Table 5.

Likelihood Ratio Test (LRT) Under Branch Models.

Model comparison	ℓ	Estimates of parameters	Free parametersa	df	2Δℓ
Test 1
Null model (M0, unrooted tree)	ℓ0 = −12, 249.40	ω = 0.336	26	1	43.14
Alternative model (branch model, chicken as foreground, unrooted tree)	ℓ1 = −12, 227.83	ω 0 = 0.293, ω1 = 1.171	27
Test 2
Null model (M0, unrooted tree)	ℓ0 = −12, 249.40	ω = 0.336	26	1	36.96
Alternative model (branch model, duck as foreground, unrooted tree)	ℓ1 = −12, 230.92	ω 0 = 0.300, ω1 = 999	27
Test 3
Null model (M0, unrooted tree)	ℓ0 = −12, 249.40	ω = 0.336	26	1	51.03
Alternative model (branch model, chicken and duck simultaneously as foreground, unrooted tree)	ℓ1 = −12, 223.89	ω 0 = 0.287, ω1 = 0.711	27
Test 4
Null model (M0, unrooted tree)	ℓ0 = −12, 249.40	ω = 0.336	26	2	17.15
Alternative model (branch model, bird clade as foreground, rooted tree)	ℓ1 = −12, 223.89	ω 0 = 0.287, ω1 = 0.711	28

Note.—The critical values are ${\chi }_{1,5\text{\%}}^2=3.84$ (i.e., 1 degree of freedom, df, at 5% significance), ${\chi }_{1,1\text{\%}}^2=6.63$ (i.e., 1 degree of freedom, df, at 1% significance), ${\chi }_{2,5\text{\%}}^2=5.99$ (i.e., 2 degree of freedom at 5% significance), and ${\chi }_{2,1\text{\%}}^2=9.21$ (i.e., 2 degree of freedom at 1% significance). The LRT statistic, 2Δℓ, is reported for all model comparisons.

^aFree parameters include branch lengths (21 for an unrooted tree and 22 for a rooted tree), the equilibrium frequencies (3), the transition/transversion rate ratio κ (1); see section “Homogenous ω across sites and taxa” to learn how to calculate the number of free parameters. There is an extra parameter for M0 (ω) and two additional ones for the branch model (one for the background branches,ω₀, and another for the branch selected as foreground,ω₁).CODEML has already been executed under the M0 model, and hence we just need to use the results already generated for this analysis (please see section “Homogenous ω across sites and taxa” for a reminder).