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 H0 is the probability T will lead to rejecting H0 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
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.
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
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.)
Consider again our one-sided Normal test T+, with null H0: μ < μ0 vs μ >μ0 and μ0 = 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 =μ0, 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 < μ0 + 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 x are not statistically significantly different from H0, and the power to detect discrepancy γ is high (low), then x constitutes good (poor) evidence that the actual effect is < γ. (See 11/9/11 post).
By taking into account the actual x0, 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 d from 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 discrepancy d to 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