The investigation of the non-linearity of the Leavitt law is a topic that began more than seven decades ago, when some of the studies in this field found that the Leavitt law has a break at about ten days. The goal of this work is to investigate a possible statistical cause of this non-linearity. By applying linear regressions to OGLE-II and OGLE-IV data, we find that, in order to obtain the Leavitt law by using linear regression, robust techniques to deal with influential points and/or outliers are needed instead of the ordinary least-squares regression traditionally used. In particular, by using $M$- and $MM$-regressions we establish firmly and without doubts the linearity of the Leavitt law in the Large Magellanic Cloud, without rejecting or excluding Cepheid data from the analysis. This implies that light curves of Cepheids suggesting blending, bumps, eclipses or period changes, do not affect the Leavitt law for this galaxy. For the SMC, including this kind of Cepheids, it is not possible to find an adequate model, probably due to the geometry of the galaxy. In that case, a possible influence of these stars could exist. ; Comment: 47 pages, 1 figure, 5 tables. Accepted for publication in ApJ