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In clinical trials, drug effect is measured by a difference between subjects who are treated by experimental drug against placebo-treated subjects. In case of binary data, with observing YES/NO on each subject in certain period of time, it is the proportion of subjects who respond in treatment group minus the proportion of responders in placebo group (for example, 50% vs. 30%). However, a greater difference was proposed by Rihmer
*et* al. (2011) [1] with their supporting arguments, in that antidepressant response and placebo response had different mechanisms and there were equal chances for antidepressant responder to be responding to placebo and not responding to placebo at all. Therefore, the authors proposed 50% - 30% * 50% when the response rate in the treatment group and the placebo group are 50% and 30% respectively, resulting in higher drug-placebo difference than traditional understanding of 50% - 30%. In this article, we tried to explain why the authors misunderstood the drug-placebo concept for evaluating drug superiority, their misunderstanding of assumptions of traditional calculation, as well as their wrong reasoning on their proposed approach. All in all, we conclude the traditional approach of 50% - 30% is the right way of evaluating drug-placebo difference and the possible methods to control impact of placebo effect are briefly discussed at the end of this article.

In clinical trials, patients are not only taking a testing drug on rigorous schedules, but also under a specific healthcare environment. Routine checks, clinical visiting and lots of psychological interviews might create a misconception to patients and clinicians and result in placebo effect. Placebo effect blunts the ability to detect drug-placebo difference in a well-controlled trail, resulting in trial failures, longer time and more resource in developing promising drugs for unmet medical needs. To deduce this trial background effects, randomization is normally applied. Subjects are randomized into either placebo or treatment group with equal probability and baseline characteristics got balanced out. With the help of randomization, only post-randomization factors and drug-placebo difference can contribute to different effects between drug and placebo groups. However, if investigators and patients have known what is given and what is taken in the trial, psychological effects will impact clinical rating scales, self-evaluation scores, compliance and patient’s willingness of coordination with trial personnel. Hence blinding is essential to get rid of above impacts on evaluating drug-placebo effect. Double- blinding is a way to exclude some of those post-randomization factors. Use of placebo is to evaluate the background effect of trial procedure on patients. Placebo is sometimes better than not treated, which is seen in most psychiatry trials depending on different disease characteristics. Placebo effect is well-known in antidepressant trials. How placebo works, how placebo effect is different from drug effect, whether there are interactions between them or not, and how these issues get accounted in statistical comparison all become interesting to the academic community. And the newly proposed method on how to calculate drug-placebo difference was one particular effort to answer one aspect of these questions. What makes anti-depressant special is that general antidepressant clinical trials, especially in short-term trials, have relatively larger placebo effect than those of other drug-testing clinical trials. Section 2 describes complexity of placebo and antidepressant mechanisms in depressive patients. Section 3 evaluates drug-placebo difference under various interaction types between placebo and antidepressant responses. Section 4 explains all the misunderstanding of drug-placebo difference and logic errors in Rihmer et al. 2011 [

Most widely used antidepressants include two classes: SSRI (selective serotonin reuptake inhibitors) and serotonin norepinephrine reuptake inhibitors. Namely, these two classes work mostly on central serotonin and norepinephrine systems (Johnson et al., 1993 [

In this section, we would like to explore the appropriate statistical evaluation for drug-placebo difference under the circumstance of placebo response in antidepressant trials. To be more complete, let’s put aside all founding in Section 2 first and explore all the scenarios, because some of these scenarios trigger Rihmer and co-authors [

Put AD and PL response in 2X2 contingency table, then the difference between drug and placebo can be viewed marginally and jointly. Marginally means whenever we consider AD response rate, we only concentrate on AD response (response = YES and response = NO corresponding to AD = 1 and AD = 0 respectively) without considering PL mechanism. Similarly, whenever looking at PL response rate, we ignore how AD works. From

Assuming two difference systems mediate PL response and AD response separately, then these two systems could: (D) totally dependent; (IND) totally independent; and (Other) some dependence in between. For totally dependence, we can further divide them into 4 subcategories (

(D1): Dependence scenario 1. Since all PL responders are AD responders, Pr(AD = 1|PL = 1) = 1. Circled cell Pr(AD = 1 and PL = 1) = Pr(AD = 1|PL = 1) * Pr(PL = 1) = 1 * 0.3 = 0.3; and then drug-placebo difference = Pr(AD = 1 and PL = 0) − Pr(AD = 0 and PL = 1) = 0.2 − 0 = 0.2 = Pr(AD = 1) − Pr(PL = 1) = 0.5 − 0.3.

(D2): Dependence scenario 2. Since all PL responders are AD non-responders, Pr(AD = 0|PL = 1) = 1. Circled cell Pr(AD = 0 and PL = 1) = Pr(AD = 0|PL = 1) * Pr(PL = 1) = 1 * 0.3 = 0.3 and drug-placebo difference = Pr(AD = 1 and PL = 0) − Pr(AD = 0 and PL = 1) = 0.5 − 0.3 = Pr(AD = 1) − Pr(PL = 1) = 0.5 − 0.3.

(D3): Dependence scenario 3. Intuitively, this can’t exist because: if all AD responders are PL responders, PL responder rate will be greater or equal to AD responder rate, which contradicts our assumption of probability of AD equal to 1 being 0.5 and PL equal to 1 being 0.3 respectively. Had we have PL responder rate exceeded AD responder rate; this would be a wrong target drug to develop since its effect is numerically inferior to placebo. Mathematically, if we have all AD responders are PL responders, conditionally probability of Pr(PL = 1|AD = 1) = 1. Therefore, Pr(AD = 1 and PL = 1) = Pr(PL = 1|AD = 1) * Pr(AD = 1) = 1 * 0.5 > Pr(PL = 1) = 0.3. This violates probability axiom, as Pr(PL = 1) = Pr(AD = 1 and PL = 1) + Pr(AD = 0 and PL = 1) and should not be less than Pr(AD = 1 and PL = 1) alone. This calculation proves our intuitive interpretation: under the condition of all AD responders are PL responders, existing of AD non-responders being PL responders will lead to greater PL response rate than AD response rate, in which is against the goal of drug development.

(D4): Dependence scenario 4. Since all AD responders are PL non-responders, Pr(PL = 0|PL = 1) = 1. Circled cell Pr(PL = 0 and AD = 1) = Pr(PL = 0|AD = 1) * Pr(AD = 1) = 1 * 0.5 = 0.5 and drug-placebo difference = Pr(AD = 1 and PL = 0) − Pr(AD = 0 and PL = 0) = 0.5 − 0.3 = 0.2 = Pr(AD = 1) − Pr(PL = 1) = 0.5 − 0.3. Graphically, dependence scenario 2 equals dependence scenario 4. Let’s try to prove it mathematically.

Claim: D2 dependence structure is the same as D4 dependence structure.

Proof: D2 = >D4

Pr(AD = 1 and PL = 1) + Pr(AD = 1 and PL = 0) + Pr(AD = 0 and PL = 1) + Pr(AD = 0 and PL = 0) = 1

Because Pr(AD = 0|PL = 1) = 1, then Pr(PL = 0|AD = 1) * Pr(AD = 1) = 1 − 1 * Pr(PL = 1) − Pr(AD = 1 and PL = 1) − Pr(AD = 0 and PL = 0) = Pr(PL = 0) − Pr(AD = 1 and PL = 1) − Pr(AD = 0 and PL = 0)

= Pr(AD = 1 and PL = 0) − Pr(AD = 1 and PL = 1)

= Pr(AD = 1) * Pr(PL = 0|AD = 1) − Pr(AD = 1) * Pr(PL = 1|AD = 1)

After Canceling Pr(AD = 1) from both sides, we have Pr(PL = 0|AD = 1) = Pr(PL = 0|AD = 1) − Pr(PL = 1|AD = 1)

Together with Pr(PL = 0 and AD = 1) + Pr(PL = 1 and AD = 1) = Pr(AD = 1)

Pr(PL = 0|AD = 1) = 1 is for D4 structure. All AD responders are PL non-responders.

Similarly, we can show D4 => D2.

In summary, under all reasonable dependence scenarios (i.e., D1 - D4 excluding D3), 4 cell probabilities are fixed and drug-placebo difference using joint probabilities is available. However, as discussed in Section 2, this drug-placebo difference is always 0.5 − 0.3, the same as that of being obtained by marginal probabilities. The other reason to have detailed discussion about above mutually exclusive and exhaustive scenarios is for later discussion about the method proposed by Rihmer et al. (2011) [

If the mechanism of placebo response is independent of that of antidepressant response, placebo responders can randomly either to be AD responder or to be AD non-responder. Similarly, AD responders have an equal chance to either be PL responder or be PL non-responder. Being a placebo responder is independent of being an AD responder. Then, under this scenario, what about drug-placebo difference? In

If neither definite dependence nor independence presents, some other structures in between play a role for mechanisms of placebo and AD responding. As in the 2X2 contingency table (

In clinical trials, we measure response on each subject, and group them into treatment versus placebo to find a measure so that superiority of drug vs. placebo can be evaluated and tested. Each joint probability is actually unobservable in the trial except under wholly independence or dependence structures. It may be possible to use another trial to test independence assumption, but normally we can just reject or fail to reject independence hypothesis. Still, we can’t prove it is indeed independent. For dependence structure, even with an external trial specifically for evaluating dependence structure, it is really hard to prove which dependence structure it is. Also, from Section 2, the presence of AD non-responder and PL responders excludes the possibility of having dependence scenario 1, which is all PL responders are AD responders; similarly, the presence of AD responders and PL responders excludes dependence scenarios 2 and 4, which are all PL responders are AD non-responders and all AD responders are PL non-responders respectively.

From general discussion in Section 2 and each specific example in Section 3, we all show that drug-placebo difference can be evaluated by marginal probability difference.

After stating and proving the right way of evaluating drug-placebo difference, we now have to discuss why the proposed method by Rihmer et al. (2011) [

We think that Rihmer and co-authors [

After the discussion of the right way of understanding and evaluating drug-placebo difference and pointing out all the flaws in Rihmer and co-authors’ wrong proposal, it seems that we are going back to the original place to favor traditional method of Pr(AD = 1) − Pr(PL = 1). Then what should we do to avoid jeopardizing a trial because of placebo effect? And should we just let it go unchecked? Of course, the answer is no. This is actually a very interesting but complicated area and not intended to be covered in this article. Here, we can briefly point out some related perspectives. To avoid failure trial due to placebo effect, we can put more efforts on innovated design and manage it more appropriate in operation. The main challenge is to lower the optimistic expectation from both patient and clinician. Since higher placebo response was found in mild-moderate depression, excluding these patients in the trial should be considered. And more scientific scoring system, more self-scoring scale, help from biomarker markers, and/or central rating could be combined to narrow the possibility of overstated expectation. Mathematically, novel designs as sequential parallel designs are also available in the literature.