If two t-statistics have the same degrees of freedom, is the p-value of the larger t-statistic greater than the p-value of the smaller t-statistic?

Question

If I'm understanding correctly, calculating p-values from a t-statistic is just an integral of the t-distribution pdf, and so if two t-stats have the same number of degrees of freedom then the larger t-stat should have a larger p-value.

Answer 1

While there is an integral involved, the p-value is not the value of the cdf in general.

Depending on the alternative you may want the area in the left-tail ($p=P(T\leq t_\text{obs})$), the right tail ($p=P(T\geq t_\text{obs})$) or both tails ($p=P(|T|\geq |t_\text{obs}|)$). (Also see here)

The most typical case is the two-sided t-test (case (c) in the image), for which larger (in absolute value) t-statistics correspond to smaller p-values, at a given value for the degrees of freedom. Only in case (a) is the p-value the cdf evaluated at the statistic, where the larger t-statistic necessarily leads to a larger p-value.