asbio (version 0.2-1)

chi.plot: Chi plots for diagnosing multivariate independence.

Description

Chi-plots (Fisher and Switzer 1983, 2001) provide a method to diagnose multivariate non-independence among Y variables.

Usage

chi.plot(Y1, Y2)

Arguments

Y1

A Y variable of interest. Must be quantitative vector.

Y2

A second Y variable of interest. Must also be a quantitative vector.

Value

• Returns a chi-plot.

Details

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 zeroes. 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 zeroes. 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 obsevation in y$_2$ that are both greater than zero be transformed to ones and all other differences be zeroes. 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: $$S = sign((\bold{z} - 0.5)(\bold{g} - 0.5))$$ $$\chi =(\bold{h} - \bold{z} \times \bold{g})/\sqrt{\bold{z} \times (1 - \bold{z}) \times \bold{g} \times (1 - \bold{g})}$$ $$\lambda = 4 \times \emph{S} \times max[(\bold{z} - 0.5)^2,(\bold{g} - 0.5)^2]$$ 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.

References

Everitt, B. (2005) 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.

See Also

bv.boxplot

Examples

Y1<-rnorm(100,15,2)
Y2<-rnorm(100,18,3.2)
chi.plot(Y1,Y2)