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Title: $p$-values and quantiles for count data Authors:  Paul Wilson - University Of Wolverhampton (United Kingdom) [presenting]
Jochen Einbeck - Durham University (United Kingdom)
Abstract: For a given continuous test statistic $T$ and observed value $t$, traditionally ``right--tail'' $p$--values are defined as $P_0(T\geq t)$, where $P_0$ is the distribution of $T$ under the null hypothesis, ``left--tail'' $p$--values being defined analogously. However, this is not the only way of defining $p$--values. For discrete data, $\hat\alpha_{T,\lambda}(t) = P_0[T>t]+\lambda P_0[T=t]$ defines the ``$\lambda$-$p$-value'' $\lambda=1$ corresponding to the ``traditional $p$-value'', and $\lambda=0.5$ to the ``mid $p$--value''. Following similar lines of reasoning, one can motivate and define the $\lambda$-\textit{quantile}. In a discrete setting, $\lambda$-$p$-values and quantiles may be shown to have superior properties to their traditional counterparts, for example, unlike traditional $p$-values, the expected value of mid-$p$-values is $0.5$ under the null hypothesis, and their use in hypothesis tests with discrete test statistics may, at least in certain cases, be shown to lead to better power and attainment rates. We explore the use of $\lambda$-$p$-values and quantiles, discussing their advantages and disadvantages.