Bayesian Testing of Scientific Expectations under Multivariate Normal Linear Models

Figures & data

Figure 1. (a) Illustration of information consistent behavior for the proposed Bayes factor when the observed effect e increases. (b) Illustration of the effect of different fractions across groups when the sample in group 1 is fixed at $N_1=5$ and the sample in group 2 increases. (c) Illustration of the difference between the adjusted prior mean and the unadjusted (OLS) prior mean. (d) Illustration of the change in evidence when increasing the number of random missing observations using list-wise deletion and sampling missings from the posterior predictive distribution.

Table 1. Unconstrained estimates and properties of the rats data with levels of enzyme type SDH (y₁), enzyme type SGOT (y₂), and enzyme type SGPT (y₃). Note. SDH = sorbitol dehydrogenase. SGOT = serum glutamic oxaloacetic transaminase. SGPT = serum gtutamic-pyruvic transaminase.

group	parameter			estimate ± standard error			sample size	fraction
	SDH	SGOT	SGPT	SDH	SGOT	SGPT
1	μ 11	μ12	μ13	22.7 ± 1.1	99.3 ± 5.0	61.9 ± 1.7	10	.175
2	μ21	μ22	μ23	22.8 ± 1.1	108.4 ± 5.0	63.8 ± 1.7	10	.175
3	μ31	μ32	μ33	23.7 ± 1.1	100.9 ± 5.0	60.2 ± 1.7	10	.175
4	μ41	μ42	μ43	27.3 ± 1.1	112.9 ± 5.0	52.9 ± 1.7	10	.175

Table 2. Bayesian model selection for the rats application. In the second, third, fourth, fifth, sixth and seventh column, the relative measures of fit of the equality constraints, complexity of the equality constraints, fit of the order constraints, complexity of the order constraints, the Bayes factors against an unconstrained alternative (Equation (7)(7) $$\begin{array}{c}{B}_{tu}^{*}=\frac{f_t^E}{c_t^E}\times \frac{f_t^O}{c_t^O}\\ =\frac{{\pi }_u({\mathbf{R}}_{t,E}\mathit{\theta }={\mathbf{r}}_{t,E}|\mathbf{Y},\mathbf{X})}{{\pi }_u^{*}({\mathbf{R}}_{t,E}\mathit{\theta }={\mathbf{r}}_{t,E}|\mathbf{Y},\mathbf{X},\mathbf{b})}\times \frac{{\text{Pr}}_u({\mathbf{R}}_{t,O}\mathit{\theta }>{\mathbf{r}}_{t,O}|{\mathbf{R}}_{t,E}\mathit{\theta }={\mathbf{r}}_{t,E},\mathbf{Y},\mathbf{X})}{{\text{Pr}}_u^{*}({\mathbf{R}}_{t,O}\mathit{\theta }>{\mathbf{r}}_{t,O}|{\mathbf{R}}_{t,E}\mathit{\theta }={\mathbf{r}}_{t,E},\mathbf{Y},\mathbf{X},\mathbf{b})}\end{array}$$(7) ), and the posterior probability are presented, respectively.

Model	$f_t^E$	$c_t^E$	$f_t^O$	$c_t^O$	Btu	$B_{t1}$	$B_{t2}$	$B_{t3}$	$P({\mathcal{H}}_t|\mathbf{y})$
${\mathcal{H}}_1$	1.62e–15	1.86e–13			.009	1.000	1795.995	0.009	.009
${\mathcal{H}}_2$			1.40e–08	2.36e–3	.000	0.001	1.000	0.000	.000
${\mathcal{H}}_3$			1.000	0.998	1.002	106.057	190478.668	1.000	.991

Table 3. Unconstrained estimates for the Sesame Street application.

parameter	estimate	standard error	correlation matrix
β11	.505	.066	1.00
β21	.188	.066	−.664 1.00
β12	.327	.062	.375 −.249 1.00
β22	.441	.062	−.249.375 −.664 1.00

Table 4. Bayesian Hypothesis Evaluation for the Sesame Street application. In the second, third, fourth, fifth, sixth and seventh column, the relative measures of fit of the equality constraints, complexity of the equality constraints, fit of the order constraints, complexity of the order constraints, the Bayes factors against an unconstrained alternative (Equation (7)(7) $$\begin{array}{c}{B}_{tu}^{*}=\frac{f_t^E}{c_t^E}\times \frac{f_t^O}{c_t^O}\\ =\frac{{\pi }_u({\mathbf{R}}_{t,E}\mathit{\theta }={\mathbf{r}}_{t,E}|\mathbf{Y},\mathbf{X})}{{\pi }_u^{*}({\mathbf{R}}_{t,E}\mathit{\theta }={\mathbf{r}}_{t,E}|\mathbf{Y},\mathbf{X},\mathbf{b})}\times \frac{{\text{Pr}}_u({\mathbf{R}}_{t,O}\mathit{\theta }>{\mathbf{r}}_{t,O}|{\mathbf{R}}_{t,E}\mathit{\theta }={\mathbf{r}}_{t,E},\mathbf{Y},\mathbf{X})}{{\text{Pr}}_u^{*}({\mathbf{R}}_{t,O}\mathit{\theta }>{\mathbf{r}}_{t,O}|{\mathbf{R}}_{t,E}\mathit{\theta }={\mathbf{r}}_{t,E},\mathbf{Y},\mathbf{X},\mathbf{b})}\end{array}$$(7) ), and the posterior probability are presented, respectively.

Model	$f_t^E$	$c_t^E$	$f_t^O$	$c_t^O$	Btu	$B_{t1}$	$B_{t2}$	$B_{t3}$	$P({\mathcal{H}}_t|\mathbf{y})$
${\mathcal{H}}_1$			0.836	0.007	115.409	1.000	2.8e44	699.11	0.999
${\mathcal{H}}_2$	0.000	0.107			0.000	0.000	1.000	0.000	0.000
${\mathcal{H}}_3$			0.164	0.993	0.165	0.001	4.0e41	1.000	0.001