Using a sample of 24 264 optically selected active galactic nuclei (AGNs) from the SDSS DR7 data base, we characterize how the profile of the O iii λ5007 emission line relates to bolometric luminosity (L AGN), Eddington ratio, radio loudness, radio luminosity (L 1.4 GHz) and optical class (i.e. broad/narrow-line Seyfert 1, type 2) to determine what drives the kinematics of this kpc-scale line emitting gas. First, we use spectral stacking to characterize how the average O iii λ5007 profile changes as a function of these five variables. After accounting for the known correlation between L AGN and L 1.4 GHz, we report that L 1.4 GHz has the strongest influence on the O iii λ5007 profile, with AGNs of moderate radio luminosity (L 1.4 GHz = 1023-1025 W Hz−1) having the broadest O iii λ5007 profiles. Conversely, we find only a modest change in the O iii λ5007 profile with increasing radio loudness and find no significant difference between the O iii λ5007 profiles of broad- and narrow-line Seyfert 1s. When binned according to Eddington ratio, only the AGNs in our highest bin (i.e. >0.3) show any signs of having broadened O iii λ5007 profiles, although the small numbers of such extreme AGNs in our sample mean we cannot rule out that other processes (e.g. radio jets) are responsible for this broadening. The O iii λ5007 profiles of type 1 and type 2 AGNs show the same trends in terms of line width, but type 1 AGNs display a much stronger 'blue wing', which we interpret as evidence of outflowing ionized gas. We perform multicomponent fitting to the Hβ, O iii λλ4959, 5007, N ii λλ6548, 6584 and Hα lines for all the AGNs in our sample to calculate the proportions of AGNs with broad O iii λ5007 profiles. The individual fits confirm the results from our stacked spectra; AGNs with L 1.4 GHz > 1023 W Hz−1 are roughly five times more likely to have extremely broad O iii λ5007 lines (full width at half-maximum, FWHMAvg > 1000 km s−1) compared to lower L 1.4 GHz AGNs, and the width of the O iii λ5007 line peaks in moderate-radio-luminosity AGNs (L 1.4 GHz ∼ 1024 W Hz−1). Our results are consistent with the most disturbed gas kinematics being induced by compact radio cores (rather than powerful radio jets), although broadened O iii λ5007 lines are also present, but much rarer, in low-L 1.4 GHz systems. Our catalogue of multicomponent fits is freely available as an online resource for statistical studies of the kinematics and luminosities of the narrow- and broad-line AGN regions and the identification of potential targets for follow-up observations at http://sites.google.com/site/sdssalpaka.