We study the relation between the density distribution of tracers for large-scale structure and the underlying matter distribution – commonly termed bias – in the Λ cold dark matter framework. In particular, we examine the validity of the local model of biasing at quadratic order in the matter density. This model is characterized by parameters b1 and b2. Using an ensemble of N-body simulations, we apply several statistical methods to estimate the parameters. We measure halo and matter fluctuations smoothed on various scales. We find that, whilst the fits are reasonably good, the parameters vary with smoothing scale. We argue that, for real-space measurements, owing to the mixing of wavemodes, no smoothing scale can be found for which the parameters are independent of smoothing. However, this is not the case in Fourier space. We measure halo and halo–mass power spectra and from these construct estimates of the effective large-scale bias as a guide for b1. We measure the configuration dependence of the halo bispectra Bhhh and reduced bispectra Qhhh for very large-scale k-space triangles. From these data, we constrain b1 and b2, taking into account the full bispectrum covariance matrix. Using the lowest order perturbation theory, we find that for Bhhh the best-fitting parameters are in reasonable agreement with one another as the triangle scale is varied, although the fits become poor as smaller scales are included. The same is true for Qhhh. The best-fitting values were found to depend on the discreteness correction. This led us to consider halo–mass cross-bispectra. The results from these statistics supported our earlier findings. We then developed a test to explore whether the inconsistency in the recovered bias parameters could be attributed to missing higher order corrections in the models. We prove that low-order expansions are not sufficiently accurate to model the data, even on scales k1∼ 0.04 h Mpc−1. If robust inferences concerning bias are to be drawn from future galaxy surveys, then accurate models for the full non-linear bispectrum and trispectrum will be essential.