For observations \(u_{i,j},\ i=1,...,N,\ j=1,2,\) the chi-plot is based on the following two quantities: the
chi-statistics
$$\chi_i = \frac{\hat{F}_{1,2}(u_{i,1},u_{i,2})
- \hat{F}_{1}(u_{i,1})\hat{F}_{2}(u_{i,2})}{
\sqrt{\hat{F}_{1}(u_{i,1})(1-\hat{F}_{1}(u_{i,1}))
\hat{F}_{2}(u_{i,2})(1-\hat{F}_{2}(u_{i,2}))}}, $$
and the lambda-statistics
$$\lambda_i = 4 sgn\left( \tilde{F}_{1}(u_{i,1}),\tilde{F}_{2}(u_{i,2}) \right)
\cdot \max\left( \tilde{F}_{1}(u_{i,1})^2,\tilde{F}_{2}(u_{i,2})^2 \right), $$
where \(\hat{F}_{1}\), \(\hat{F}_{2}\) and
\(\hat{F}_{1,2}\) are the empirical distribution functions
of the uniform random variables \(U_1\) and \(U_2\) and of
\((U_1,U_2)\), respectively. Further,
\(\tilde{F}_{1}=\hat{F}_{1}-0.5\) and
\(\tilde{F}_{2}=\hat{F}_{2}-0.5\).
These quantities only depend on the ranks of the data and are scaled to the
interval \([0,1]\). \(\lambda_i\) measures a distance of a data point
\(\left(u_{i,1},u_{i,2}\right)\) to the center of the
bivariate data set, while \(\chi_i\) corresponds to a correlation
coefficient between dichotomized values of \(U_1\) and \(U_2\). Under
independence it holds that \(\chi_i \sim
\mathcal{N}(0,\frac{1}{N})\) and \(\lambda_i \sim
\mathcal{U}[-1,1]\) asymptotically, i.e., values of
\(\chi_i\) close to zero indicate independence---corresponding to
\(F_{1, 2}=F_{1}F_{2}\).
When plotting these quantities, the pairs of \(\left(\lambda_i, \chi_i
\right)\) will tend to be located above zero for
positively dependent margins and vice versa for negatively dependent
margins. Control bounds around zero indicate whether there is significant
dependence present.
If mode = "lower" or "upper", the above quantities are
calculated only for those \(u_{i,1}\)'s and \(u_{i,2}\)'s which are
smaller/larger than the respective means of
u1\(=(u_{1,1},...,u_{N,1})\) and
u2\(=(u_{1,2},...,u_{N,2})\).