## Books & intersection-union [General Statistics]

» Honestly, I feel like a caveman […]

C’mon! Your skills in maths are impressive. If you want to dive deeper into the matter:

- Chow SC, Liu JP.
*Design and Analysis of Bioavailability and Bioequivalence Studies.*Boca Raton: CRC Press; 3^{rd}ed. 2009.

- Patterson SD, Jones B.
*Bioequivalence and Statistics in Clinical Pharmacology.*Boca Raton: CRC Press; 2^{nd}ed. 2016.

- Hauschke D, Steinijans VW, Pigeot I.
*Bioequivalence Studies in Drug Development.*Chichester: John Wiley; 2007.

- Jones B, Kenward MG.
*Design and Analysis of Cross-Over Trials.*Boca Raton: CRC Press; 3^{rd}ed. 2015.

- Julious SA.
*Sample Sizes for Clinical Trials.*Boca Raton: CRC Press; 2010.

- Senn S.
*Cross-over Trials in Clinical Research.*Chichester: John Wiley; 2^{nd}ed. 2002.

- Wellek S.
*Testing statistical hypotheses of equivalence.*Boca Raton: CRC Press; 2003.

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 *i*th 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?

Cheers,

Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

Science Quotes

### Complete thread:

- What is the largest α (Alpha) & β (Beta) allowed by FDA? victor 2019-11-16 21:57
- What do you want to achieve? Helmut 2019-11-17 01:26
- I'm seeking to understand the math behind our current regulation victor 2019-11-17 10:53
- Some answers Helmut 2019-11-17 14:35
- Wow! Amazing answers! victor 2019-11-18 08:26
- More answers Helmut 2019-11-18 15:09
- Wow! More amazing answers! victor 2019-11-18 20:16
- Books & intersection-unionHelmut 2019-11-19 12:01
- My progress on IUT so far victor 2019-11-22 01:28
- Update: Counterexamples victor 2019-11-23 09:05

- My progress on IUT so far victor 2019-11-22 01:28

- Books & intersection-unionHelmut 2019-11-19 12:01

- Wow! More amazing answers! victor 2019-11-18 20:16

- More answers Helmut 2019-11-18 15:09

- Wow! Amazing answers! victor 2019-11-18 08:26

- Some answers Helmut 2019-11-17 14:35

- I'm seeking to understand the math behind our current regulation victor 2019-11-17 10:53

- What do you want to achieve? Helmut 2019-11-17 01:26