The Gromov-Bishop volume comparison theorem says that if we have a lower bound for the Ricci curvature on $(M,g)$, then its geodesic ball has volume not greater than the geodesic ball with the same radius in the corresponding space form. Since the proof only uses the bound for the Ricci curvature in the radial direction, I am sure there is a counterexample if we merely assume a lower bound for the scalar curvature. However, I couldn't construct any counterexample, and couldn't find any reference either. Any idea?
Chaprer 3 of Bray's dissertation discusses this problem for 3-manifolds. A comparison theorem analogous to Bishop–Gromov is shown to follow from a lower bound on scalar curvature plus a (weak) lower bound on the Ricci curvature.
Specifically, comparing a metric $g$ with the spherical metric $g_0$, which has $R_0 = 6$ and $Ric_0 = 2 g_0$, one requires $R_g \geq 6$ and $Ric_g \geq 2\epsilon g$, where $\epsilon<1$ is a given constant (which can be estimated).