Abstract
The notion ofhypothesis is fundamental to science. Typically it refers to an idea that might plausibly be true, and that is to be examined or “tested” with some experimental data. Sometimes, the expectation is that the data will conform to the hypothesis. In other situations, the hypothesis is introduced with the goal of refuting it.
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Notes
- 1.
This order of presentation is the one followed by Fisher in his immensely influential Statistical Methods for Research Workers, but it seems to have been abandoned later in the twentieth century as the Neyman-Pearson approach became dominant.
- 2.
Our characterization of \(p<.05\) as “modest evidence” is consistent with Fisher’s view. In particular, he felt \(p=.05\) was inconclusive. See the footnote on p. 298.
- 3.
In this example we use the notation \(p\) in two different ways: at first \(p\) stands for the probability that P.S. would choose the non-burning house, and then later it stands for the \(p\)-value. These are both such common notations that we felt we couldn’t change either of them. We hope our double use of \(p\) is not confusing.
- 4.
The logic of the procedure does not demand that we use \(\theta _0\) in place of \(\hat{\theta }\). The justification of the large-sample significance test, the Theorem in Section 7.3.5 that says \(Z\) is approximately \(N(0,1)\), is not refined enough to distinguish between the two alternative choices for \(\textit{SE}(T_n)\) (both would satisfy the theorem). However, because we are doing the calculation under the assumption that \(\theta =\theta _0\), it makes some sense to use the value \(\theta =\theta _0\) in computing the standard error.
- 5.
We discuss this distinction again in Section 13.1.
- 6.
Welch provided an approximate distribution from which \(p\)-values could be computed, which is more accurate than the normal.
- 7.
This may be considered an abuse of the notation because we usually consider \(H_0\) to be a fixed, non-random entity, so we are not really “conditioning” on it in the usual sense developed in Chapter 3. The exception occurs under the Bayesian interpretation given in Section 10.4.5, where \(H_0\) is formally considered to be an event. In that scenario the probability in (10.24) does become a conditional probability.
- 8.
This generalizes (and follows from) Eq. (A.29), which says that the maximal length of the sum of two unit vectors is 2 and it occurs when the vectors are equal.
- 9.
See pages 114 and 128 of the fourteenth (1970) edition of Fisher (1925).
- 10.
- 11.
On the other hand, we should recall that the \(p\)-value we obtained for the data \(x=14\) was \(p=.0076\) based on \(\chi ^2_{obs}\) and the chi-squared distribution while the exact \(p\)-value was \(p=.0127\). The discrepancy between approximate and exact values is a bit larger; the approximation apparently gets worse as we move further out into the tails.
- 12.
Specifically, both groups followed the distribution specified by the empirical cdf based on the 60 data values. This is an example ofbootstrap sampling and will lead to abootstrap test discussed in Chapter 11.
- 13.
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Kass, R.E., Eden, U.T., Brown, E.N. (2014). Models, Hypotheses, and Statistical Significance. In: Analysis of Neural Data. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9602-1_10
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