chi.plot {asbio} R Documentation

## 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. ... Additional arguments from 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 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.

### Value

Returns a chi-plot.

### Author(s)

Ken Aho and Tom Taverner (Tom provided modified original code to eliminate looping)

### References

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)