9 Equivalence Trials

Equivalence trials: aim to show the therapeutic equivalence of two treatments, typically an experimental treatment to a standard.

Absolute equivalence can never be demonstrated so trialists aim to demonstrate similarity based upon limits of equivalence (a clinically irrelevant difference).

Conventional significance test have little relevance: failure to detect a difference does not imply equivalence.

Comparison with superiority trial

• •

  “swap the null hypothesis and the alternative”

• •

  “relax former null hypothesis to an interval”

Important to specify boundaries (limits of equivalence) in advance!

Why do you think this is?

The concept of similarity may be one sided or two-sided: two-sided $\to$ ’equivalence’; one-sided $\to$‘non-inferiority’.

Instead of non-inferiority, some usage of the terms

• •

  one-sided equivalence

• •

  therapeutic equivalence.

Bioequivalence trials

typically aim to show the pharmacokinetic profile of an experimental treatment is similar to a reference product.

Pharmacokinetics

study of absorption, distribution and elimination of a pharmaceutical in the body as a function of dose and time.

Often bioequivalence based upon a single-dose cross-over on healthy volunteers.

Bioequivalence studies aim to indirectly demonstrate therapeutic equivalence!

Analysis is based upon the construction of a $(1-\alpha )\%$ confidence interval for the parameter of interest.

Suppose interest lies in parameter $\theta$ (for example $\theta ={\mu }_T-{\mu }_C$).

Hypotheses:

Where ${\theta }_1$ and ${\theta }_2$ are the pre-specified limits of equivalence

Compute confidence interval for $\theta$

• •

  lower confidence bound for $\theta$ at level $1-\alpha$, say ${\theta }_L(x,1-\alpha )$; upper confidence bound for $\theta$ at level $1-\alpha$, say ${\theta }_U(x,1-\alpha )$

• •

  reject $H_{\mathrm{0}}$, if $\mathrm{(}{\theta }_L\mathbf{}\mathrm{(}x\mathrm{,}\mathrm{1}\mathrm{-}\alpha \mathrm{)}\mathrm{,}{\theta }_U\mathbf{}\mathrm{(}x\mathrm{,}\mathrm{1}\mathrm{-}\alpha \mathrm{)}\mathrm{)}\mathrm{\subseteq }\mathrm{(}{\theta }_{\mathrm{1}}\mathrm{,}{\theta }_{\mathrm{2}}\mathrm{)}$
  

• •

  Example asthma trial later.

Unnumbered Figure: Link

CI $({\theta }_L(x,1-\alpha ),{\theta }_U(x,1-\alpha ))$ has level $1-2\alpha$
However, actual significance level of test procedure $\le \alpha$
Procedure equivalent to two one-sided tests

Equivalence Test: Two one-sided test to compare the difference of two Means.

Data

(independent observations!)

• •

  $X_{iT}\sim N({\mu }_T,{\sigma }^2),i=1,\mathrm{\dots },r\cdot n$

• •

  $X_{iC}\sim N({\mu }_C,{\sigma }^2),i=1,\mathrm{\dots },(1-r)\cdot n$

Hypotheses

• •

  $H_0:{\mu }_T-{\mu }_C\le {\theta }_1\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}{\mu }_T-{\mu }_C\ge {\theta }_2$  vs.
   

• •

  for example ${\theta }_1=-D$ and ${\theta }_2=D$ with $D\ge 0$

Test statistic

$$T_i=\sqrt{r(1-r)n}\frac{{\overline{X}}_T-{\overline{X}}_C-{\theta }_i}{S},i=1,2$$

with

$$S^2=\frac{1}{n-2}\sum {(X_{ij}-{\overline{X}}_i)}^2$$

Rejection (2 one-sided tests)

$$T_1\ge t_{n-2,1-\alpha }\text{𝐚𝐧𝐝}{T}_2\le t_{n-2,\alpha }$$

Confidence interval approach preferred!:

$${\overline{X}}_T-{\overline{X}}_C\pm t_{n-2,1-2\alpha }.SE({\overline{X}}_T-{\overline{X}}_C)$$

A bioequivalence trial using a cross-over design.

References:

• •

  Buehrens K et al (1991) Arzneimittel-Forschung / Drug Research 41: 250-253.

• •

  Bock J (1998), Section 5.2.

Objective: bioequivalence of two “Allopurinol” compounds (“Cellidrin” (T) and standard drug (R))

Endpoint

• •

  area under the serum concentration curve (AUC) [$\mu$g/ml h]

• •

  (approximate) log normal distribution.

Design

• •

  $2\times 2$ crossover trial

• •

  wash-out phase of 14 days.

Log-transform the AUC’s to achieve approximate normality $\to$ working upon the log-ratio scale.

Standard equivalence margins for bioequivalence: 0.80 and 1.25 for the ratio of the population means.

Results

• •

  $n=12$ healthy males

• •

  mean of the difference of the logarithms: $\overline{d}=-0.0446$, $s_d=0.1719$

• •

  90% CI for the mean of the difference [-0.1337; 0.0445]

• •

  90% CI for the ratio on the original scale [0.875; 1.046] $\Rightarrow$ equivalence.

Assumptions: no period effect, no carry-over.
Comments?

Sometimes interest lies in the ratio of two group means.

Hypotheses

• •

  $H_0:{\mu }_T/{\mu }_C\le {\theta }_1\text{or}{\mu }_T/{\mu }_C\ge {\theta }_2$  vs.  

• •

  for example ${\theta }_1=B$ and ${\theta }_2=1/B$ with .

Sasabuchi test statistic

$$T_i=\frac{{\overline{X}}_T-{\theta }_i{\overline{X}}_C}{S\sqrt{\frac{1-r+r{\theta }_i^2}{r(1-r)n}}},i=1,2.$$

Rejection (2 one-sided tests)

$$T_1\ge t_{n-2,1-\alpha }\mathrm{𝐚𝐧𝐝}{T}_2\le t_{n-2,\alpha }.$$

A $(1-\alpha )\%$ confidence interval for ratio of group means is preferable.

Ratio

of two means (${\mu }_T\ne {\mu }_C$)

$$n\approx \frac{1+(B^2-1)r}{r(1-r)}\frac{{({\mathrm{\Phi }}^{-1}(\alpha )+{\mathrm{\Phi }}^{-1}(\beta ))}^2}{{({\mu }_T/{\mu }_C-B)}^2}\frac{{\sigma }^2}{{\mu }_c^2}.$$

Difference

of two means (${\mu }_T\ne {\mu }_C$)

$$n\approx \frac{1}{r(1-r)}\frac{{({\mathrm{\Phi }}^{-1}(\alpha )+{\mathrm{\Phi }}^{-1}(\beta ))}^2}{{({\mu }_T-{\mu }_C-D)}^2}{\sigma }^2.$$

For ${\mu }_T={\mu }_C$: $\beta :=\beta /2.$

Reference: Kieser M, Hauschke D (1999). Journal of Biopharmaceutical Statistics 9: 641–650.
For exact sample sizes see Hauschke D, Kieser M, Diletti E, Burke M (1999). Statistics in Medicine 18: 93–105.

References

• •

  CPMP. Replacement of Chlorofluorocarbons (CFCs) in metered dose inhalation products. Note for Guidance III/5378/93; 1993.

• •

  Kunkel G, Schlaefke S, Koehler S. European Respiratory Journal 1999;14 (Suppl 30):105s (Abstract).

Background

• •

  Montreal protocol on substances that deplete the ozon layer

• •

  suitable alternatives to CFCs in metered-dose inhalers

• •

  to prove equivalence:

  • –

    pharmaceutical equivalence: e.g. amount of active ingredient delivered and deposition of the emitted dose

  • –

    clinical equivalence: …to demonstrate that the change …has no adverse effect on the benefit/risk ratio to the patients in comparison with the existing CFC-containing products.

Design

controlled, randomised, double-blind trial in patients with asthma.

Objective

clinical equivalence of two inhalation devices for beclomethasone

• •

  experimental device: new dry powder inhaler which is CFC-free

• •

  control device: std metered dose inhaler using CFC-containing propellants.

Endpoint

• •

  forced expiratory volume in one second (FEV${}_1$)

• •

  measured after 4, 8, and 12 weeks of treatment

• •

  primary endpoint: mean of these three measurements

• •

  ratio of means.

Equivalence margins

$(0.85;1.18)$.

Results

90% confidence interval of $(0.94;1.11)$ $\Rightarrow$ equivalence.

Sample size calculation

• •

  $\frac{\sigma }{{\mu }_C}=0.28$, $\alpha =0.05$, $1-\beta =0.80$, and $\frac{{\mu }_T}{{\mu }_C}$ within the interval $(0.96;1.05)$

• •

  ${\mu }_T/{\mu }_C=1$: $B=0.85\to 52$ ($B=1.18\to 50$)

• •

  ${\mu }_T/{\mu }_C=0.96$: $B=0.85\to 70$ ($B=1.18\to 24$)

• •

  ${\mu }_T/{\mu }_C=1.05$: ($B=0.85\to 21$) $B=1.18\to 69$

$\Rightarrow$ 70 patients per group.

Note: ${\mu }_T/{\mu }_C=0.95$ requires 84 patients!
(using exact formula: 85 patients).

Exact sample sizes with nQuery.