A novel correction method for modelling parameter-driven autocorrelated time series with count outcome

Xu, Xiao-Han; Zhan, Zi-Shu; Shi, Chen; Xiao, Ting; Ou, Chun-Quan

doi:10.1186/s12889-024-18382-4

BMC Public Health

Table 1 Point estimates and type I error rates for UB-corrected method and MSRC method

From: A novel correction method for modelling parameter-driven autocorrelated time series with count outcome

N	\(\boldsymbol{\sigma}_{\boldsymbol{\alpha}}^{\boldsymbol{2}}\)	\(\boldsymbol{\rho }_{\boldsymbol{\alpha }}\)	Level_bias	Trend_bias	\(\boldsymbol{\sigma }_{\boldsymbol{UB}}^{\boldsymbol{2}}\)^a	UB-corrected				MSRC
						\(\boldsymbol{\rho}_{\boldsymbol{UB}}\)^a	Type I error rate			\(\boldsymbol{\rho}_{\boldsymbol{MSRC}}\)^a	Type I error rate
						\(\boldsymbol{\rho}_{\boldsymbol{UB}}\)^a	Level change	Trend change	Both change	\(\boldsymbol{\rho}_{\boldsymbol{MSRC}}\)^a	Level change	Trend change	Both change
20	0.5	0.2	0.005	-0.007	0.461	0.190	0.088	0.092	0.106	0.190	0.088	0.092	0.106
		0.4	0.018	0.059	0.399	0.259	0.114	0.130	0.159	0.259	0.114	0.130	0.159
		0.6	0.004	0.006	0.306	0.341	0.153	0.203	0.237	0.341	0.153	0.203	0.237
		0.8	0.005	-0.024	0.175	0.415	0.168	0.255	0.292	0.415	0.168	0.255	0.292
	1	0.2	0.021	-0.070	0.918	0.171	0.078	0.084	0.094	0.171	0.078	0.084	0.094
		0.4	0.023	0.022	0.789	0.242	0.114	0.135	0.153	0.242	0.114	0.135	0.153
		0.6	0.015	-0.138	0.623	0.315	0.152	0.202	0.233	0.315	0.152	0.202	0.233
		0.8	0.030	-0.076	0.356	0.392	0.167	0.278	0.308	0.391	0.167	0.278	0.308
180	0.5	0.2	0.001	-0.022	0.503	0.200	0.049	0.052	0.050	0.215	0.048	0.051	0.049
		0.4	0.005	-0.022	0.501	0.387	0.055	0.059	0.063	0.409	0.052	0.057	0.060
		0.6	0.004	-0.006	0.503	0.581	0.068	0.068	0.076	0.619	0.060	0.060	0.066
		0.8	0.007	-0.017	0.489	0.766	0.088	0.095	0.112	0.809	0.068	0.069	0.078
	1	0.2	0.007	0.009	1.003	0.206	0.052	0.050	0.051	0.228	0.050	0.048	0.049
		0.4	0.009	-0.019	1.005	0.397	0.053	0.052	0.057	0.421	0.050	0.050	0.055
		0.6	0.011	-0.016	1.011	0.586	0.070	0.064	0.073	0.626	0.060	0.057	0.065
		0.8	0.017	-0.006	0.993	0.767	0.089	0.101	0.113	0.808	0.068	0.079	0.086
340	0.5	0.2	0.000	0.007	0.500	0.199	0.052	0.052	0.056	0.215	0.050	0.051	0.054
		0.4	0.006	-0.007	0.501	0.397	0.053	0.056	0.060	0.430	0.048	0.051	0.055
		0.6	0.002	-0.008	0.503	0.591	0.058	0.060	0.065	0.643	0.043	0.044	0.046
		0.8	0.003	0.000	0.500	0.783	0.074	0.075	0.086	0.825	0.051	0.050	0.052
	1	0.2	-0.004	0.009	1.000	0.204	0.054	0.050	0.054	0.232	0.053	0.048	0.052
		0.4	0.002	-0.014	1.000	0.402	0.053	0.056	0.056	0.440	0.048	0.051	0.050
		0.6	-0.002	0.004	1.001	0.594	0.062	0.063	0.070	0.649	0.046	0.048	0.051
		0.8	0.017	-0.001	1.011	0.786	0.074	0.074	0.086	0.827	0.051	0.049	0.053

UB-corrected Unbiased correction, MSRC Maximum significant ρ correction
^aIn order to compare with the true values of the nuisance parameters, the estimators \({\hat{\sigma }}_{w,UB}^{2}\), \({\widehat{\rho }}_{w,UB}\), and \({\widehat{\rho }}_{w,MSRC}\)for the process \({w}_{t}\) were all transformed into estimators \({\hat{\sigma }}_{UB}^{2}\), \({\widehat{\rho }}_{UB}\), and \({\widehat{\rho }}_{MSRC}\) corresponding to the latent process \({\alpha }_{t}\) by an exponential function

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ISSN: 1471-2458

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