On association in regression: the coefficient of determination revisited

Universal coefficients of determination are investigated which
quantify the strength of the relation between a vector of dependent
variables Y and a vector of independent covariates X. They are
defined as measures of dependence between Y and X through theta(x),
with theta(x) parameterizing the conditional distribution of Y
given X=x. If theta(x) involves unknown coefficients gamma the
definition is conditional on gamma, and in practice gamma,
respectively the coefficient of determination has to be estimated.
The estimates of quantities we propose generalize R^2 in classical
linear regression and are also related to other definitions
previously suggested. Our definitions apply to generalized
regression models with arbitrary link functions as well as
multivariate and nonparametric regression. The definition and use
of the proposed coefficients of determination is illustrated for
several regression problems with simulated and real data sets.