chi.plot {asbio} | R Documentation |
Chi-plots (Fisher and Switzer 1983, 2001) provide a method to diagnose multivariate non-independence among Y variables.
chi.plot(Y1, Y2)
Y1 |
A Y variable of interest. Must be quantitative vector. |
Y2 |
A second Y variable of interest. Must also be a quantitative vector. |
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((z - 0.5)(g - 0.5))
chi = (h - z x g)/sqrt(z) x (1 - z) x g x (1 - g)
lambda = 4 x S x max[(z - 0.5)^2,(g - 0.5)^2]
We plot the resulting paired chi and λ values for values of λ less than 4(1/(n - 1) - 0.5)^2. Values outside of frac{1.78}{sqrt{n}} can be considered non-independent.
Returns a chi-plot.
Ken Aho
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.
Y1<-rnorm(100,15,2) Y2<-rnorm(100,18,3.2) chi.plot(Y1,Y2)