Summary We show a novel approach for k-FWER control which does not involve any correction, but only testing the hypotheses along a (possibly data-driven) order until a suitable number of p-values are found above the uncorrected α level. p-values can arise from any linear model in a parametric or nonparametric setting. The approach is not only very simple and computationally undemanding, but also the data-driven order enhances power when the sample size is small (and also when k and/or the number of tests is large). We illustrate the method on an original study about gene discovery in multiple sclerosis, in which were involved a small number of couples of twins, discordant by disease. The methods are implemented in an R package (someKfwer), freely available on CRAN.