The peak-background split argument is commonly used to relate the abundance of dark matter haloes to their spatial clustering. Testing this argument requires an accurate determination of the halo mass function. We present a maximum-likelihood method for fitting parametric functional forms to halo abundances which differs from previous work because it does not require binned counts. Our conclusions do not depend on whether we use our method or more conventional ones. In addition, halo abundances depend on how haloes are defined. Our conclusions do not depend on the choice of link length associated with the friends-of-friends halo finder, nor do they change if we identify haloes using a spherical overdensity algorithm instead. The large-scale halo bias measured from the matter–halo cross spectrum b× and the halo autocorrelation function bξ (on scales k∼ 0.03 h Mpc−1 and r∼ 50 h−1 Mpc) can differ by as much as 5 per cent for haloes that are significantly more massive than the characteristic mass M*. At these large masses, the peak-background split estimate of the linear bias factor b1 is 3–5 per cent smaller than bξ, which is 5 per cent smaller than b×. We discuss the origin of these discrepancies: deterministic non-linear local bias, with parameters determined by the peak-background split argument, is unable to account for the discrepancies we see. A simple linear but non-local bias model, motivated by peaks theory, may also be difficult to reconcile with our measurements. More work on such non-local bias models may be needed to understand the nature of halo bias at this level of precision.