A two-way enriched clinical trial design: combining advantages of placebo lead-in and randomized withdrawal

Abstract

A new clinical trial design, designated the two-way enriched design (TED), is introduced, which augments the standard randomized placebo-controlled trial with second-stage enrichment designs in placebo non-responders and drug responders. The trial is run in two stages. In the first stage, patients are randomized between drug and placebo. In the second stage, placebo non-responders are re-randomized between drug and placebo and drug responders are re-randomized between drug and placebo. All first-stage data, and second-stage data from first-stage placebo non-responders and first-stage drug responders, are utilized in the efficacy analysis. The authors developed one, two and three degrees of freedom score tests for treatment effect in the TED and give formulae for asymptotic power and for sample size computations. The authors compute the optimal allocation ratio between drug and placebo in the first stage for the TED and compare the operating characteristics of the design to the standard parallel clinical trial, placebo lead-in and randomized withdrawal designs. Two motivating examples from different disease areas are presented to illustrate the possible design considerations.

Keywords

placebo lead-in placebo response placebo run-in randomized withdrawal sequential parallel comparison design (SPCD)

1 Introduction and motivating examples

The placebo-controlled trial is a gold standard in the clinical assessment of drugs intended for symptomatic improvement of diseases such as pain, anxiety, depression and sleep. However, in many of these disease areas, there is a belief that the standard trial has not adequately discriminated active compounds. One significant problem in many of these disease areas is a high rate of placebo response.^1,2 A common practice in such trials is to use a placebo lead-in design (also referred to as placebo run-in), where all patients receive placebo first, after which, patients who have not responded to placebo are randomized to placebo versus drug; patients who respond to placebo in this phase are usually excluded from further trial participation. The main goal of using a placebo lead-in is to reduce the placebo response rate, and thereby hopefully increase the treatment effect.

A second proposal for improving the standard parallel trial is to conduct a randomized withdrawal design (also referred to as randomized discontinuation).³ In the randomized withdrawal design, patients are treated with the drug of interest to select those patients who respond to the drug. These patients are subsequently randomized to either stay on drug or switch to placebo. The outcome variable is some measure of maintenance of response. The logic behind the design is that a patient who has shown symptomatic improvement to an active drug is more at risk to lose that benefit when switched to placebo as opposed to remaining on drug. Randomized withdrawal designs are often suggested when the disease is considered very heterogeneous and the patient population which can benefit from the drug is only a subset of the entire patient population.

Both the placebo lead-in design and the randomized withdrawal design are enrichment designs⁴ in that the population randomized in the trial is a subset of the population recruited into the trial. In the placebo lead-in design, the subset is the population of placebo non-responders (and hence, ideally, placebo responders are eliminated from the design). In the randomized withdrawal design, the subset is the subset of drug responders (and hence, ideally, drug non-responders are eliminated from the design). The goal in both designs is to create subsets of the recruited population which will have a larger drug effect than the overall population.

The benefit of the placebo lead-in is debatable. A meta-analysis of 101 studies in depression showed that the placebo lead-in did not lower the placebo response rate, nor increase the drug–placebo difference.⁵ The failure of the standard placebo lead-in may have been due to the short duration of the period or the fact that investigators and patients may not have been blinded to this design feature. Because the placebo lead-in has not produced noticeable improvement, a new design called the sequential parallel comparison design (SPCD)^6,7 proposed to assign patients to both drug and placebo in an initial stage, reassign placebo non-responders in a second stage and observe the outcome in each stage. The data utilized in the efficacy analysis in this design consist of all first-stage data, and second-stage data only from patients who fail to respond to placebo in the first stage. Patients who receive drug in the first stage often participate in Stage 2 to maintain blinding, however, they are not included in the efficacy analysis.⁷ We propose a design that is an extension of the basic SPCD that, in addition, randomizes drug responders between drug and placebo and utilizes information on this comparison in the overall drug–placebo comparison. The patients who respond to drug are examined to see whether the maintenance of response is different between those patients switched to placebo versus those who remain on drug. We refer to the new design the two-way enriched design (TED). The new TED is depicted in Figure 1. The premise for our TED design is that a drug which is significantly superior to placebo in achieving short-term efficacy will also be superior to placebo in the maintenance of efficacy in those patients who respond to drug. This premise, of course, needs to be examined on a case-by-case basis and may be influenced by considerations such as chronicity of the disease and time period associated with achieving response. We now give two motivating examples in which we believe our new design would be very useful.

Figure 1.

The two-way enriched design.

Motivating Example 1: An early-phase clinical development team at a pharmaceutical company is charged with assessing efficacy of a new drug for generalized anxiety disorder (GAD). The new drug candidate has a different mechanism of action than benzodiazepines (currently used in the treatment of GAD) and is hypothesized to have a similar efficacy profile to benzodiazepines but significantly less abuse potential. The team needs to design an early Phase 2 study with as small a total sample size as possible to determine if there is sufficient evidence of efficacy to continue development of the compound. Abuse potential is to be explored in separate studies. High placebo response is an issue in GAD¹. The GAD is a chronic disease and it is believed that if an effective drug is withdrawn from a patient, symptoms of the disease quickly return.

Motivating Example 2: Ulcerative colitis is a form of inflammatory bowel disease. The disease, which is characterized by inflammation of parts or even the whole colon, usually requires medical treatment to go into remission. In clinical trials investigating induction and maintenance regimen to induce remission in patients with active ulcerative colitis, considerable placebo rates of remission up to 40% were observed.⁸ This could be due to the fact that some patients suffer additionally of irritable bowel syndrome, which is a non-inflammatory condition of the intestinal tract. These patients have high bowel frequency, which responds well to placebo. Both single-stage induction of remission trials and randomized withdrawal trials⁹ are trial designs used for the evaluation of the clinical efficacy of novel drug regimen in ulcerative colitis. The proposed TED might be a more effective alternative to randomized withdrawal or parallel single-stage design.

We will return to these motivating examples later in Section 3. The remainder of the manuscript is organized in the following fashion. In Section 2, we develop a variety of score tests for the TED. The different score tests arise because of different possible assumptions one is willing to make about the underlying parameters. The asymptotic distribution of the score tests under the null and under local alternatives is derived. In Section 3, in addition to examining the motivating examples, we also examine the robustness of the various tests when the assumptions being made are not correct and compare various designs in terms of required sample size. In Section 4, we discuss testing of the separate null hypotheses underlying the TED.

2 Score tests for the TED

2.1 Description of the design

The trial is conducted in two stages. Patients are randomized to one of four sequences placebo–placebo, placebo–drug, drug–placebo and drug–drug. In the TED, first-stage placebo responders and first-stage drug non-responders are not included in the efficacy analysis in Stage 2 (although patients might be included in the trial for blinding purposes) because it is unlikely to observe a treatment effect in these patients. The time point of switching from placebo to drug and from drug to placebo does not need to be the same between patients initially treated with placebo compared to the patients initially randomized to drug. Another way to implement the design is to randomize patients between drug and placebo in the first stage and then re-randomize placebo non-responders to receive drug or placebo and re-randomize drug non-responders to receive drug or placebo. Placebo responders and drug non-responders can also be re-randomized to drug or placebo but their data is not included in primary efficacy analysis. The two approaches are equivalent if randomization is via independent Bernoulli random variables within each stage. Simple randomization to four groups in the beginning of the trial is logistically easier and we will assume this approach in the remainder of the article.

Let the total sample size in the trial be n with n₁ subjects in the placebo–placebo group, n₂ subjects in the placebo–drug group, n₃ subjects in the drug–placebo group and n₄ subjects in the drug–drug group; n₁ + n₂ + n₃ + n₄=n. Define b=(n₁ + n₂)/n, as the proportion assigned to placebo in the first stage. If n₁=n₂ and n₃=n₄, the allocation ratio to the four sequences is b/2:b/2:[(1 – b)/2]:[(1 – b)/2. The range for b is 0 ≤ b ≤ 1, with b=1 corresponding to the placebo lead-in design and b=0 corresponding to a randomized withdrawal design. Further notation is introduced in Table 1. Note that p₁=Pr (drug response in Stage 1) and q₁=Pr (placebo response in Stage 1) are marginal probabilities of response and p₂=Pr (drug response in Stage 2 | placebo non-responder in Stage 1), q₂=Pr (placebo response in Stage 2 | placebo non-responder in Stage 1), p₃=Pr (drug response in Stage 2 | drug responder in Stage 1) and q₃=Pr (placebo response in Stage 2 | drug responder in Stage 1) are conditional probabilities. Let s₂ be the probability that a first-stage placebo non-responder continues to the second stage, and s₃ be the probability that a first-stage drug responder continues to the second stage. We assume possibly different retention rates in placebo non-responders, s₂, and drug responders, s₃. Since subjects are independent, it follows that:

n_{11} + n_{12} ~ Bin (n_{1} - n_{13}, s_{2}), n_{21} + n_{22} ~ Bin (n_{2} - n_{23}, s_{2}), n_{32} + n_{33} ~ Bin (n_{3} - n_{31}, s_{3}), n_{42} + n_{43} ~ Bin (n_{4} - n_{41}, s_{3}) .

Table 1.

The two-way enriched design^a

Treatment		Response		Count	Probability
Period 1	Period 2	Period 1	Period 2	Count	Probability
Placebo	Placebo (n₁)	No	Yes	n ₁₁	s₂(1 – q₁)q₂
		No	No	n ₁₂	s₂(1 – q₁)(1 – q₂)
		Yes	•	n ₁₃	q ₁
		No	missing	n ₁₄	(1 – s₂)(1 – q₁)
Placebo	Drug(n₂)	No	Yes	n ₂₁	s₂(1 – q₁)p₂
		No	No	n ₂₂	s₂(1 – q₁)(1 – p₂)
		Yes	•	n ₂₃	q ₁
		No	missing	n ₂₄	(1 – s₂)(1 – q₁)
Drug	Placebo (n₃)	No	•	n ₃₁	1 – p₁
		Yes	Yes	n ₃₂	s ₃ p ₁ q ₃
		Yes	No	n ₃₃	s₃ p₁(1 – q₃)
		Yes	missing	n ₃₄	(1 – s₃)p₁
Drug	Drug (n₄)	No	•	n ₄₁	1 – p₁
		Yes	Yes	n ₄₂	s ₃ p ₁ p ₃
		Yes	No	n ₄₃	s₃ p₁(1 –p₃)
		Yes	missing	n ₄₄	(1 – s₃)p₁

Note. p₁: Pr (drug response in Stage 1), q₁: Pr (placebo response in Stage 1), p₂: Pr (drug response in Stage 2 | placebo non-responder in Stage 1), q₂=Pr (placebo response in Stage 2 | placebo non-responder in Stage 1), p₃=Pr (drug response in Stage 2 | drug responder in Stage 1), q₃=Pr (placebo response in Stage 2 | drug responder in Stage 1), s₂ is the proportion of placebo non-responders in Stage 1 who participate in Stage 2, s₃ is the proportion of drug responders in Stage 1 who participate in Stage 2.

Responses denoted ‘•’ are not included in the analysis by design, n₁₄ and n₂₄ are placebo non-responders and n₃₄ and n₄₄ are drug responders, who dropout and do not participate in Stage 2.

The dropout process is assumed random and independent of future outcomes. The joint likelihood for (p₁, q₁, p₂, q₂, p₃, q₃, s₂, s₃) is then:

L_{0} (p_{1}, q_{1}, p_{2}, q_{2}, p_{3}, q_{3}, s_{2}, s_{3}) \propto p_{1}^{n_{32}} (1 - p_{1}) n_{31} q_{1}^{n_{23} + n_{13}} (1 - q_{1}) n_{1} + n_{2} - n_{23} - n_{13} p_{2}^{n_{21}} (1 - p_{2}) n_{22} \times q_{2}^{n_{11}} (1 - q_{2}) n_{12} p_{3}^{n_{42}} (1 - p_{3}) n_{43} q_{3}^{n_{32}} (1 - q_{3}) n_{33} s_{2}^{n_{11} + n_{12} + n_{21} + n_{22}} (1 - s_{2}) n_{1} + n_{2} - n_{13} - n_{23} - n_{11} - n_{12} - n_{21} - n_{22} \times s_{3}^{n_{32} + n_{33} + n_{42} + n_{43}} (1 - s_{3}) n_{3} + n_{4} - n_{31} - n_{41} - n_{32} - n_{33} - n_{42} - n_{43} .

2.2 Score test with one degree of freedom

Suppose that investigators have confidence in their knowledge of treatment effects in the enriched portion of the design relative to the treatment effect in the overall population. Define the treatment effects Δ₁=p₁ – q₁, Δ₂=p₂ – q₂ and Δ₃=p₃ – q₃. Define ρ₂ and ρ₃ such that Δ₁ρ₂=Δ₂ and Δ₁ρ₃=Δ₃. The test uses two test parameters r₂ and r₃ that are known constants. We restrict r_i to be 0 ≤ r_i < +∞, i=2, 3. The test is derived under the assumption that r₂=ρ₂ and r₃=ρ₃. The parameters (p₁, q₁, p₂, q₂, p₃, q₃, s₂, s₃) are transformed to (Δ₁, q₁, q₂, q₃, s₂, s₃). Then p₁=Δ₁ + q₁, p₂=r₂Δ₁ + q₂ and p₃=r₃Δ₁ + q₃ and the re-parametrized likelihood is:

L_{0} (Δ_{1}, q_{1}, q_{2}, q_{3}, s_{2}, s_{3}) \propto (Δ_{1} + q_{1}) n_{32} (1 - Δ_{1} - q_{1}) n_{31} q_{1}^{n_{23} + n_{13}} (1 - q_{1}) n_{1} + n_{2} - n_{23} - n_{13} (r_{2} Δ_{1} + q_{2}) n_{21} (1 - r_{2} Δ_{1} - q_{2}) n_{22} \times q_{2}^{n_{11}} (1 - q_{2}) n_{12} (r_{3} Δ_{1} + q_{3}) n_{42} (1 - r_{3} Δ_{1} - q_{3}) n_{43} q_{3}^{n_{32}} (1 - q_{3}) n_{33} s_{2}^{n_{11} + n_{12} + n_{21} + n_{22}} (1 - s_{2}) n_{1} + n_{2} - n_{13} - n_{23} - n_{11} - n_{12} - n_{21} - n_{22} \times s_{3}^{n_{32} + n_{33} + n_{42} + n_{43}} (1 - s_{3}) n_{3} + n_{4} - n_{31} - n_{41} - n_{32} - n_{33} - n_{42} - n_{43} .

We are interested in testing H₀: Δ₁=0. Under H₀, Δ₁=p₁ – q₁=0, (p₂ – q₂)=r₂ (p₁ – q₁)=r₂Δ₁=0 and (p₃ – q₃)=r₃ (p₁ – q₁)=r₃Δ₁=0 and therefore H₀ implies that treatment effects are 0 in the Stage 1 population and the two Stage 2 sub-populations.

Maximum likelihood estimates under H₀ obtained by setting Δ₁=0 and solving the likelihood equations for q₁, q₂, q₃, s₂ and s₃ are:

{\tilde{q}}_{1} = \frac{n_{13} + n_{23} + n_{3} - n_{31} + n_{4} - n_{41}}{n}, {\tilde{q}}_{2} = \frac{n_{11} + n_{21}}{n_{11} + n_{12} + n_{21} + n_{22}}, {\tilde{q}}_{3} = \frac{n_{32} + n_{42}}{n_{32} + n_{33} + n_{42} + n_{43}}, {\tilde{s}}_{2} = \frac{n_{11} + n_{12} + n_{21} + n_{22}}{n_{1} + n_{2} - n_{13} - n_{23}}, {\tilde{s}}_{3} = \frac{n_{32} + n_{33} + n_{42} + n_{43}}{n_{3} + n_{4} - n_{31} - n_{41}} .

(1)

When deriving the test statistic, one can use expected or observed information to estimate the variance. Previous researchers^10,11 have used expected information. We obtained both tests: based on expected and observed information. The tests based on expected information (Appendix 1) have more compact formulae than their observed information counterparts (Appendix 2). However, asymptotic power formulae derived from tests based on observed information are usually more accurate. This may be due to the fact that the expected information is averaged over the random sample size, while the observed information essentially conditions on its observed value.¹²

The derivation of the score test statistic with observed information is given in Appendix 2. The test statistic is:

T_{1} = (\frac{n_{3} + n_{4} - n_{31} - n_{41}}{{\tilde{q}}_{1}} - \frac{n_{31} + n_{41}}{1 - {\tilde{q}}_{1}} + \frac{r_{2} n_{21}}{{\tilde{q}}_{2}} - \frac{r_{2} n_{22}}{1 - {\tilde{q}}_{2}} + \frac{r_{3} n_{42}}{{\tilde{q}}_{3}} - \frac{r_{3} n_{43}}{1 - {\tilde{q}}_{3}}) 2 / (v_{1} + r_{2}^{2} v_{2} + r_{3}^{2} v_{3}),

where

v_{1} = \frac{(\frac{n_{13} + n_{23}}{{\tilde{q}}_{1}^{2}} + \frac{n_{1} + n_{2} - n_{13} - n_{23}}{(1 - {\tilde{q}}_{1}) 2}) (\frac{n_{3} + n_{4} - n_{31} - n_{41}}{{\tilde{q}}_{1}^{2}} + \frac{n_{31} + n_{41}}{(1 - {\tilde{q}}_{1}) 2})}{\frac{n_{13} + n_{23}}{{\tilde{q}}_{1}^{2}} + \frac{n_{1} + n_{2} - n_{13} - n_{23}}{(1 - {\tilde{q}}_{1}) 2} + \frac{n_{3} + n_{4} - n_{31} - n_{41}}{{\tilde{q}}_{1}^{2}} + \frac{n_{31} + n_{41}}{(1 - {\tilde{q}}_{1}) 2}}, v_{2} = \frac{(\frac{n_{21}}{{\tilde{q}}_{2}^{2}} + \frac{n_{22}}{(1 - {\tilde{q}}_{2}) 2}) (\frac{n_{11}}{{\tilde{q}}_{2}^{2}} + \frac{n_{12}}{(1 - {\tilde{q}}_{2}) 2})}{\frac{n_{11} + n_{21}}{{\tilde{q}}_{2}^{2}} + \frac{n_{12} + n_{22}}{(1 - {\tilde{q}}_{2}) 2}}, v_{3} = \frac{(\frac{n_{42}}{{\tilde{q}}_{3}^{2}} + \frac{n_{43}}{(1 - {\tilde{q}}_{3}) 2}) (\frac{n_{32}}{{\tilde{q}}_{3}^{2}} + \frac{n_{33}}{(1 - {\tilde{q}}_{3}) 2})}{\frac{n_{32} + n_{42}}{{\tilde{q}}_{3}^{2}} + \frac{n_{33} + n_{43}}{(1 - {\tilde{q}}_{3}) 2}} .

(2)

The asymptotic distribution of T₁ under H₀ is chi-square with one degree of freedom.

If r₂=r₃=0, the test is equivalent to the score test that uses data from Stage 1 only. If r₃=0, data from the randomized discontinuation part are not used in the analyses and the test is the score test for the SPCD. If both r₂ and r₃ are large, T₁ is close to the score test statistic that uses data from Stage 2 only. If r₂ is rather large compared to r₃, the test statistics addresses comparison in placebo non-responders only; if r₃ is rather large compared to r₂, the test statistics addresses comparison in randomized withdrawal group only.

Under a local alternative with true parameters (p₁, q₁, p₂, q₂, p₃, q₃, s₂, s₃), the limiting distribution of the test statistic is non-central chi-square with df=1 and non-centrality parameter is nγ₁ where:

γ_{1} = [(1 - b) (\frac{p_{1}}{η_{1}} - \frac{1 - p_{1}}{1 - η_{1}}) + r_{2} 0.5 {bs}_{2} (1 - q_{1}) (\frac{p_{2}}{η_{2}} - \frac{1 - p_{2}}{1 - η_{2}}) + r_{3} (0.5 - 0.5 b) s_{3} p_{1} (\frac{p_{3}}{η_{3}} - \frac{1 - p_{3}}{1 - η_{3}})] 2 / (w_{1} + r_{2}^{2} w_{2} + r_{3}^{2} w_{3}),

where

η_{1} = (1 - b) p_{1} + {bq}_{1}, η_{2} = 0.5 p_{2} + 0.5 q_{2}, η_{3} = 0.5 p_{3} + 0.5 q_{3},

(3)

w_{1} = \frac{(\frac{{bq}_{1}}{η_{1}^{2}} + \frac{b (1 - q_{1})}{(1 - η_{1}) 2}) (\frac{(1 - b) p_{1}}{η_{1}^{2}} + \frac{(1 - b) (1 - p_{1})}{(1 - η_{1}) 2})}{\frac{{bq}_{1} + (1 - b) p_{1}}{η_{1}^{2}} + \frac{b (1 - q_{1}) + (1 - b) (1 - p_{1})}{(1 - η_{1}) 2}}, w_{2} = \frac{s_{2} 0.5 b (1 - q_{1}) (\frac{p_{2}}{η_{2}^{2}} + \frac{1 - p_{2}}{(1 - η_{2}) 2}) (\frac{q_{2}}{η_{2}^{2}} + \frac{1 - q_{2}}{(1 - η_{2}) 2})}{\frac{q_{2} + p_{2}}{η_{2}^{2}} + \frac{2 - q_{2} - p_{2}}{(1 - η_{2}) 2}}, w_{3} = \frac{s_{3} (0.5 - 0.5 b) p_{1} (\frac{p_{3}}{η_{3}^{2}} + \frac{1 - p_{3}}{(1 - η_{3}) 2}) (\frac{q_{3}}{η_{3}^{2}} + \frac{1 - q_{3}}{(1 - η_{3}) 2})}{\frac{q_{3} + p_{3}}{η_{3}^{2}} + \frac{2 - q_{3} - p_{3}}{(1 - η_{3}) 2}} .

(4)

Therefore, the sample size required to achieve a power of 1 – β at (p₁, q₁, p₂, q₂, s₂, s₃) with a two-sided level α test is:

n = \frac{(Z_{1 - β} + Z_{1 - α / 2}) 2}{γ_{1}},

where Z_x denotes the xth quantile of the standard normal distribution. The performance of the test when r₂ and r₃ are misspecified, r₂ ≠ ρ₂ and/or r₃ ≠ ρ₃, is addressed in Section 3.

2.3 Score test with two degrees of freedom

Suppose that the investigator is willing to make assumptions regarding the treatment effect in the enriched population of placebo non-responders but is unwilling to make a similar assumption regarding the treatment effect in the enriched population of drug responders. This corresponds mathematically to removing the constraint regarding relationship between treatment effects Δ₁ and Δ₃. Changing parameters from (p₁, q₁, p₂, q₂, p₃, q₃, s₂, s₃) to (Δ₁, Δ₃, q₁, q₂, q₃, s₂, s₃) by setting p₁=Δ₁ + q₁, p₂=r₂Δ₁ + q₂ and p₃=Δ₃ + q₃ the likelihood is:

L_{0} (Δ_{1}, Δ_{3}, q_{1}, q_{2}, q_{3}, s_{2}, s_{3}) \propto (Δ_{1} + q_{1}) n_{32} (1 - Δ_{1} - q_{1}) n_{31} q_{1}^{n_{23} + n_{13}} (1 - q_{1}) n_{1} + n_{2} - n_{23} - n_{13} (r_{2} Δ_{1} + q_{2}) n_{21} (1 - r_{2} Δ_{1} - q_{2}) n_{22} \times q_{2}^{n_{11}} (1 - q_{2}) n_{12} (Δ_{3} + q_{3}) n_{42} (1 - Δ_{3} - q_{3}) n_{43} q_{3}^{n_{32}} (1 - q_{3}) n_{33} s_{2}^{n_{11} + n_{12} + n_{21} + n_{22}} (1 - s_{2}) n_{1} + n_{2} - n_{13} - n_{23} - n_{11} - n_{12} - n_{21} - n_{22} \times s_{3}^{n_{32} + n_{33} + n_{42} + n_{43}} (1 - s_{3}) n_{3} + n_{4} - n_{31} - n_{41} - n_{32} - n_{33} - n_{42} - n_{43} .

We are interested in testing H₀: Δ₁=Δ₃=0. The maximum likelihood estimates of parameters under H₀ are as given in Equation (1). The observed information score test statistic is:

T_{2} = (\frac{n_{3} + n_{4} - n_{31} - n_{41}}{{\tilde{q}}_{1}} - \frac{n_{31} + n_{41}}{1 - {\tilde{q}}_{1}} + \frac{r_{2} n_{21}}{{\tilde{q}}_{2}} - \frac{r_{2} n_{22}}{1 - {\tilde{q}}_{2}}) 2 / (v_{1} + r_{2}^{2} v_{2}) + (\frac{n_{42}}{{\tilde{q}}_{3}} - \frac{n_{43}}{1 - {\tilde{q}}_{3}}) 2 / v_{3},

where v₁, v₂ and v₃ are defined in Equation (2).

The distribution of T₂ under H₀ is chi-square with two degrees of freedom (df). Under local alternatives, the limiting distribution of the test statistic is non-central chi-square with df=2 and non-centrality parameter is nγ₂ where:

γ_{2} = [(1 - b) (\frac{p_{1}}{η_{1}} - \frac{1 - p_{1}}{1 - η_{1}}) + r_{2} 0.5 {bs}_{2} (1 - q_{1}) (\frac{p_{2}}{η_{2}} - \frac{1 - p_{2}}{1 - η_{2}})] 2 / (w_{1} + r_{2}^{2} w_{2}) + [(0.5 - 0.5 b) s_{3} p_{1} (\frac{p_{3}}{η_{3}} - \frac{1 - p_{3}}{1 - η_{3}})] 2 / w_{3},

where η₁, η₂ and η₃ are defined in Equation (3) and w₁, w₂ and w₃ are defined in Equation (4).

Define g as the 1 – α quantile of a chi-square random variable with df=2, for example, if α=0.05, g=5.99. Let λ to be the non-centrality parameter such that $⪻ [χ_{2, λ}^{2} \leq g] = β$ , where $χ_{2, λ}^{2}$ denotes a random variable distributed as non-central chi-square with df=2 and non-centrality parameter λ. For example, when β=0.1, α=0.05: λ=12.65; λ=9.63 when β=0.2 and λ=7.70 when β=0.3. Then, the sample size required to achieve power 1 – β at (p₁, q₁, p₂, q₂, p₃, q₃, s₂, s₃) with a two-sided level α test is n=λ/γ₂.

A score test for the situation where the constraint is removed from Δ₁ and Δ₂ but the constraint between Δ₁ and Δ₃ remains can be constructed similarly.

2.4 The score test with three df

The last scenario we consider is the situation in which the investigator is unwilling to make any assumptions regarding the treatment effect in the enriched populations compared to the treatment effect in the overall population. In this case, no relationship is assumed between Δ₁, Δ₂ and Δ₃. The likelihood is:

L_{0} (Δ_{1}, Δ_{2}, Δ_{3}, q_{1}, q_{2}, q_{3}, s_{2}, s_{3}) \propto (Δ_{1} + q_{1}) n_{32} (1 - Δ_{1} - q_{1}) n_{31} q_{1}^{n_{23} + n_{13}} (1 - q_{1}) n_{1} + n_{2} - n_{23} - n_{13} (Δ_{2} + q_{2}) n_{21} (1 - Δ_{2} - q_{2}) n_{22} \times q_{2}^{n_{11}} (1 - q_{2}) n_{12} (Δ_{3} + q_{3}) n_{42} (1 - Δ_{3} - q_{3}) n_{43} q_{3}^{n_{32}} (1 - q_{3}) n_{33} s_{2}^{n_{11} + n_{12} + n_{21} + n_{22}} \times (1 - s_{2}) n_{1} + n_{2} - n_{13} - n_{23} - n_{11} - n_{12} - n_{21} - n_{22} s_{3}^{n_{32} + n_{33} + n_{42} + n_{43}} (1 - s_{3}) n_{3} + n_{4} - n_{31} - n_{41} - n_{32} - n_{33} - n_{42} - n_{43},

and H₀: Δ₁=Δ₂=Δ₃=0. The maximum likelihood estimates under H₀ are as given in Equation (1). The observed information score test statistic is:

T_{3} = (\frac{n_{3} + n_{4} - n_{31} - n_{41}}{{\tilde{q}}_{1}} - \frac{n_{31} + n_{41}}{1 - {\tilde{q}}_{1}}) 2 / v_{1} + (\frac{n_{21}}{{\tilde{q}}_{2}} - \frac{n_{22}}{1 - {\tilde{q}}_{2}}) 2 / v_{2} + (\frac{n_{42}}{{\tilde{q}}_{3}} - \frac{n_{43}}{1 - {\tilde{q}}_{3}}) 2 / v_{3} .

The distribution of T₃ under H₀ is chi-square with three df.

Under local alternatives, the limiting distribution of the test statistic is non-central chi-square with df=3 and non-centrality parameter is nγ₃ where:

γ_{3} = [(1 - b) (\frac{p_{1}}{η_{1}} - \frac{1 - p_{1}}{1 - η_{1}})] 2 / w_{1} + [0.5 {bs}_{2} (1 - q_{1}) (\frac{p_{2}}{η_{2}} - \frac{1 - p_{2}}{1 - η_{2}})] 2 / w_{2} + [(0.5 - 0.5 b) s_{3} p_{1} (\frac{p_{3}}{η_{3}} - \frac{1 - p_{3}}{1 - η_{3}})] 2 / w_{3},

where η₁, η₂ and η₃ are defined in Equation (3) and w₁, w₂ and w₃ are defined in Equation (4).

Let g be the 1 – α quantile of a chi-square random variable with df=3, for example, if α=0.05, g=7.81. Let λ to be the non-centrality parameter such that $⪻ [χ_{3, λ}^{2} \leq g] = β$ , where $χ_{3, λ}^{2}$ denotes a random variable distributed as non-central chi-square with df=3 and non-centrality parameter is λ. For α=0.05: λ=14.17 when β=0.1; λ=10.90 when β=0.2; λ=8.79 when β=0.3. Then, the sample size required to achieve power 1 – β at (p₁, q₁, p₂, q₂, p₃, q₃, s₂, s₃) with a two-sided level α test is n=λ/γ₃.

3 Comparison of test statistics and designs

Several choices need to be made in the TED: the allocation proportion to placebo in the first stage, b, the test statistic and the test parameters r₁ and r₂, if the one df statistic is used, and the total sample size. For any given p₁, q₁, p₂, q₂, p₃, q₃, r₂, r₃, s₂ and s₃, the optimal r₂=(p₂ – q₂)/(p₁ – q₁), r₃=(p₃ – q₃)/(p₁ – q₁) and the optimal allocation proportion b that maximizes the power can be computed via grid search. If no information is available regarding true response rates, we recommend using r₂=r₃=1 and b=0.5. Table 2 displays total sample size required to achieve 80% power for various response scenarios for the optimal and recommended sets of design parameters for one, two and three df tests. The sample sizes reported in this and other tables are calculated as 2⌈0.5bn⌉ + 2⌈0.5(1 – b)n⌉, where n is the required sample size computed as described in Section 2 and ⌈x⌉ is the smallest integer greater than or equal to x. Several observations can be made from Table 2. As expected, the smallest sample size is for the one df test when parameters are selected in the optimal way. If r₂ is close to ρ₂ and r₃ is close to ρ₃, required sample size for one df is smaller than for the two and three df tests. However, if at least one r_i is far from ρ _i , i=2, 3, as in Scenario 7, where r₂=r₃=1 are far from ρ₂=ρ₃=+∞, the three df test yields smaller sample size than one and two df tests with recommended parameters.

Table 2.

Total sample size required to achieve 80% power when the two-way enriched design is used with one, two and three df tests with two-sided type I error rate of 0.05^a

p ₁	q ₁	p ₂	q ₂	p ₃	Q ₃	1 df		2 df		3 df
p ₁	q ₁	p ₂	q ₂	p ₃	Q ₃	(b, r₁, r₂)*	(b, r₁, r₂)	(b, r₁)*	(b, r₁)	b*	b
0.4	0.3	0.4	0.3	0.9	0.8	(0.48,1,1)*	(0.5,1,1)	(0.48,1)*	(0.5,1)	0.48*	0.5
						412	412	504	504	570	572
0.5	0.3	0.4	0.2	0.9	0.8	(0.54,1,0.5)*	(0.5,1,1)	(0.54,1)*	(0.50,1)	0.54*	0.5
						122	128	152	152	170	170
0.5	0.3	0.4	0.1	0.9	0.8	(0.72,1.5,0.5)*	(0.5,1,1)	(0.72,1.5)*	(0.5,1)	0.74*	0.5
						76	96	92	108	104	114
0.4	0.3	0.4	0.3	0.9	0.7	(0.26,1,2)*	(0.5,1,1)	(0.26,1)*	(0.5,1)	0.26*	0.5
						246	312	300	328	340	372
0.5	0.3	0.4	0.2	0.9	0.7	(0.48,1,1)*	(0.5,1,1)	(0.48,1)*	(0.5,1)	0.48*	0.5
						104	104	126	128	142	142
0.5	0.3	0.4	0.1	0.9	0.7	(0.64,1.5,1)*	(0.5,1,1)	(0.64,1.5)*	(0.5,1)	0.66*	0.5
						72	80	88	96	98	100
0.4	0.4	0.4	0.3	0.9	0.8	(0,100,100)*	(0.5,1,1)	(0,100)*	(0.5,1)	0.00	0.50
						926	2612	1136	1916	1292	1446
0.4	0.3	0.4	0.3	0.9	0.9	(0.58,1,0)*	(0.5,1,1)	(0.58,1)*	(0.5,1)	0.58*	0.5
						522	728	638	648	724	734
0.4	0.3	0.4	0.4	0.9	0.8	(0.40,0,1)*	(0.5,1,1)	(0.40,0)*	(0.5,1)	0.40*	0.5
						500	644	610	772	690	716

Note. df: degrees of freedom.

Values indicated by * are the optimal values for the underlying parameter set, other sets are recommended values, if no information regarding ρ₂ and ρ₃ is available. If optimal r₂ is infinite, r₂ is set to 100; similarly for r₃. Retention rates are assumed to be equal to 1.

The sequential parallel comparison design, the placebo lead-in, the randomized withdrawal and the parallel design with equal allocation can be considered to be special cases of the TED. Another special case which has not been published in the literature is the case in which only the randomized withdrawal enrichment is chosen to augment the initial stage of the TED. Choosing b, r₂ and r₃ in the one df test for the TED appropriately (Table 3) will yield a one df test for one of the above designs.

Table 3.

Relationship among the two-way enriched design and its special cases^a

	b	r ₂	r ₃
Two-way enriched design	[0, 1]	[0, +∞]	[0, +∞]
Sequential parallel comparison design	[0, 1]	[0, +∞]	0
Placebo run-in	1	[0, +∞]	0
Randomized withdrawal	0	0	[0, +∞]
Parallel design with equal allocation	0.5	0	0
Parallel design plus randomized withdrawal	[0, 1]	0	[0, +∞]

By setting test parameters r₂ and r₃ in the one degrees of freedom test for two-way enriched design to the values listed below corresponding one df tests for special case designs can be obtained.

Table 4 displays total sample size required to achieve 80% power for various designs. As the TED uses more data than other designs, it requires the smallest total sample size. Computations similar to those in Table 4 can be used as a tool to choose the best design for the study. We illustrate this by returning to our motivating examples.

Table 4.

Total sample size required to achieve 80% power with one df test with two-sided type I error rate of 0.05^a

p ₁	q ₁	p ₂	q ₂	p ₃	q ₃	TED		SPCD	Equal	Lead-in	RWithdr
p ₁	q ₁	p ₂	q ₂	p ₃	q ₃	(b, r₂, r₃)*	(b, r₂, r₃)	(b, r₂)*	Equal	Lead-in	RWithdr
0.4	0.3	0.4	0.3	0.9	0.8	(0.48,1,1)*	(0.5,1,1)	(0.56,1)*
						412	412	522	712	1016	926
0.5	0.3	0.4	0.2	0.9	0.8	(0.54,1,0.5)*	(0.5,1,1)	(0.58,1)*
						122	128	132	186	228	742
0.5	0.3	0.4	0.1	0.9	0.8	(0.72,1.5,0.5)*	(0.5,1,1)	(0.76,1.5)*
						76	96	80	186	80	742
0.4	0.3	0.4	0.3	0.9	0.7	(0.26,1,2)*	(0.5,1,1)	(0.56,1)*
						246	312	522	712	1016	270
0.5	0.3	0.4	0.2	0.9	0.7	(0.48,1,1)*	(0.5,1,1)	(0.58,1)*
						104	104	132	186	228	216
0.5	0.3	0.4	0.1	0.9	0.7	(0.64,1.5,1)*	(0.5,1,1)	(0.76,1.5)*
						72	80	80	186	80	252
0.4	0.4	0.4	0.3	0.9	0.8	(0,100,100)*	(0.5,1,1)	(1,100)*
						926	2612	1186	—	1186	926
0.4	0.3	0.4	0.3	0.9	0.9	(0.58,1,0)*	(0.5,1,1)	(0.56,1)*
						522	728	522	712	1016	—
0.4	0.3	0.4	0.4	0.9	0.8	(0.40,0,1)*	(0.5,1,1)	(0.48,0)*
						500	644	712	712	—	926

Note. df: degrees of freedom.

The designs are: the two-way enriched design (TED); the sequential parallel comparison design (SPCD); a single-stage design with equal allocation (Equal), that is, first-stage data only are used; the placebo lead-in design (Lead-in), that is, Stage 2 data from placebo non-responders only are used and randomized withdrawal (RWithdr), that is, Stage 2 data from drug responders are used. Values indicated by * are the optimal values for the underlying parameter set, other sets are recommended values for one df test, if no information regarding ρ₂ and ρ₃ is available. If optimal r₂ is infinite, r₂ is set to 100; similarly for r₃. Retention rates are assumed to be equal to 1.

Motivating Example 1: For GAD, it is feasible to run either a 4-week traditional design or a 6-week TED or SPCD design with each stage being 3 weeks. Investigation of the literature suggests that the placebo response rate in the overall population will be around 0.4 with Δ₁=0.2 yielding p₁=0.6 and q₁=0.4. For the placebo non-responders, p₂ is assumed to be 0.4 and q₂=0.2 and for the randomized withdrawal enriched population, p₃=0.9 and q₃=0.7. We feel that r₂=r₃=1 will be good estimates of the treatment effects in the enriched population and are comfortable using these values with a single degree of freedom test. The retention rates are assumed to be 1 for both placebo non-responders and drug responders. Under these scenarios, the sample sizes for TED with equal allocation to placebo and drug in Stage 1, for the SPCD with allocation ratio of 4:3 to placebo versus drug and for the traditional parallel trial are 104, 144 and 194, respectively, for α=0.05 and β=0.20. In this example, there is a clear reduction in sample size for the TED over both the SPCD and the traditional parallel trial. If there is no early dropout in the trials, the total patient exposure time for the TED, SPCD and the traditional parallel trial are 624 weeks, 864 weeks and 776 weeks, respectively. Since in most two-stage trials, such as SPCD trials, all patients usually continue in the second stage irrespective of whether or not their data are used in the efficacy analysis, we report total exposure time rather than exposure time in the efficacy analyzable set.

Motivating Example 2: A randomized withdrawal in ulcerative colitis usually has a 6–12–weeks-long induction of remission stage followed by the maintenance stage that is 24 weeks or longer. A possible modification of this design is to assign some patients to placebo in Stage 1 and then re-randomize placebo non-responders to drug and placebo, subsequently adding this treatment comparison to the efficacy analysis, that is, the TED without first-stage comparison. Such a trial might be more efficient than a randomized withdrawal design. Another possibility is to increase the length of the first stage to 24 weeks to allow for comparison of placebo and drug in Stage 1, followed by the comparisons of placebo and drug in placebo non-responders and drug responders. Assume that the response rates to drug and placebo are p₁=p₂=p₃=0.6 and q₁=q₂=q₃=0.4 and that patients are allocated equally to drug and placebo in the first stage. The sample sizes for TED, the TED without first-stage comparison and the randomized withdrawal are 124, 328 and 328, respectively, for α=0.05 and β=0.20. The sample sizes are based on one df test with r₂=r₃=1 used in TED and one df test with r₂=r₃=1000 used in the TED without first-stage comparison, and assuming retention rates of 1. Though treatment differences are the same in each sub-population, the sample size for the TED is much smaller than for the other two designs. The total exposure time for the TED (24 weeks in both stages), the TED without first-stage comparison (Stage 1: 12 weeks; Stage 2: 24 weeks) and the randomized withdrawal designs (Stage 1: 12 weeks; Stage 2: 24 weeks) are 5952 weeks, 11808 weeks and 6288 weeks, respectively.

We performed simulations of the one, two and three df tests (results are available from the authors) and concluded that the type I error rate for the tests is generally well preserved even for small sample sizes. Our simulation study also showed that the asymptotic power formulae for tests based on observed information provide good approximation of power and required sample size. The asymptotic power formulae for tests based on expected information might underestimate required sample size, especially if the required sample size is not large. For example, in the scenario with response rates p₁=0.5, q₁=0.3, p₂=0.4, q₂=0.1, p₃=0.9 and q₃=0.7, according to the asymptotic formula, the test based on observed information requires 80 subjects to achieve 80% power with one df test r₂=r₃=1 and b=0.5 (Table 1). The simulated power is 0.82. At the same time, the asymptotic power formulae based on expected information calls for 76 total subjects. The test based on expected information yields 0.78 power with 76 subjects. When used with the same number of subjects, the power of the test based on observed information is slightly higher on average than power of the test based on expected information.

4 Testing individual hypotheses about treatment differences

The traditional single-stage randomized study of a drug versus placebo tests the hypothesis H₀₁: Δ₁=0. Enriched designs test hypotheses that are different from H₀₁. The placebo lead-in tests the hypothesis H₀₂: Δ₂=0, the randomized withdrawal tests H₀₃: Δ₃=0. The TED tests H₀: {#x00394;₁=0}∩{#x00394;₂=0}∩{#x00394;₃=0} which is a subset of H₀₁. The null hypothesis H₀ represents a drug that is ineffective not only in Stage 1 but also in patients who did not improve on placebo in Stage 1, and also ineffective in maintaining response in patients who initially responded to drug. This null hypothesis is appropriate when the investigator has strong belief that an active drug in the overall population implies that the drug is superior to placebo in the subset of placebo non-responders and also superior to placebo in maintenance of response.

When TED or other enrichment designs are used, it might be of interest to test not only H₀ but also H₀₁, H₀₂ and H₀₃ to shed light on why H₀ was rejected. Let u₁, u₂ and u₃ be two-sided p-values for testing H₀₁, H₀₂ and H₀₃ obtained from an α-level test. One can use the Bonferroni procedure, and reject a null hypothesis H₀ _i , if 3u_i < α, i=1, 2, 3. If one of H₀₁, H₀₂ and H₀₃ is rejected, H₀ is rejected as well, which leads to an alternative way of testing H₀. Another way to test H₀₁, H₀₂ and H₀₃ is to use the closed testing principle¹³: which states that a hypothesis is rejected if all intersection hypotheses containing the hypothesis of interest are rejected. For example, to reject H₀₁, one needs to reject H₀=H₀₁∩H₀₂∩H₀₃, H₀₁∩H₀₂, H₀₁∩H₀₃ and H₀₁. Any of these intersections can be tested using the score test we developed with appropriately chosen test parameters. Closed testing procedures are usually more powerful for testing H₀₁, H₀₂ and H₃₀ simultaneously compared to the Bonferroni procedure. Table 5 displays power for testing H₀, H₀₁, H₀₂ and H₀₃ based on closed testing principle in conjunction with the score test. The numbers in parentheses are power values for the Bonferroni procedure. The total sample size used is the sample size required to achieve 80% with 0.05-level one df test with r₂=r₃=1. The power for the closed testing principle and the score test is higher than for the Bonferroni procedure in all scenarios except Scenario 7, where parameters used in one df score test, r₂=r₃=1, are very far from the true ρ₂=ρ₃=+∞.

Table 5.

Power for testing H₀: {#x00394;₁=0}∩{#x00394;₂=0}∩{#x00394;₃=0} and individual hypotheses H₀₁: Δ₁=0, H₀₂: Δ₂=0 and H₀₃: Δ₃=0 using closed testing principle^a

p ₁	q ₁	p ₂	q ₂	p ₃	q ₃	n	H ₀	H₀₁: Δ₁=0	H₀₂: Δ₂=0	H₀₃: Δ₃=0
0.4	0.3	0.4	0.3	0.9	0.8	412	0.79 (0.55)	0.51 (0.39)	0.19 (0.14)	0.22 (0.16)
0.5	0.3	0.4	0.2	0.9	0.8	128	0.80 (0.60)	0.54 (0.48)	0.20 (0.21)	0.14 (0.07)
0.5	0.3	0.4	0.1	0.9	0.8	96	0.82 (0.67)	0.43 (0.37)	0.42 (0.47)	0.12 (0.05)
0.4	0.3	0.4	0.3	0.9	0.7	312	0.80 (0.60)	0.42 (0.30)	0.16 (0.11)	0.40 (0.39)
0.5	0.3	0.4	0.2	0.9	0.7	104	0.81 (0.57)	0.51 (0.39)	0.23 (0.18)	0.25 (0.17)
0.5	0.3	0.4	0.1	0.9	0.7	80	0.82 (0.63)	0.42 (0.30)	0.43 (0.42)	0.21 (0.12)
0.4	0.4	0.4	0.3	0.9	0.8	2612	0.79 (0.94)	0.02 (0.02)	0.29 (0.71)	0.39 (0.80)
0.4	0.3	0.4	0.3	0.9	0.9	728	0.80 (0.76)	0.60 (0.67)	0.15 (0.24)	0.03 (0.02)
0.4	0.3	0.4	0.4	0.9	0.8	644	0.80 (0.70)	0.60 (0.61)	0.02 (0.02)	0.18 (0.25)

The H₀ is tested using the one df test with recommended parameters r₂=r₃ =1, H₀₁ is tested with r₂=r₃=0, H₀₂ is tested with r₂=100, r₃=0, and H₀₃ with r₂=0, r₃=100. Numbers in parentheses are the power for the Bonferroni procedure where each p-value for testing H₀₁, H₀₂ and H₀₃ is compared with α/3 and H₀ is rejected, if any of H₀₁, H₀₂ and H₀₃ $H_{03}$ are rejected.

5 Discussion

Clearly, the TED is particularly well suited to chronic conditions in which achieving response does not imply that the response will be maintained without treatment. For conditions in which response is terminal (i.e. death and pregnancy), designs for absorbing binary endpoints^14,15 are more appropriate. Two-stage designs like the SPCD and the TED require longer exposure for patients participating in both stages than the traditional parallel trial, however, since the SPCD or TED require smaller sample size compared to the parallel trial, the total length of the trial in calendar time would typically be shorter. Thus, in considering the TED, it is important to try to minimize the duration of the two stages in order to limit dropout of patients. Typically, the costs in clinical trials are more dependent on the number of enrolled subjects as opposed to the duration of the trial, therefore, it is often true that a smaller trial with longer duration is less costly than a larger trial with shorter duration. In trials like the TED, blinding of design aspects such as the definition of response and the changeover point/points to both investigators and patients is important. The definition of response needs to be clinically relevant if the ultimate outcome, for example, remission, is not feasible in short-term studies. Such details could be provided to institutional review boards but withheld from the site investigators.

Enriched trials have an important role in early phase studies described as proof of concept intended to detect any positive signal of a drug. In these early phases, minimizing sample size is important in order to be able to control costs. In later-phase studies, enriched trials may be more controversial because they may raise questions regarding the extrapolation of the results of such trials. However, we believe that the use of enriched trials in the confirmatory setting needs to be evaluated on a case-by-case basis, and that there will be situations where enriched trials are warranted. As one example, consider the situation of paediatric depression. Paediatric clinical trials of depression are required for any approved antidepressant and regulatory authorities enroll such trials to be both placebo and active controlled. However, these trials are difficult to enrol and are known to have high placebo response rate with a corresponding large number of failed trials.¹⁶ Because of this, there is only one drug, fluoxetine, which is indicated for depression in children under the age of 12 in the United States. The lack of a larger number of approved antidepressants for children is not an ideal situation because not all children will respond to fluoxetine. In this disease, it would seem reasonable to design either a SPCD- or TED-enriched trial as opposed to a larger placebo- and active-controlled traditional trial. The use of TED might improve enrolment as a larger proportion of patients, 1 – b/2, in a TED trial will receive experimental drug during the trial compared to a typical parallel trial. Ultimately, for diseases in which a drug will likely help only a small proportion of the population, regulators and sponsors need to decide whether an enriched trial will yield more information than the traditional trial. We believe that our TED is a new useful tool for enriched clinical trials and its application will hopefully allow medical researchers to better understand its actual potential.

Footnotes

Acknowledgements

Ivanova’s work was supported in part by the National Institutes of Health [RO1 CA120082-01A1]. The authors thank Hans Herfarth, Charles Beasley, David DeBrota and anonymous reviewers for helpful comments.

Appendix 1

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