The p-value would have been lower if...

One is often asked, it seems, to extend someone a p-value on credit. "The p-value would be lower if we'd had more subjects." "The p-value would have been lower if we'd had a stronger manipulation." "The p-value would have been lower with a cleaner measurement, a continuous instead of a dichotomous outcome, the absence of a ceiling effect..."

These claims could be true, or they could be false, conditional on one thing: Whether the null hypothesis is true or false. This is, of course, a tricky thing to condition on. The experiment itself should be telling us the evidence for or against the null hypothesis.

So now we see that these statements are very clearly begging the question. Perhaps the most accurate formulation would be, "I would have stronger evidence that the null were false if the null were false and I had stronger evidence." It is perfectly circuitous.

When I see a claim like this, I imagine a cockney ragamuffin pleading, "I'll have the p-value next week, bruv, sware on me mum." But one can't issue an IOU for evidence.