Testing Normal Means: The Reconcilability of the 𝑃 Value and the Bayesian Evidence

Table 1

𝑃 value and lim𝑃(𝐻₀∣𝑥) for testing the Behrens-Fisher problem for different 𝑚 and 𝑛.

	𝑚=2, 𝑛=2
𝑐−(𝑥−𝑦)	−2.5000	−2.1000	−1.8000	−1.6000	−1.3000	−0.9000	−0.5000	−0.3000	−0.1000
𝑝(𝑥)	0.1165	0.1405	0.1785	0.1880	0.2330	0.3110	0.3850	0.4155	0.4745
lim𝑃(𝐻0∣𝑥)	0.0195	0.0420	0.0505	0.0705	0.1280	0.1945	0.3140	0.3770	0.4675
	𝑚=5, 𝑛=5
𝑐−(𝑥−𝑦)	−2.5000	−2.1000	−1.8000	−1.6000	−1.3000	−0.9000	−0.5000	−0.3000	−0.1000
𝑝(𝑥)	0.0385	0.0520	0.0760	0.0995	0.1405	0.2250	0.3460	0.3845	0.4705
lim𝑃(𝐻0∣𝑥)	0.0060	0.0195	0.0400	0.0465	0.0765	0.1630	0.2840	0.3640	0.4630
	𝑚=1, 𝑛=8
𝑐−(𝑥−𝑦)	−2.5000	−2.1000	−1.8000	−1.6000	−1.3000	−0.9000	−0.5000	−0.3000	−0.1000
𝑝(𝑥)	0.1165	0.1405	0.1785	0.1880	0.2330	0.3110	0.3850	0.4155	0.4745
lim𝑃(𝐻0∣𝑥)	0.0195	0.0420	0.0505	0.0705	0.1280	0.1945	0.3140	0.3770	0.4675
	𝑚=3, 𝑛=5
𝑐−(𝑥−𝑦)	−2.5000	−2.1000	−1.8000	−1.6000	−1.3000	−0.9000	−0.5000	−0.3000	−0.1000
𝑝(𝑥)	0.0600	0.0755	0.1030	0.1360	0.1765	0.2615	0.3495	0.4355	0.4595
lim𝑃(𝐻0∣𝑥)	0.0070	0.0205	0.0405	0.0545	0.0845	0.1715	0.3005	0.3760	0.4670
	𝑚=12, 𝑛=3
𝑐−(𝑥−𝑦)	−2.5000	−2.1000	−1.8000	−1.6000	−1.3000	−0.9000	−0.5000	−0.3000	−0.1000
𝑝(𝑥)	0.0835	0.1130	0.1260	0.1585	0.2075	0.2975	0.3700	0.4245	0.4995
lim𝑃(𝐻0∣𝑥)	0.0185	0.0315	0.0515	0.0730	0.1025	0.1810	0.2905	0.3635	0.4585