Table 5 Rank of 5 two-variable models including the best single-variable model to predict success of copulation using Akaike’s information criterion (AIC) AIC 389.69 390.84 390.85 388.87 388.87 379.69 D AICi 10 11.15 11.16 9.18 9.18 0 Likelihood Akaike’s weight Evidence ratio 0.007 0.004 0.004 0.010 0.010 1 0.007 0.004 0.004 0.010 0.010 0.967 148.4 263.7 265.1 98.5 98.5 1 380 Model Swelling size syn Swelling size agg Swelling size call Swelling size ini Swelling size Swelling size sex ratio Presented are AIC, D AICi, Akaike’s weight and evidence ratio (best model in comparison with model i). The best model is the model with the lowest AIC (sex ratio and relative swelling size) and, consequently, with a D AICi of 0 Primates (2011) 52:373–383 Fig. 4 Best two-variables model: relationship between relative swelling size and predicted probability of successful copulation if the sex ratio is 2, 3, and 6 females. Shown are predicted mean values. Model variables: relative swelling size: b = 0.03 (95% bca C.I.: 0.011–0.045), exp(b) = 1.03 (95% bCI: 1.011–0.1.046), z = 3.36, p = 0.001; sex ratio: b = -0.49 (95% bca CI: -0.63 - [-0.31]), exp(b) = 0.60 (95% bCI: 0.530–0.730), z = -4.19, p\0.001; constant = -0.88 (95% bCI: -2.14 - 0.43). Random effects variable ‘‘Female iden- tity’’: SD = 0.42, SE = 1.32 determinant of copulation success. In the two-variable approach, relative swelling size with sex ratio was the best predictor of copulation success. The phase of maximum receptivity for a baboon female is supposed to be when she has a maximally swollen per- ineum (Wildt et al. 1977). For this reason, males are expected to use swelling size as an indicator of female receptivity, and to ﬁnd swollen females more attractive than females with none or smaller swellings (Girolami and Bielert 1987). Our results are consistent with these expectations, because the probability of success was more than twice as high when females had large swellings than when they had swellings smaller than half of the maximum swelling. Additionally, females generally copulated at higher rates when they had large swellings. It is therefore likely that males use the swollen perineum as a signal of higher probability of ovulation. Proximately hamadryas males may simply ﬁnd larger swellings more attractive therefore copulate and ejaculate more often, similarly to 123 chacma baboon males in the study by Bielert and van der Walt (1982). Variation in the sex ratio of OMUs was almost as good as swelling size at explaining variation in copulation suc- cess. Furthermore, a model that included both sex ratio and relative swelling size performed substantially better than models that included relative swelling size or sex ratio alone. Both factors therefore independently help to explain variation in copulation success. Males have a limited budget of sperm, time, and energy (Nakatsuru and Kramer 1982; Dunbar 1988, 1992). Therefore, when several females are simultaneously in oestrus, males may have to divide these resources, possibly by reducing the number of successful copulations per female. Contrary to our expectation, oestrus synchrony had no effect on copulation success. In our study, however, overall synchrony was rather low (\20% and in all cases only one additional female within the OMU was simulta- neously in oestrus). We cannot, therefore, exclude the possibility that oestrus synchrony affects copulation suc- cess in large OMUs when more than two females are simultaneously oestrous. We know that synchrony (or asynchrony) can be part of a female reproductive strategy (e.g. Lemur katta, Pereira 1991). However, synchrony can occur by chance, as was demonstrated for hamadryas baboons by Tobler et al. (2010). In contrast with the operational sex-ratio, the socio- nomic sex ratio has a clear effect on copulation success in our study colony. This, however, might be an effect of the particular hamadryas social system, in which male leaders are most likely to have to invest more time and energy in monitoring and herding their females because of the per- manent close spatial presence of several male competitors (Kummer 1968; Swedell 2006; Swedell and Schreier 2009; Pines et al. 2011). In other species which live in single-male groups, leader males are normally not permanently surrounded by a large number of male competitors (Kirkpatrick 2011; Jaffe 2011; Robbins 2011) and females might have an interest in