The means of the Bayesian and ML estimates, the square root of the MSE (⁠⁠), and the 2.5% and 97.5% percentiles of estimates from the 10,000 simulations are presented in tables 1 and 2. Those for the ML method are calculated after the infinite estimates are removed. We see that for highly similar (t = 0.1) and highly divergent (t = 5) sequences, the prior has a noticeable impact. For example, when t = 0.1 the mean of Bayesian estimates of ω is 0.02 when the true ω = 0.01 and is 1.591 when the true ω = 2.0. The mean MLEs are in comparison closer to the true values than the means of Bayesian estimates. However, the means for the MLEs are calculated after data sets in which = ∞ are excluded, whereas those same data sets are included in the calculation of the Bayesian estimates. Similar patterns are observed concerning estimates of t. Moreover, for small and intermediate ω and t, ML and Bayesian methods have similar MSE, but for large ω and t, the Bayesian has smaller MSE indicating that in those cases Bayesian estimates are preferable to the MLEs.

Table 1.

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Summary Statistics of Bayesian (top, underlined) and ML (bottom) Estimates of from 10,000 Simulated Data Sets.

.	ω = 0.01 .					ω = 0.1 .				ω = 0.5 .					ω = 2 .
	Mean .	.	2.5% .	97.5% .	N0 .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	N∞ .	Mean .	.	2.5% .	97.5% .	N∞ .	P+ .
t = 0.1	0.020	0.014	0.007	0.044	0	0.118	0.045	0.052	0.214	0.543	0.160	0.301	0.904	0	1.591	0.546	0.966	2.359	0	35.1
	0.011	0.009	0	0.033	2861	0.103	0.039	0.041	0.194	0.528	0.172	0.278	0.936	0	2.365	1.484	1.015	5.626	3	60.7
t = 0.5	0.012	0.005	0.005	0.021	0	0.104	0.018	0.072	0.141	0.511	0.076	0.379	0.677	0	1.878	0.329	1.360	2.543	0	98.3
	0.010	0.004	0.003	0.019	15	0.101	0.018	0.069	0.138	0.506	0.076	0.374	0.674	0	2.064	0.424	1.409	3.031	0	98.9
t = 1	0.011	0.003	0.006	0.018	0	0.102	0.014	0.076	0.132	0.506	0.062	0.397	0.637	0	1.922	0.278	1.466	2.497	0	99.9
	0.010	0.003	0.005	0.017	0	0.100	0.014	0.075	0.130	0.503	0.062	0.393	0.635	0	2.038	0.326	1.508	2.764	0	100
t = 5	0.014	0.005	0.009	0.022	0	0.129	0.038	0.089	0.183	0.526	0.109	0.348	0.755	0	1.876	0.374	1.331	2.642	0	97.4
	0.010	0.005	0	0.019	370	0.101	0.034	0.037	0.171	0.515	0.981	0.226	0.762	44	2.120	1.398	1.400	3.228	0	98.6

.	ω = 0.01 .					ω = 0.1 .				ω = 0.5 .					ω = 2 .
	Mean .	.	2.5% .	97.5% .	N0 .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	N∞ .	Mean .	.	2.5% .	97.5% .	N∞ .	P+ .
t = 0.1	0.020	0.014	0.007	0.044	0	0.118	0.045	0.052	0.214	0.543	0.160	0.301	0.904	0	1.591	0.546	0.966	2.359	0	35.1
	0.011	0.009	0	0.033	2861	0.103	0.039	0.041	0.194	0.528	0.172	0.278	0.936	0	2.365	1.484	1.015	5.626	3	60.7
t = 0.5	0.012	0.005	0.005	0.021	0	0.104	0.018	0.072	0.141	0.511	0.076	0.379	0.677	0	1.878	0.329	1.360	2.543	0	98.3
	0.010	0.004	0.003	0.019	15	0.101	0.018	0.069	0.138	0.506	0.076	0.374	0.674	0	2.064	0.424	1.409	3.031	0	98.9
t = 1	0.011	0.003	0.006	0.018	0	0.102	0.014	0.076	0.132	0.506	0.062	0.397	0.637	0	1.922	0.278	1.466	2.497	0	99.9
	0.010	0.003	0.005	0.017	0	0.100	0.014	0.075	0.130	0.503	0.062	0.393	0.635	0	2.038	0.326	1.508	2.764	0	100
t = 5	0.014	0.005	0.009	0.022	0	0.129	0.038	0.089	0.183	0.526	0.109	0.348	0.755	0	1.876	0.374	1.331	2.642	0	97.4
	0.010	0.005	0	0.019	370	0.101	0.034	0.037	0.171	0.515	0.981	0.226	0.762	44	2.120	1.398	1.400	3.228	0	98.6

Note.—The Fequal model is used for codon frequencies. Results for ML have been calculated after removing infinite estimates. For ω = 0.1, there were no data sets with 0 or infinite estimates. N₀ is the number of replicates with = 0, whereas N_∞ is the number of replicates with = ∞. P₊ is the proportion of replicates with significant evidence for positive selection indicated by P(ω > 1 | x) > 0.95 in the Bayesian method or by a significant LRT at the 5% level (one-sided with critical value 2.71) in the likelihood method.

Table 1.

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Summary Statistics of Bayesian (top, underlined) and ML (bottom) Estimates of from 10,000 Simulated Data Sets.

.	ω = 0.01 .					ω = 0.1 .				ω = 0.5 .					ω = 2 .
	Mean .	.	2.5% .	97.5% .	N0 .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	N∞ .	Mean .	.	2.5% .	97.5% .	N∞ .	P+ .
t = 0.1	0.020	0.014	0.007	0.044	0	0.118	0.045	0.052	0.214	0.543	0.160	0.301	0.904	0	1.591	0.546	0.966	2.359	0	35.1
	0.011	0.009	0	0.033	2861	0.103	0.039	0.041	0.194	0.528	0.172	0.278	0.936	0	2.365	1.484	1.015	5.626	3	60.7
t = 0.5	0.012	0.005	0.005	0.021	0	0.104	0.018	0.072	0.141	0.511	0.076	0.379	0.677	0	1.878	0.329	1.360	2.543	0	98.3
	0.010	0.004	0.003	0.019	15	0.101	0.018	0.069	0.138	0.506	0.076	0.374	0.674	0	2.064	0.424	1.409	3.031	0	98.9
t = 1	0.011	0.003	0.006	0.018	0	0.102	0.014	0.076	0.132	0.506	0.062	0.397	0.637	0	1.922	0.278	1.466	2.497	0	99.9
	0.010	0.003	0.005	0.017	0	0.100	0.014	0.075	0.130	0.503	0.062	0.393	0.635	0	2.038	0.326	1.508	2.764	0	100
t = 5	0.014	0.005	0.009	0.022	0	0.129	0.038	0.089	0.183	0.526	0.109	0.348	0.755	0	1.876	0.374	1.331	2.642	0	97.4
	0.010	0.005	0	0.019	370	0.101	0.034	0.037	0.171	0.515	0.981	0.226	0.762	44	2.120	1.398	1.400	3.228	0	98.6

.	ω = 0.01 .					ω = 0.1 .				ω = 0.5 .					ω = 2 .
	Mean .	.	2.5% .	97.5% .	N0 .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	N∞ .	Mean .	.	2.5% .	97.5% .	N∞ .	P+ .
t = 0.1	0.020	0.014	0.007	0.044	0	0.118	0.045	0.052	0.214	0.543	0.160	0.301	0.904	0	1.591	0.546	0.966	2.359	0	35.1
	0.011	0.009	0	0.033	2861	0.103	0.039	0.041	0.194	0.528	0.172	0.278	0.936	0	2.365	1.484	1.015	5.626	3	60.7
t = 0.5	0.012	0.005	0.005	0.021	0	0.104	0.018	0.072	0.141	0.511	0.076	0.379	0.677	0	1.878	0.329	1.360	2.543	0	98.3
	0.010	0.004	0.003	0.019	15	0.101	0.018	0.069	0.138	0.506	0.076	0.374	0.674	0	2.064	0.424	1.409	3.031	0	98.9
t = 1	0.011	0.003	0.006	0.018	0	0.102	0.014	0.076	0.132	0.506	0.062	0.397	0.637	0	1.922	0.278	1.466	2.497	0	99.9
	0.010	0.003	0.005	0.017	0	0.100	0.014	0.075	0.130	0.503	0.062	0.393	0.635	0	2.038	0.326	1.508	2.764	0	100
t = 5	0.014	0.005	0.009	0.022	0	0.129	0.038	0.089	0.183	0.526	0.109	0.348	0.755	0	1.876	0.374	1.331	2.642	0	97.4
	0.010	0.005	0	0.019	370	0.101	0.034	0.037	0.171	0.515	0.981	0.226	0.762	44	2.120	1.398	1.400	3.228	0	98.6

Note.—The Fequal model is used for codon frequencies. Results for ML have been calculated after removing infinite estimates. For ω = 0.1, there were no data sets with 0 or infinite estimates. N₀ is the number of replicates with = 0, whereas N_∞ is the number of replicates with = ∞. P₊ is the proportion of replicates with significant evidence for positive selection indicated by P(ω > 1 | x) > 0.95 in the Bayesian method or by a significant LRT at the 5% level (one-sided with critical value 2.71) in the likelihood method.

Table 2.

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Summary Statistics of Bayesian (top, underlined) and ML (bottom) Estimates of t from 10,000 Simulated Data Sets.

.	t = 0.1 .				t = 0.5 .				t = 1 .				t = 5 .
	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	N∞ .
ω = 0.01	0.102	0.015	0.074	0.134	0.504	0.045	0.421	0.596	1.013	0.100	0.837	1.223	3.910	1.322	2.600	5.506	0
	0.100	0.015	0.072	0.132	0.503	0.045	0.419	0.595	1.011	0.100	0.836	1.222	7.572	8.922	2.676	43.744	244
ω = 0.1	0.102	0.015	0.075	0.133	0.503	0.041	0.427	0.587	1.007	0.077	0.865	1.171	4.406	0.869	3.317	5.795	0
	0.100	0.015	0.073	0.131	0.502	0.041	0.425	0.585	1.006	0.077	0.864	1.170	5.629	2.700	3.373	11.506	24
ω = 0.5	0.102	0.015	0.075	0.132	0.503	0.036	0.436	0.574	1.004	0.057	0.895	1.118	5.158	1.469	4.249	6.368	0
	0.100	0.015	0.073	0.130	0.501	0.036	0.434	0.572	1.002	0.057	0.894	1.116	5.440	2.601	4.228	7.979	43
ω = 2	0.102	0.015	0.075	0.131	0.501	0.035	0.434	0.571	1.001	0.056	0.895	1.112	4.988	0.737	4.274	6.035	0
	0.100	0.014	0.073	0.129	0.500	0.035	0.433	0.571	1.002	0.056	0.895	1.114	5.119	0.726	4.323	6.401	3

.	t = 0.1 .				t = 0.5 .				t = 1 .				t = 5 .
	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	N∞ .
ω = 0.01	0.102	0.015	0.074	0.134	0.504	0.045	0.421	0.596	1.013	0.100	0.837	1.223	3.910	1.322	2.600	5.506	0
	0.100	0.015	0.072	0.132	0.503	0.045	0.419	0.595	1.011	0.100	0.836	1.222	7.572	8.922	2.676	43.744	244
ω = 0.1	0.102	0.015	0.075	0.133	0.503	0.041	0.427	0.587	1.007	0.077	0.865	1.171	4.406	0.869	3.317	5.795	0
	0.100	0.015	0.073	0.131	0.502	0.041	0.425	0.585	1.006	0.077	0.864	1.170	5.629	2.700	3.373	11.506	24
ω = 0.5	0.102	0.015	0.075	0.132	0.503	0.036	0.436	0.574	1.004	0.057	0.895	1.118	5.158	1.469	4.249	6.368	0
	0.100	0.015	0.073	0.130	0.501	0.036	0.434	0.572	1.002	0.057	0.894	1.116	5.440	2.601	4.228	7.979	43
ω = 2	0.102	0.015	0.075	0.131	0.501	0.035	0.434	0.571	1.001	0.056	0.895	1.112	4.988	0.737	4.274	6.035	0
	0.100	0.014	0.073	0.129	0.500	0.035	0.433	0.571	1.002	0.056	0.895	1.114	5.119	0.726	4.323	6.401	3

Note.—The Fequal model is used for codon frequencies. Results for ML have been calculated after removing the infinite estimates. For t = 0.1, 0.5, and 1, there were no data sets with 0 or infinite estimate. N_∞ is the number of replicates with = ∞.

Table 2.

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Summary Statistics of Bayesian (top, underlined) and ML (bottom) Estimates of t from 10,000 Simulated Data Sets.

.	t = 0.1 .				t = 0.5 .				t = 1 .				t = 5 .
	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	N∞ .
ω = 0.01	0.102	0.015	0.074	0.134	0.504	0.045	0.421	0.596	1.013	0.100	0.837	1.223	3.910	1.322	2.600	5.506	0
	0.100	0.015	0.072	0.132	0.503	0.045	0.419	0.595	1.011	0.100	0.836	1.222	7.572	8.922	2.676	43.744	244
ω = 0.1	0.102	0.015	0.075	0.133	0.503	0.041	0.427	0.587	1.007	0.077	0.865	1.171	4.406	0.869	3.317	5.795	0
	0.100	0.015	0.073	0.131	0.502	0.041	0.425	0.585	1.006	0.077	0.864	1.170	5.629	2.700	3.373	11.506	24
ω = 0.5	0.102	0.015	0.075	0.132	0.503	0.036	0.436	0.574	1.004	0.057	0.895	1.118	5.158	1.469	4.249	6.368	0
	0.100	0.015	0.073	0.130	0.501	0.036	0.434	0.572	1.002	0.057	0.894	1.116	5.440	2.601	4.228	7.979	43
ω = 2	0.102	0.015	0.075	0.131	0.501	0.035	0.434	0.571	1.001	0.056	0.895	1.112	4.988	0.737	4.274	6.035	0
	0.100	0.014	0.073	0.129	0.500	0.035	0.433	0.571	1.002	0.056	0.895	1.114	5.119	0.726	4.323	6.401	3

.	t = 0.1 .				t = 0.5 .				t = 1 .				t = 5 .
	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	Mean .	.	2.5% .	97.5% .	N∞ .
ω = 0.01	0.102	0.015	0.074	0.134	0.504	0.045	0.421	0.596	1.013	0.100	0.837	1.223	3.910	1.322	2.600	5.506	0
	0.100	0.015	0.072	0.132	0.503	0.045	0.419	0.595	1.011	0.100	0.836	1.222	7.572	8.922	2.676	43.744	244
ω = 0.1	0.102	0.015	0.075	0.133	0.503	0.041	0.427	0.587	1.007	0.077	0.865	1.171	4.406	0.869	3.317	5.795	0
	0.100	0.015	0.073	0.131	0.502	0.041	0.425	0.585	1.006	0.077	0.864	1.170	5.629	2.700	3.373	11.506	24
ω = 0.5	0.102	0.015	0.075	0.132	0.503	0.036	0.436	0.574	1.004	0.057	0.895	1.118	5.158	1.469	4.249	6.368	0
	0.100	0.015	0.073	0.130	0.501	0.036	0.434	0.572	1.002	0.057	0.894	1.116	5.440	2.601	4.228	7.979	43
ω = 2	0.102	0.015	0.075	0.131	0.501	0.035	0.434	0.571	1.001	0.056	0.895	1.112	4.988	0.737	4.274	6.035	0
	0.100	0.014	0.073	0.129	0.500	0.035	0.433	0.571	1.002	0.056	0.895	1.114	5.119	0.726	4.323	6.401	3

Note.—The Fequal model is used for codon frequencies. Results for ML have been calculated after removing the infinite estimates. For t = 0.1, 0.5, and 1, there were no data sets with 0 or infinite estimate. N_∞ is the number of replicates with = ∞.