It soon appeared that comparisons between two rival hypotheses were far more informative, in particular because little can be said about error rates if the null hypothesis is in fact false. Neyman and Pearson (1928, 1933, and 1967) devised the so-called likelihood ratio test, a test that compares the likelihoods of two rivaling hypotheses. Let \(h\) and \(h'\) be the null and the alternative hypothesis respectively. We can compare these hypotheses by the following test function \(F\) over the sample space:  \[ F(s) = \begin{cases} 1 \quad \text{if } \frac{P_{h'}(s)}{P_{h}(s)} &gt; r,\\ 0 \quad \text{otherwise,} \end{cases} \]  where \(P_{h}\) and \(P_{h'}\) are the probability distributions over the sample space determined by the statistical hypotheses \(h\) and \(h'\) respectively. If \(F(s) = 1\) we decide to reject the null hypothesis \(h\), else we accept \(h\) for the time being and so disregard \(h'\).