This paper describes a new determination of the primordial helium abundance (YP), based on the abundance analysis of five metal-poor extragalactic H II regions. For three regions of the sample (SBS 0335-052, I Zw 18, and H29) we present tailored photoionization models based on improved calculations with respect to previous models. In particular, we use the photoionization models to study quantitatively the effect of collisional excitation of Balmer lines on the determination of the helium abundance (Y) in the individual regions. This effect is twofold: first, the intensities of the Balmer lines are enhanced with respect to the pure recombination value, mimicking a higher hydrogen abundance; second, the observed reddening is larger than the true extinction, as a result of the differential effect of collisions on different Balmer lines. In addition to these effects, our analysis takes into account the following features of H II regions: (1) the temperature structure, (2) the density structure, (3) the presence of neutral helium, (4) the collisional excitation of the He I lines, (5) the underlying absorption of the He I lines, and (6) the optical thickness of the He I lines. The object that shows the highest increase in Y after the inclusion of collisional effects in the analysis is SBS 0335-052, whose helium abundance has been revised by ΔY=+0.0107. The revised Y-values for the five objects in our sample yield an increase of +0.0035 in YP, giving YP=0.2391+/-0.0020.