BMC Medical Research Methodology

Table 2 Bayesian selection criteria and posterior estimates of the ZAG-RE and ZABP-RE models fitted to pharmaceutical expenditure (PE-2018) data

From: Zero-augmented beta-prime model for multilevel semi-continuous data: a Bayesian inference

	Criterion	ZAG-RE				ZABP-RE
	DIC₃	6754.54				6369.19
	LPML	− 1124.56				− 1124.28
	WAIC	6769.13				6398.17
	LOO-CV	6772.95				6403.25
	Compute time	4195 s				4338 s
Model	Parameter	Posterior features
Model	Parameter	Mean	SD	2.5%	97.5%	Mean	SD	2.5%	97.5%
Zero-part	Intercept	1.645	0.051	1.546	1.744	1.234	0.051	1.522	1.714
	NDPP	0.117	0.007	0.103	0.132	0.117	0.003	0.093	0.135
	Age, year	−0.010	0.001	− 0.012	− 0.009	− 0.011	0.001	−0.012	− 0.008
	Male	−0.183	0.032	−0.119	− 0.246	− 0.184	0.024	− 0.116	−0.241
Continuous-part	Intercept	−2.320	0.050	−2.419	− 2.222	−0.988	0.015	−1.069	−0.906
	Total	0.063	0.002	0.058	0.068	0.013	0.00	0.013	0.013
	ICPD	−0.061	0.003	−0.066	− 0.056	−0.012	0.00	−0.013	− 0.12
	NDPP	−0.001	0.005	−0.010	0.008	−0.038	0.003	−0.045	−0.031
	NOP	0.110	0.017	0.077	0.142	0.220	0.013	0.194	0.246
	Age, year	0.003	0.001	0.002	0.005	0.006	0.001	0.004	0.007
	Male	−0.019	0.025	−0.067	0.029	−0.092	0.023	−0.137	− 0.047
	ψ	0.905	0.014	0.877	0.933	0.499	0.006	0.487	0.512
Variance component	\({\sigma}_{p_1}\)	0.144	0.015	0.117	0.176	0.148	0.011	0.112	0.153
	\({\sigma}_{p_2}\)	0.020	0.011	0.008	0.045	0.019	0.011	0.008	0.041
	\({\sigma}_{p_1{p}_2}\)	−0.004	0.014	−0.029	0.020	−0.004	0.008	−0.022	0.020
	\({\sigma}_{c_1}\)	0.718	0.216	0.404	1.250	0.754	0.214	0.397	1.241
	\({\sigma}_{c_2}\)	13.53	16.51	1.097	56.570	10.16	4.891	1.090	35.525
	\({\sigma}_{c_1{c}_2}\)	−1.698	1.249	−5.018	0.154	−1.700	0.891	−3.851	0.151

SD, 2.5 and 97.5% represents standard deviation and percentiles from the posterior distributions of parameters, respectively. Computational time in second
DIC₃ deviance information criterion, LPML log pseudo marginal likelihood, WAIC watanabe-akaike information criterion, LOO-CV leave-one-out cross-validation, ZAG-RE zero-augmented gamma regression, ZABP-RE zero-augmented beta-prime regression

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