This post may be seen to continue the discussion in May 17 post on Reforming the Reformers.
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 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 no greater than 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):