The investigation of the nonlinearity of the Leavitt law (LL) is a topic that began more than seven decades ago, when some of the studies in this field found that the LL has a break at about 10 days. The goal of this work is to investigate a possible statistical cause of this nonlinearity. By applying linear regressions to OGLE-II and OGLE-IV data, we find that to obtain the LL 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 doubt the linearity of the LL 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 LL for this galaxy. For the Small Magellanic Cloud, when including Cepheids of this kind, it is not possible to find an adequate model, probably because of the geometry of the galaxy. In that case, a possible influence of these stars could exist.