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We've all seen how $p$-values can be misinterpreted to draw false inferences. However, I'm interested in learning more about theoretical/mathematical/philosophical objections to the hypothesis-testing-paradigm, ie. I'd like to learn why someone would prefer (say) a likelihood ratio over a hypothesis test.
I've read some articles by Andrew Gelman on this topic, but I'm trying to find more literature. Any help would be greatly appreciated.
Well, try virtually any book about Bayesian statistics. I haven't find one that doesn't have at least a few paragraphs debunking practice of significance testing. Really. And Gelman's books are good example of that. I used to remember relevant titles (I still remember my first one: E.T. Jaynes "Probability theory - the logic of science". Jaynes definitely wrote with passion, and the controversy over significance testing is something, that really made me do statistics professionally ;-) ), but soon I got overwhelmed by the abundance of literature on this topic.
Try also the phrase +bayesian significance testing fisher in Google Scholar and I guarantee you will find at least dozen good references (and some interesting info too).
This paper characterizes the publication/scientific practice in psychology as being a ritual of finding p<.05. They argue against hypothesis testing without solid theoretical basis. Note, though, that it is--if at all--more philosophical than anything else. But maybe it helps.
I'm reading an article right now called "Principles of Inference and Their Consequences" by D.G. Mayo and M. Kruse. It's a good article so far about how hypothesis testing can violate principles like the likelihood principle(LP). They go through a concrete example of coin-tossing and show how the concept of statistical significance violates the LP.
