Table 3.

Empirical power of score test statistics to detect various forms ${\omega }_i^{\ast }$ of heterogeneity coupled with a nondegenerate Poisson model with mean ${\mu }_i^{\ast }=exp\{{\beta }_0^{\ast }-1.45x_i\}$, x_i ~ Uniform(0, 1), at 5% significance level

${\beta }_0^{\ast }$	n = 50			n = 100			n = 200
	−0.75	0	0.75	−0.75	0	0.75	−0.75	0	0.75
${\omega }_i^{\ast }=0.25$
vdB test	0.060	0.096	0.313	0.086	0.147	0.578	0.122	0.294	0.895
J&H test	0.069	0.087	0.249	0.094	0.118	0.481	0.123	0.238	0.851
Prop. c-a test	0.071	0.086	0.247	0.094	0.122	0.478	0.124	0.241	0.849
${\omega }_i^{\ast }=0.45+0.15x_i$
vdB test	0.100	0.221	0.698	0.156	0.388	0.960	0.219	0.684	1.000
J&H test	0.102	0.190	0.623	0.178	0.362	0.927	0.237	0.608	0.999
Prop. c-a test	0.105	0.191	0.617	0.179	0.363	0.933	0.235	0.603	0.999
${\omega }_i^{\ast }=\frac{exp\{-1.2+1.1x_i\}}{1+exp\{-1.2+1.1x_i\}}$
vdB test	0.059	0.109	0.373	0.090	0.213	0.704	0.124	0.343	0.953
J&H test	0.080	0.112	0.318	0.103	0.192	0.662	0.136	0.297	0.929
Prop. c-a test	0.087	0.112	0.317	0.105	0.194	0.655	0.135	0.292	0.924
${\omega }_i^{\ast }=\frac{{\pi }_i^{\ast }-exp\{-{\mu }_i^{\ast }\}}{1-exp\{-{\mu }_i^{\ast }\}},{\pi }_i^{\ast }=exp\{-exp\{1.3-2.4x_i\}\}$
vdB test	0.166	0.293	0.257	0.295	0.395	0.413	0.392	0.612	0.590
J&H test	0.236	0.454	0.386	0.383	0.628	0.544	0.506	0.847	0.731
Prop. c-a test	0.710	0.845	0.957	0.944	0.981	0.997	0.997	1.000	1.000
${\omega }_i^{\ast }=\frac{exp\{-15+30x_i\}}{1+exp\{-15+30x_i\}}$
vdB test	0.026	0.067	0.095	0.039	0.079	0.131	0.051	0.090	0.189
J&H test	0.054	0.107	0.166	0.082	0.141	0.271	0.090	0.180	0.448
Prop. c-a test	0.068	0.151	0.430	0.122	0.238	0.758	0.131	0.431	0.967
${\omega }_i^{\ast }=\mathrm{\Phi }(-15+30x_i)$
vdB test	0.042	0.070	0.099	0.045	0.069	0.153	0.057	0.083	0.221
J&H test	0.051	0.097	0.160	0.075	0.114	0.278	0.104	0.167	0.465
Prop. c-a test	0.062	0.142	0.485	0.095	0.222	0.739	0.142	0.409	0.962

Note: vdB, van den Broek, test with df = 1; J&H, Jansakul & Hinde, test with df = 2; Prop. c-a, Proposed covariate-adjusted, test with df = 2.