  We investigate testing of the hypothesis of independence between a covariate
and the marks in a marked point process. It would be rather straightforward if
the (unmarked) point process were independent of the covariate and the marks.
In practice, however, such an assumption is questionable and possible
dependence between the point process and the covariate or the marks may lead to
incorrect conclusions. Therefore, we propose to investigate the complete
dependence structure in the triangle points--marks--covariates together. We
take advantage of the recent development of the nonparametric random shift
methods, namely the new variance correction approach, and propose tests of the
null hypothesis of independence between the marks and the covariate and between
the points and the covariate. We present a detailed simulation study showing
the performance of the methods and provide two theorems establishing the
appropriate form of the correction factors for the variance correction.
Finally, we illustrate the use of the proposed methods in two real
applications.
