CI inclusion ≠ TOST [Regulatives / Guidelines]

posted by Helmut Homepage – Vienna, Austria, 2020-12-04 20:30 (137 d 16:04 ago) – Posting: # 22115
Views: 1,328

Hi ElMaestro,

» does anyone know what this sentence means:

Yes. :-D

» "The calculation of the 90% confidence interval (CI) of the mean test/comparator ratio for the primary PK parameters‚…

$$\small{H_0:\frac{\mu_\textrm{T}}{\mu_\textrm{R}}\notin \left [ \theta_1, \theta_2 \right]\:vs\:H_1:\theta_1<\frac{\mu_\textrm{T}}{\mu_\textrm{R}}<\theta_2}\tag{1}$$

» … should not be confused with the two one-sided t-tests employed to reject the null hypothesis of non-equivalence…

$$\small{H_\textrm{0L}:\frac{\mu_\textrm{T}}{\mu_\textrm{R}} \leq \theta_1\:vs\:H_\textrm{1L}:\frac{\mu_\textrm{T}}{\mu_\textrm{R}}>\theta_1}\tag{2a}$$ $$\small{H_\textrm{0U}:\frac{\mu_\textrm{T}}{\mu_\textrm{R}} \geq \theta_2\:vs\:H_\textrm{1U}:\frac{\mu_\textrm{T}}{\mu_\textrm{R}}<\theta_2}\tag{2b}$$
» … The end result is the same, but these are not the same calculations." :confused:

Exactly! :thumb up:

For decades global guidelines ask for for the confidence interval inclusion approach \(\small{(1)}\). In Schuirmann’s famous TOST procedure \(\small{(2)}\) one gets two p-values; one for \(\small{(2\textrm{a})}\) and another one for \(\small{(2\textrm{b})}\). Nothing else. BE is concluded if both Nulls are rejected.
I think that statisticians of the WHO are fed up reading in \(\frac{\mathfrak{protocols}}{\mathfrak{reports}}\) …

Bioequivalence \(\frac{\textrm{will be}}{\textrm{has been}}\) assessed by the Two-One-Sided Tests procedure (Schuirmann 1987).

… only to find the 90% CI in the report.
BTW, only once I have seen TOST performed (in 1991). Lead to a deficiency letter: “The applicant should provide the 90% CI.”

Chow and Liu1 erred when stating

The two one-sided t tests procedure is operationally equivalent* to the classic (shortest) confidence interval approach; that is, if the classic (1–α)×100% confidence interval for µTµR is within (θL, θU), then both H01 and H02 are also rejected at the α level by the two one-sided t tests procedure.

I would not go that far like Brown et al.2 stating that

This similarity [between level α TOSTs and a 1–2α CI] is somewhat of a fiction, based more on an algebraic coincidence rather than a statistical equivalence.
[my insert]

More details given by Berger and Hsu.3 Already in the abstract:

The misconception that size-α bioequivalence tests generally correspond to 100(1–2α)% confidence sets […] lead[s] to incorrect statistical practices, and should be abandoned.


When reviewing stuff I insist in deleting the – all too common – TOST-statement as well (i.e., claiming \(\small{(2)}\) whilst performing \(\small{(1)}\)).



  1. Chow S-C, Liu J-p. Design and Analysis of Bioavailability and Bioequivalence Studies. Boca Raton: Chapman & Hall / CRC Press; 3rd ed. 2009. p. 98.
  2. Brown LD., Casella G, Hwang JTG. Optimal Confidence Sets, Bioequivalence, and the Limaçon of Pascal. J Am Stat Assoc. 1995;90(431):880–9. doi:10.2307/2291322. [image] Open access.
  3. Berger RL, Hsu JC. Bioequivalence Trials, Intersection–Union Tests and Equivalence Confidence Sets. Stat Sci. 1996;11(4):283–319. doi:10.1214/ss/1032280304. [image] Open access.

Dif-tor heh smusma 🖖
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
21,419 posts in 4,475 threads, 1,510 registered users;
online 10 (0 registered, 10 guests [including 4 identified bots]).
Forum time: Wednesday 13:34 CEST (Europe/Vienna)

In the Middles Ages the lingua franca of science was Latin.
Nowadays the language of science is bad English.    Anonymous

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