- Berkson critiqued the p value but did not offer a real alternative.
- Fisher pointed out that it may be difficult to limit what we call ‘null’ and ‘alternative’ hypothesis.
- Goodman presented an alternative to the P-value:
The likelihood ratio: The LR is defined as the ratio of the data’s probability under two hypotheses:
- The P-value is not comparative, so it is not an evidential measure and does not behave like one.
- It is often a monotonic function of the maximum LR, the LR of the hypothesis with maximum likelihood (i.e. the MLE) versus the null.
- how do we pre-specify alternative hypotheses? The easiest way is to choose a single non-null parameter value, e.g. the ‘minimum important difference’.
- weighted likelihood ratio: averaging the likelihood function over all values of the treatment difference included in the alternative hypothesis.
- Using the relationship between the fixed sample size P-value and the maximum LR will provide better results.