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In a recent article discussing the demerits of relying on the p-value for statistical inference, called "Matrixx v. Siracusano and Student v. Fisher Statistical significance on trial" (DOI: 10.1111/j.1740-9713.2011.00511.x), Stephen T. Ziliak opposes the use of p-values. In the concluding paragraphs he says:

The data is the one thing that we already do know, and for certain. What we actually want to know is something quite different: the probability of a hypothesis being true (or at least practically useful), given the data we have. We want to know the probability that the two drugs are different, and by how much, given the available evidence. The significance test – based as it is on the fallacy of the transposed conditional, the trap that Fisher fell into – does not and cannot tell us that probability. The power function, the expected loss function, and many other decision-theoretic and Bayesian methods descending from Student and Jeffreys, now widely available and free on-line, do.

What is the power function, the expected loss function and "other decision-theoretic and Bayesian methods? Are these methods widely used and available in R? How are these new suggested methods implemented? How, for instance, would I use these methods to test my hypothesis in a dataset I would otherwise use conventional 2-sample t-tests and p-values?

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There are a lot of papers arguing against the use of $p$-values alone, but it really depends on the context, IMO. Could you add more information on what you're interested in (cf. your last sentence)? –  chl Nov 3 '11 at 23:06
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I don't have access to the article, but this argument indicates a rather flawed understanding of what's going on. Despite a flawed understanding, the conclusion that other statistics are worth consideration is reasonable. Expected loss function is simply an estimate of the expected value of the loss function (e.g. squared error, logistic, etc.). –  Iterator Nov 4 '11 at 21:00

4 Answers 4

up vote 7 down vote accepted

This sounds like another strident paper by a confused individual. Fisher didn't fall into any such trap, though many students of statistics do.

Hypothesis testing is a decision theoretic problem. Generally, you end up with a test with a given threshold between the two decisions (hypothesis true or hypothesis false). If you have a hypothesis which corresponds to a single point, such as $\theta=0$, then you can calculate the probability of your data resulting when it's true. But what do you do if it's not a single point? You get a function of $\theta$. The hypothesis $\theta\not= 0$ is such a hypothesis, and you get such a function for the probability of producing your observed data given that it's true. That function is the power function. It's very classical. Fisher knew all about it.

The expected loss is a part of the basic machinery of decision theory. You have various states of nature, and various possible data resulting from them, and some possible decisions you can make, and you want to find a good function from data to decision. How do you define good? Given a particular state of nature underlying the data you have obtained, and the decision made by that procedure, what is your expected loss? This is most simply understood in business problems (if I do this based on the sales I observed in the past three quarters, what is the expected monetary loss?).

Bayesian procedures are a subset of decision theoretic procedures. The expected loss is insufficient to specify uniquely best procedures in all but trivial cases. If one procedure is better than another in both state A and B, obviously you'll prefer it, but if one is better in state A and one is better in state B, which do you choose? This is where ancillary ideas like Bayes procedures, minimaxity, and unbiasedness enter.

The t-test is actually a perfectly good solution to a decision theoretic problem. The question is how you choose the cutoff on the $t$ you calculate. A given value of $t$ corresponds to a given value of $\alpha$, the probability of type I error, and to a given set of powers $\beta$, depending on the size of the underlying parameter you are estimating. Is it an approximation to use a point null hypothesis? Yes. Is it usually a problem in practice? No, just like using Bernoulli's approximate theory for beam deflection is usually just fine in structural engineering. Is having the $p$-value useless? No. Another person looking at your data may want to use a different $\alpha$ than you, and the $p$-value accommodates that use.

I'm also a little confused on why he names Student and Jeffreys together, considering that Fisher was responsible for the wide dissemination of Student's work.

Basically, the blind use of p-values is a bad idea, and they are a rather subtle concept, but that doesn't make them useless. Should we object to their misuse by researchers with poor mathematical backgrounds? Absolutely, but let's remember what it looked like before Fisher tried to distill something down for the man in the field to use.

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+1 for actually answering the question, and an additional (but virtual) +1 for challenging the quotation, which is provocative but problematic. I see you are a recent participant here but have already contributed many answers: many thanks and welcome (a bit belatedly) to our site! –  whuber Nov 8 '11 at 21:25
    
Thanks very much for your detailed answer. It helps to think about alternative strategies that are suggested in that paper critically. I asked this question because some colleagues used this paper to say that we shouldn't be looking at p-values at all and I realized that I didn't understand what these alternatives actually meant. Thanks for your clarification! –  Ariel Dec 20 '11 at 23:51

The ez package provides likelihood ratios when you use the ezMixed() function to do mixed effects modelling. Likelihood ratios aim to quantify evidence for a phenomenon by comparing the likelihood (given the observed data) of two models: a "restricted" model that restricts the influence of the phenomenon to zero and an "unrestricted" model that permits non-zero influence of the phenomenon. After correcting the observed likelihoods for the models' differential complexity (via Akaike's Information Criterion, which is asymptotically equivalent to cross-validation), the ratio quantifies the evidence for the phenomenon.

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All those techniques are available in R in the same sense that all of algebra is available in your pencil. Even p-values are available through many many different functions in R, deciding which function to use to get a p-value or a Bayesian posterior is more complex than a pointer to a single function or package.

Once you learn about those techniques and decide what question you actually want the answer too then you can see (or we can provide more help) how to do it using R (or other tools). Just saying that you want to minimize your loss function, or to get a posterior distribution is about as useful as replying "food" when asked what you want to eat for dinner.

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I recommend focusing on things like confidence intervals and model-checking. Andrew Gelman has done great work on this. I recommend his textbooks but also check out the stuff he's put online, e.g. http://andrewgelman.com/2011/06/the_holes_in_my/

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