If p-values are random, why not decide using the test statistic instead? I am currently reading the paper

Duncan J Murdoch, Yu-Ling Tsai & James Adcock (2008) P-Values are Random Variables, The American Statistician, 62:3, 242-245, DOI: 10.1198/000313008X332421

In this paper, the authors argue that a p-value is itself a random test statistic. Moreover, given some test statistic $T$ that takes in an i.i.d. sample $\mathbf{X}$ and outputs a real number, a p-value is the probability integral transform of $T(\mathbf{X})$. That is, if $T(\mathbf{X})$ has a cumulative distribution function $F_{\tau}$, then the corresponding p-value is $F_{\tau}(T(\mathbf{X}))$. We can then decide whether to reject a null hypothesis based on this p-value. For example, a decision rule could be to reject the null hypothesis if the corresponding p-value is less than 0.05.
However, because $F_{\tau}$ is monotone increasing, I am not sure why we need to compute a p-value in the first place to decide whether or not to reject a null hypothesis. Can't the value of the test statistic $T(\mathbf{X})$ be used to decide? For example, if the decision rule is to reject the null hypothesis when the p-value is less than 0.05, then, if the inverse of $F_{\tau}$ exists, we can obtain the threshold value for $T(\mathbf{X})$ beyond which the null hypothesis is rejected. Furthermore, we should be able to compute type I and II error rates just using this threshold value for $T(\mathbf{X})$.

You're absolutely right. To perform a test of a fixed size, we don't need p-values at all. We may just define a rejection region of that size in terms of the test-statistic.
Possible advantages of reporting p, though, include that

*

*readers may (if they reader wish) interpret p as a measure of evidence a la Fisher

*readers may bring their own error rate, $\alpha$, to compare with your p

*it may be computationally easier to compute p and compare it with the desired error rate, than invert the desired error rate to find the boundaries of the critical region of the test-statistic

*readers may perform further calculations with p, such as adjusting p for multiple comparisons or converting it to a bound on a Bayes factor etc

