A metric has positive sectional curvature if and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$

Question

It is well known that

In dimension three a metric has positive sectional curvature if and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$. where $r$ denotes the scalar curvature.

Is there a higher dimensional analogues of this theorem?