Objective Bayesian hypothesis testing and estimation for the intraclass model

Figures & data

Figure 1. The intrinsic discrepancy in (8(8) ) as a function of for n=1, k=4 and .

Table 1. The MAE of the Bayesian estimators for ρ based on 10,000 replications in the simulation study.

		n=5				n=20
	Prior	Intrinsic	Mean	Median	Mode	Intrinsic	Mean	Median	Mode
		0.148	0.155	0.149	0.162	0.058	0.060	0.058	0.059
		0.164	0.163	0.166	0.185	0.060	0.060	0.060	0.062
0		0.242	0.213	0.243	0.333	0.108	0.105	0.108	0.115
		0.294	0.263	0.296	0.379	0.114	0.111	0.114	0.121
0.3		0.268	0.230	0.268	0.377	0.119	0.115	0.119	0.129
		0.315	0.276	0.315	0.412	0.124	0.119	0.124	0.133
0.8		0.148	0.157	0.148	0.151	0.057	0.059	0.057	0.057
		0.142	0.141	0.141	0.153	0.056	0.056	0.056	0.058

Table 2. The MAE of the MLE for ρ based on 10,000 replications in the simulation study.

	n=5	n=20
	0.137	0.058
0	0.198	0.102
0.3	0.231	0.118
0.8	0.236	0.074

Table 3. The estimated tail probabilities of posterior distributions based on 10,000 replications in the simulation study.

		n=5		n=20
	Prior
		0.0453	0.9127	0.0477	0.9010
		0.0497	0.9145	0.0425	0.9166
0		0.0460	0.9069	0.0535	0.8977
		0.0842	0.8598	0.0617	0.8879
0.3		0.0439	0.9119	0.0453	0.9054
		0.1021	0.8357	0.0614	0.8816
0.8		0.0441	0.9087	0.0484	0.9001
		0.1341	0.7825	0.0779	0.8587

Figure 2. Sampling distribution of under obtained from the 5000 simulations with for different sample sizes when testing .

Figure 3. Sampling distribution of under obtained from 5000 simulations with for different sample sizes when testing .

Figure 4. The marginal posterior density for ρ based on two objective priors (left), and the intrinsic statistic with the non-rejection regions corresponding to the threshold values (right) for the orthodontic data in Frees (2004): (a) marginal posterior distribution and (b) intrinsic statistic.

Table 4. The orthodontic data from Frees (2004).

	Age of girls				Age of boys
Number	8	10	12	14	8	10	12	14
1	21	20	21.5	23	26	25	29	31
2	21	21.5	24	25.5	21.5	22.5	23	26.5
3	20.5	24	24.5	26	23	22.5	24	27.5
4	23.5	24.5	25	26.5	25.5	27.5	26.5	27
5	21.5	23	22.5	23.5	20	23.5	22.5	26
6	20	21	21	22.5	24.5	25.5	27	28.5
7	21.5	22.5	23	25	22	22	24.5	26.5
8	23	23	23.5	24	24	21.5	24.5	25.5
9	20	21	22	21.5	23	20.5	31	26
10	16.5	19	19	19.5	27.5	28	31	31.5
11	24.5	25	28	28	23	23	23.5	25
12					21.5	23.5	24	28
13					17	24.5	26	29.5
14					22.5	25.5	25.5	26
15					23	24.5	26	30
16					22	21.5	23.5	25

Table 5. Estimations of ρ for the orthodontic data from Frees (2004).

Priors	Intrinsic	Mean	Median	Mode
	0.603	0.598	0.601	0.608
	0.622	0.616	0.620	0.627

Frees, E. W. (2004). Longitudinal and panel data: Analysis and applications in the social sciences. New York: Cambridge University Press.

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