Table 2.

The mean estimates (standard deviation) over 1000 replicates when data sets were generated from the Dirichlet‐multinomial mixed model with categorical‐specific random effect with common variance

	${\mathit{\lambda }}_{\mathbf{2}}^{\mathit{F}}=$ 0.5	${\mathit{\lambda }}_{\mathbf{2}}^{\mathit{E}}=$ −1	${\mathit{\lambda }}_{\mathbf{3}}^{\mathit{E}}=$ 0.1	${\mathit{\lambda }}_{\mathbf{22}}^{\mathit{E}\mathit{F}}=$ 0.8	${\mathit{\lambda }}_{\mathbf{32}}^{\mathit{E}\mathit{F}}=$ −2	$\mathbf{\text{log}}\left({\mathit{\sigma }}_{\mathit{u}}\right)=$ −1.309	$\mathbf{\text{log}}(\mathit{\theta })=$ −2.302	Loglik
Subj‐sp	0.418(0.130)	−0.959(0.059)	0.096(0.050)	0.765(0.071)	−1.900(0.075)	−1.680(0.754)	−1.886(0.094)	−3918.581(18.333)
Cat‐sp	0.438(0.172)	−1.007(0.116)	0.108(0.109)	0.809(0.084)	−2.017(0.086)	−0.704(0.005)	−2.366(0.125)	−3961.971(14.865)
	${\mathit{\lambda }}_{\mathbf{2}}^{\mathit{F}}=$ 0.5	${\mathit{\lambda }}_{\mathbf{2}}^{\mathit{E}}=$ −1	${\mathit{\lambda }}_{\mathbf{3}}^{\mathit{E}}=$ 0.1	${\mathit{\lambda }}_{\mathbf{22}}^{\mathit{E}\mathit{F}}=$ 0.8	${\mathit{\lambda }}_{\mathbf{32}}^{\mathit{E}\mathit{F}}=$−2	$\mathbf{\text{log}}\left({\mathit{\sigma }}_{\mathit{u}}\right)=$ −0.693	$\mathbf{\text{log}}(\mathit{\theta })=$ −2.302	Loglik
Sub‐sp	0.359(0.130)	−0.882(0.080)	0.096(0.077)	0.695(0.088)	−1.739(0.095)	−0.973(0.334)	−1.320(0.099)	−4064.461(18.503)
Cat‐sp	0.462(0.170)	−1.000(0.122)	0.110(0.117)	0.794(0.085)	−1.997(0.083)	−0.607(0.028)	−2.253(0.129)	−3988.959(16.392)
	${\mathit{\lambda }}_{\mathbf{2}}^{\mathit{F}}=$ 0.5	${\mathit{\lambda }}_{\mathbf{2}}^{\mathit{E}}=$ −1	${\mathit{\lambda }}_{\mathbf{3}}^{\mathit{E}}=$ 0.1	${\mathit{\lambda }}_{\mathbf{22}}^{\mathit{E}\mathit{F}}=$ 0.8	${\mathit{\lambda }}_{\mathbf{32}}^{\mathit{E}\mathit{F}}=$ −2	$\mathbf{\text{log}}\left({\mathit{\sigma }}_{\mathit{u}}\right)=$ −0.223	$\mathbf{\text{log}}(\mathit{\theta })=$ −2.302	Loglik
Sub‐sp	0.300(0.132)	−0.766(0.099)	0.092(0.105)	0.602(0.11)	−1.509(0.121)	−0.766(0.196)	−0.698(0.112)	−4171.966(17.482)
Cat‐sp	0.455(0.194)	−1.004(0.173)	0.099(0.17)	0.795(0.089)	−1.985(0.096)	−0.237(0.049)	−2.235(0.165)	−4011.262(19.136)
	${\mathit{\lambda }}_{\mathbf{2}}^{\mathit{F}}=$ 0.5	${\mathit{\lambda }}_{\mathbf{2}}^{\mathit{E}}=$ −1	${\mathit{\lambda }}_{\mathbf{3}}^{\mathit{E}}=$ 0.1	${\mathit{\lambda }}_{\mathbf{22}}^{\mathit{E}\mathit{F}}=$ 0.8	${\mathit{\lambda }}_{\mathbf{32}}^{\mathit{E}\mathit{F}}=$ −2	$\mathbf{\text{log}}\left({\mathit{\sigma }}_{\mathit{u}}\right)=$ 0	$\mathbf{\text{log}}(\mathit{\theta })=$ −2.302	Loglik
Sub‐sp	0.270(0.129)	−0.691(0.112)	0.092(0.117)	0.541(0.122)	−1.376(0.133)	−0.699(0.187)	−0.367(0.117)	−4193.591(19.342)
Cat‐sp	0.449(0.200)	−1.003(0.213)	0.100(0.197)	0.795(0.088)	−1.985(0.101)	−0.020(0.046)	−2.225(0.177)	−3993.517(23.146)

Each rows started with Sub‐sp represents the estimates (standard deviation) when data sets were fitted with the DMM model with subject‐specific random effect and rows started with Cat‐sp represents the estimates (standard deviation) when data sets were fitted with DMM model with categorical‐specific random effect having common variance.

λ,σ _u,θ are as explained in Figure 1.

Loglik represents the loglikelihood value obtained using the corresponding model.

Rows in gray represent the estimation when the standard deviation of the normally distributed random effect is small.