CI inclusion ≠ TOST [BE/BA News]

posted by Helmut Homepage – Vienna, Austria, 2020-12-04 21:30 (722 d 13:26 ago) – Posting: # 22115
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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.

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