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 $T(\mathbf{X})$.


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.

We do not need to compute a p-value for hypothesis testing. We can do indeed what you suggest and compute a threshold value. An example is a likelihood ratio test as described by Neyman and Pearson. The example below from wikipedia uses a threshold value $\eta$ for the likelihood ratio

Define the rejection region of the null hypothesis for the Neyman–Pearson (NP) test as $$R_{NP} = \left\{ x: \frac{\mathcal{L}(\theta_0|x)}{\mathcal{L}(\theta_1|x)} \leq \eta \right\}$$ where $\eta$ is chosen so that $P( R_{NP}  |  \theta_0 ) = \alpha$ .


Why still compute the p-values?
Because it is not all about hypothesis testing.
For instance, often you want to provide a more detailed value than just a 'pass' or 'not-pass'.
