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