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We perform a detailed analysis of Cepheids in NGC 4258, Magellanic Clouds andMilky Way in order to verify the reliability of the theoretical scenario basedon a large set of nonlinear convective pulsation models. We derive Wesenheitfunctions from the synthetic BVI magnitudes of the pulsators and we show thatthe sign and the extent of the metallicity effect on the predictedPeriod-Wesenheit (P-W) relations change according to the adopted passbands.These P-W relations are applied to measured BVI magnitudes of NGC 4258,Magellanic and Galactic Cepheids available in the literature. We find thatMagellanic and Galactic Cepheids agree with the metallicity dependence of thepredicted P-W relations. Concerning the NGC 4258 Cepheids, the results stronglydepend on the adopted metallicity gradient across the galactic disc. The mostrecent nebular oxygen abundances support a shallower gradient and provide ametallicity dependence that agrees well with current pulsation predictions.Moreover, the comparison of Cepheid distances based on VI magnitudes withdistance estimates based on the revised TRGB method for external galaxies, onthe HST trigonometric parallaxes for Galactic Cepheids, and on eclipsingbinaries in the Magellanic Clouds seems to favor the metallicity correctionpredicted by pulsation models. The sign and the extent of the metallicitydependence of the Period-Wesenheit and of the Period-Luminosity relationschange according to the adopted passbands. Therefore, distances based ondifferent methods and/or bands should not be averaged. The use of extragalacticCepheids to constrain the metallicity effect requires new accurate andextensive nebular oxygen measurements.

Cepheids in external galaxies. I. The maser-host galaxy NGC 4258 and the metallicity dependence of P-L and P-W relations