## Books & intersection-union [General Sta­tis­tics]

Hi Victor,

❝ Honestly, I feel like a caveman […]

C’mon! Your skills in maths are impressive. If you want to dive deeper into the matter:
1. Chow SC, Liu JP. Design and Analysis of Bioavailability and Bioequivalence Studies. Boca Raton: CRC Press; 3rd ed. 2009.
2. Patterson SD, Jones B. Bioequivalence and Statistics in Clinical Pharmacology. Boca Raton: CRC Press; 2nd ed. 2016.
3. Hauschke D, Steinijans VW, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: John Wiley; 2007.
4. Jones B, Kenward MG. Design and Analysis of Cross-Over Trials. Boca Raton: CRC Press; 3rd ed. 2015.
5. Julious SA. Sample Sizes for Clinical Trials. Boca Raton: CRC Press; 2010.
6. Senn S. Cross-over Trials in Clinical Research. Chichester: John Wiley; 2nd ed. 2002.
7. Wellek S. Testing statistical hypotheses of equivalence. Boca Raton: CRC Press; 2003.
The last one is demanding but contains a proof of the intersection-union principle in Chapter 7 (Multisample tests for equivalence). Excerpt (he uses $$H$$ for the Null and $$K$$ for the alternative hypothesis, respectively):

The proof of the result is almost trivial, at least if one is willing to adopt some piece of the basic formalism customary in expositions of the abstract theory of statistical hypothesis testing methods. […] The condition we have to verify, reads […] as follows:
$$E_{(\eta_1,\ldots,\eta_q)}(\phi)\leq\alpha\;\textrm{for all}\;(\eta_1,\ldots,\eta_q)\in H\tag{7.3}$$ where $$E_{(\eta_1,\ldots,\eta_q)}(\cdot)$$ denotes the expected value computed under the parameter constellation $$(\eta_1,\ldots,\eta_q)$$. […]
In order to apply the result to multisample equivalence testing problems, let $$\theta_j$$ be the parameter of interest (e.g., the expected value) for the ith distribution under comparison, and require of a pair $$(i,j)$$ of distributions equivalent to each other that the statement $$K_{(i,j)}:\,\rho(\theta_i,\theta_j)<\epsilon,\tag{7.4}$$ holds true with $$\rho(\cdot,\cdot)$$ denoting a suitable measure of distance between parameters. Suppose furthermore that for each $$(i,j)$$ a test $$\phi_{(i,j)}$$ of $$H_{(i,j)}:\,\rho(\theta_i,\theta_j)\geq \epsilon$$ versus $$K_{(i,j)}:\,\rho(\theta_i,\theta_j)< \epsilon$$ is available whose rejection probability is $$\leq \alpha$$ at any point $$(\theta_1,\ldots,\theta_k)$$ in the full parameter space such that $$\rho(\theta_i,\theta_j)\geq \epsilon$$. Then, by the intersection-union principle, deciding in favour of “global quivalence” if and only if equivalence can be established for all $$(_{2}^{k})$$ possible pairs, yields a valid level-$$\alpha$$ test for $$H:\,\underset{i<j}{\max}\{\rho(\theta_i,\theta_j)\}\geq \epsilon\;\textrm{vs.}\;K:\,\underset{i<j}{\max}\{\rho(\theta_i,\theta_j)\}<\epsilon\tag{7.5}$$

❝ I thought I'd end with a nice little picture for memory :p

Nice picture! Do you know Anscombe’s quartet?

Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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