We investigate the effect of statistical properties of the surface roughness on its superhydrophobic properties. In particular we focus on the liquid-solid interfacial structure and its dependence on the coupled effect of surface statistical properties and drop pressure. We find that, for self-affine fractal surfaces with Hurst exponent $H>0.5$, the transition to the Wenzel state firstly involves the short wavelengths of the roughness and, then, gradually moves to larger and larger scales. However, as the drop pressure is increased, at a certain point of the loading history, an abrupt transition to the Wenzel state occurs. This sudden transition identifies the critical drop pressure $p_{W}$, which destabilizes the composite interface. We find that $p_{W}$ can be strongly enhanced by increasing the mean square slope of the surface, or equivalently the Wenzel roughness parameter $r_{W}$. Our investigation shows that, even in the case of randomly rough surface, $r_{W}$ is still the most crucial parameter in determining the superhydrophobicity of the surface. We also propose an analytical approach, which allows us to show that, for any given value of Young's contact angle $\theta_{Y}$, a threshold value $\left( r_{W}\right) _{th}=1/\left( -\cos\theta_{Y}\right) $ exists, above which the composite interface is strongly stabilized and the surface presents robust superhydrophobic properties. Interestingly, this threshold value is identical to the one that would be obtained in pure Wenzel regime to guarantee perfect superhydrophobicity.