1\) and the first observation in \(\bold{y}_2\) that are both greater than zero be transformed to ones and all other differences be zeros. Take the sum of the transformed values, and let this sum divided by (1 - n) be be the first element in the 1 x n vector h. Find the rest of the elements (2,..,n) in h using the same process. We let:
$$S = sign((\bold{z} - 0.5)(\bold{g} - 0.5))$$ $$\chi =(\bold{h} - \bold{z} \times \bold{g})/\sqrt{\bold{z} \times (1 - \bold{z}) \times \bold{g} \times (1 - \bold{g})}$$ $$\lambda = 4 \times S \times max[(\bold{z} - 0.5)^2,(\bold{g} - 0.5)^2]$$
We plot the resulting paired \(\chi\) and \(\lambda\) values for values of \(\lambda\) less than \(4(1/(n - 1) - 0.5)^2\). Values outside of \(\frac{1.78}{\sqrt{n}}\) can be considered non-independent.
Everitt, B. (2006) R and S-plus Companion to Multivariate Analysis. Springer.
Fisher, N. I, and Switzer, P. (1985) Chi-plots for assessing dependence. Biometrika, 72: 253-265.
Fisher, N. I., and Switzer, P. (2001) Graphical assessment of dependence: is a picture worth 100 tests? The American Statistician, 55: 233-239.
bv.boxplot
Y1<-rnorm(100, 15, 2)
Y2<-rnorm(100, 18, 3.2)
chi.plot(Y1, Y2)
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