8.3 Posterior hypothesis testing

Hypothesis testing:

\[H_0: \pi \geqslant 0.2\] \[H_a: \pi < 0.2\]

One-sided tests

To evaluate exactly how plausible it is that \(\pi<0.2\):

\[P(\pi < 0.2|Y=14)\]

pbeta(0.20, 18, 92)

## [1] 0.8489856

\[P(H_0|Y=14)=0.151\] \[P(H_a|Y=14)=0.849\]

\[\text{posterior odds}=\frac{P(H_a|Y=14)}{P(H_0|Y=14)} \approx 5.62\]

pbeta(0.20, 18, 92) / (1 - pbeta(0.20, 18, 92))

## [1] 5.621883

\[\text{prior odds}=\frac{P(H_a)}{P(H_0)} \approx 0.093\]

pbeta(0.20, 4, 6) / (1 - pbeta(0.20, 4, 6))

## [1] 0.09366321

\[\text{Bayes Factor}=\frac{\text{posterio odds}}{\text{prior odds}}\]

BF <- (pbeta(0.20, 18, 92) / (1 - pbeta(0.20, 18, 92))) / (pbeta(0.20, 4, 6) / (1 - pbeta(0.20, 4, 6)))
BF

## [1] 60.02232

\[BF=\left\{\begin{matrix} 1 & H_a\text{constant} \\ >1 & H_a\text{increased} \\ <1 & H_a\text{decreased} \\ \end{matrix}\right.\]

Two-sided tests

There’s not one recipe for success

\[H_0: \pi = 0.3\] \[H_a: \pi \neq 0.3\]