*In preparation for a new post that takes up some of the recent battles on reforming or replacing p-values, I reblog an older post on power, one of the most misunderstood and abused notions in statistics. (I add a few “notes on howlers”.) The power of a test T in relation to a discrepancy from a test hypothesis H _{0} is the probability T will lead to rejecting H_{0} when that discrepancy is present. Power is sometimes misappropriated to mean something only distantly related to the probability a test leads to rejection; but I’m getting ahead of myself. This post is on a classic fallacy of rejection.* Continue reading

# CIs and tests

## How to avoid making mountains out of molehills (using power and severity)

## Continued:”P-values overstate the evidence against the null”: legit or fallacious?

Since the comments to my previous post are getting too long, I’m reblogging it here to make more room. I say that the issue raised by J. Berger and Sellke (1987) and Casella and R. Berger (1987) concerns evaluating the evidence in relation to a given hypothesis (using error probabilities). Given the information that *this* hypothesis H* was randomly selected from an urn with 99% true hypothesis, we wouldn’t say this gives a great deal of evidence for the truth of H*, nor suppose that H* had thereby been well-tested. (H* might concern the existence of a standard model-like Higgs.) I think the issues about “science-wise error rates” and long-run performance in dichotomous, diagnostic screening should be taken up separately, but commentators can continue on this, if they wish (perhaps see this related post). Continue reading

## “P-values overstate the evidence against the null”: legit or fallacious? (revised)

** 0. July 20, 2014: **Some of the comments to this post reveal that using the word “fallacy” in my original title might have encouraged running together the current issue with the fallacy of transposing the conditional. Please see a newly added Section 7.

**1. What you should ask…**

Discussions of P-values in the Higgs discovery invariably recapitulate many of the familiar criticisms of P-values (some “howlers”, some not). When you hear the familiar refrain, “We all know that P-values overstate the evidence against the null hypothesis”, denying the P-value aptly measures evidence, what you should ask is:

“What do you mean by overstating the evidence against a hypothesis?”

## A. Spanos: “Recurring controversies about P values and conﬁdence intervals revisited”

**Aris Spanos**

Wilson E. Schmidt Professor of Economics

*Department of Economics, Virginia Tech*

**Recurring controversies about P values and conﬁdence intervals revisited*
**

*Ecological Society of America (ESA) ECOLOGY*

Forum—P Values and Model Selection (pp. 609-654)

Volume 95, Issue 3 (March 2014): pp. 645-651

*INTRODUCTION*

The use, abuse, interpretations and reinterpretations of the notion of a *P* value has been a hot topic of controversy since the 1950s in statistics and several applied ﬁelds, including psychology, sociology, ecology, medicine, and economics.

The initial controversy between Fisher’s signiﬁcance testing and the Neyman and Pearson (N-P; 1933) hypothesis testing concerned the extent to which the pre-data Type I error probability α can address the arbitrariness and potential abuse of Fisher’s *post-data threshold *for the *P *value. Continue reading

## Anything Tests Can do, CIs do Better; CIs Do Anything Better than Tests?* (reforming the reformers cont.)

Having reblogged the 5/17/12 post on “reforming the reformers” yesterday, I thought I should reblog its follow-up: 6/2/12.

Consider again our one-sided Normal test T+, with null *H*_{0}: μ < μ_{0} vs μ >μ_{0} and μ_{0} = 0, α=.025, and σ = 1, but let n = 25. So M is statistically significant only if it exceeds .392. Suppose M (the sample mean) just misses significance, say

Mo = .39.

The flip side of a *fallacy of rejection* (discussed before) is a *fallacy of acceptance*, or the fallacy of misinterpreting statistically insignificant results. To avoid the age-old fallacy of taking a statistically insignificant result as evidence of zero (0) discrepancy from the null hypothesis μ =μ_{0}, we wish to identify discrepancies that can and cannot be ruled out. For our test T+, we reason from insignificant results to inferential claims of the form:

μ < μ_{0} + γ

Fisher continually emphasized that failure to reject was not evidence for the null. Neyman, we saw, in chastising Carnap, argued for the following kind of power analysis:

** Neymanian Power Analysis (Detectable Discrepancy Size DDS)**: If data

**are not statistically significantly different from**

*x**H*

_{0}, and the power to detect discrepancy γ is high (low), then

**constitutes good (poor) evidence that the actual effect is < γ. (See 11/9/11 post).**

*x*By taking into account the actual **x**_{0}, a more nuanced post-data reasoning may be obtained.

“In the Neyman-Pearson theory, sensitivity is assessed by means of the power—the probability of reaching a preset level of significance under the assumption that various alternative hypotheses are true. In the approach described here, sensitivity is assessed by means of the distribution of the random variable P, considered under the assumption of various alternatives. “ (Cox and Mayo 2010, p. 291):

This may be captured in :

FEV(ii): A moderate p-value is evidence of the absence of a discrepancy

dfrom Ho only if there is a high probability the test would have given a worse fit with H0 (i.e., a smaller p value) were a discrepancydto exist. (Mayo and Cox 2005, 2010, 256).

This is equivalently captured in the Rule of Acceptance (Mayo (EGEK) 1996, and in the severity interpretation for acceptance, SIA, Mayo and Spanos (2006, p. 337):

SIA: (a): If there is a very high probability that [the observed difference] would have been larger than it is, were μ > μ1, then μ < μ1 passes the test with high severity,…

But even taking tests and CIs just as we find them, we see that CIs do not avoid the fallacy of acceptance: they do not block erroneous construals of negative results adequately. Continue reading