Books & intersection-union [General Sta­tis­tics]

posted by Helmut Homepage – Vienna, Austria, 2019-11-19 11:01  – Posting: # 20827
Views: 1,246

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 $$H:\,\underset{i<j}{\max}\{\rho(\theta_i,\theta_j)\}\geq \epsilon\;\textrm{vs.}\;K:\,\underset{i<j}{\max}\{\rho(\theta_i,\theta_j)\}<\epsilon\tag{7.5}$$

:-D

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

Nice picture! Do you know [image] Anscombe’s quartet?

Cheers,
Helmut Schütz
[image]

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

Complete thread:

Activity
 Admin contact
20,255 posts in 4,263 threads, 1,398 registered users;
online 12 (0 registered, 12 guests [including 8 identified bots]).
Forum time (Europe/Vienna): 10:43 UTC

You should treat as many patients as possible with the new drugs
while they still have the power to heal.    Armand Trousseau

The Bioequivalence and Bioavailability Forum is hosted by
BEBAC Ing. Helmut Schütz
HTML5