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. Author manuscript; available in PMC: 2012 May 1.
Published in final edited form as: Theor Popul Biol. 2011 Jan 26;79(3):102–113. doi: 10.1016/j.tpb.2011.01.002

Figure .6.

Figure .6

Normalizing constants on a log scale for antagonistic (cyan), independence (black) and synergistic (red) models for a range of σ values, as obtained by an adjusted Monte Carlo method using equation 14 and Appendix A. The effect of choosing an appropriate σind to minimize the adjustment by the estimate of E[eσepi(x)σind(x)] in equation 13 can be seen by comparing the difference between the constant obtained by choosing the optimal σind and choosing σepi = σind. For example, for the constant desired with σsyn = 32, the corresponding optimal value for σind(x) is 51.3. If σind = 32 were used, the estimate of the expected value E[eσepi(x)σind(x)] in the approximation of c(θ, σ) would have to adjust for a large discrepancy (approximately 1 on the log scale). Similarly, for the constant desired with σant = 80, the optimum is σind = 32, and if σind = 80 were used, the adjustment by the expectation in the estimate would be approximately 2 (log scale). Choosing the optimal value of σind minimizes the effect of the adjustment.