2026-08-12
\[ X_{Ti}\overset{iid}{\sim}N(\mu_T,\sigma^2), \qquad X_{Cj}\overset{iid}{\sim}N(\mu_C,\sigma^2) \]
\[ \delta=\mu_T-\mu_C, \qquad Y=\bar X_T-\bar X_C \]
Let
\[ \kappa=\frac1{n_T}+\frac1{n_C}, \qquad \nu=n_T+n_C-2. \]
Then
\[ Y\sim N(\delta,\sigma^2\kappa), \qquad \frac{\nu S_p^2}{\sigma^2}\sim\chi^2_\nu, \]
with \(Y\perp S_p^2\).
The program advances only when
\[ \boxed{Y>c}. \]
For an observed Go result \((y,s_p^2)\),
\[ L_c(\delta,\sigma) = \frac{f_Y(y\mid\delta,\sigma)f_{S_p^2}(s_p^2\mid\sigma)} {P_{\delta,\sigma}(Y>c)} \propto \frac{ \sigma^{-(\nu+1)} \exp\!\left[-\dfrac{\nu s_p^2+(y-\delta)^2/\kappa}{2\sigma^2}\right] }{\bar\Phi(a)}, \]
where \(a=(c-\delta)/(\sigma\sqrt\kappa)\). The MCLE maximizes \(L_c\) over \(\delta\in\mathbb R\) and \(\sigma>0\).
As \(y\downarrow c\), the MCLE satisfies
\[ \widehat a_{\rm MCLE} \sim \frac{s_p\sqrt\kappa}{y-c} \longrightarrow\infty, \qquad \bar\Phi(\widehat a_{\rm MCLE})\longrightarrow0. \]
As \(y\downarrow c\),
\[ \widehat\delta_{\rm MCLE} =c-\frac{s_p^2\kappa}{y-c} +o\{(y-c)^{-1}\} \longrightarrow-\infty. \]
Simultaneously,
\[ \widehat p_{\rm Go} =\bar\Phi(\widehat a) \longrightarrow0. \]
The fitted model explains an observed Go result using an effect under which Go would have been essentially impossible.
With 10,000 selected replicates per true effect, the estimated probability is 4.33% at \(\delta=0\) and 1.29% at \(\delta=0.35\).
Kirby et al. report
\[ \widehat\delta_{\rm Kirby} =\max(0,\widetilde\delta_{\rm MCLE}). \]
Full conditional log likelihood
\[ \ell_c(\delta,\sigma) =-(\nu+1)\log\sigma -\frac{\nu s_p^2+(y-\delta)^2/\kappa}{2\sigma^2} -\color{#B33A3A}{\log\bar\Phi(a)}. \]
Tempered conditional log likelihood
\[ \ell_\gamma(\delta,\sigma) =-(\nu+1)\log\sigma -\frac{\nu s_p^2+(y-\delta)^2/\kappa}{2\sigma^2} -\color{#006D77}{\gamma\log\bar\Phi(a)}, \qquad 0\leq\gamma<1. \]
Let \(\lambda(a)=\phi(a)/\bar\Phi(a)\) and \(m_\gamma(a)=\gamma\lambda(a)-a\). For every fixed \(0\leq\gamma<1\), \(m_\gamma\) has a unique finite zero \(a_\gamma\):
\[ \boxed{a_\gamma=\gamma\lambda(a_\gamma)}. \]
As \(y\downarrow c\),
\[ \widehat a\longrightarrow a_\gamma, \qquad \widehat\sigma_\gamma^2 \longrightarrow\frac{\nu}{\nu+1}s_p^2, \]
and
\[ \boxed{ \widehat\delta_\gamma \longrightarrow c-a_\gamma s_p\sqrt{\frac{\kappa\nu}{\nu+1}} }. \]
At \(y=c\) and fixed \(\sigma\), the \(a\)-dependent likelihood factor is
\[ \frac{\phi(a)}{\bar\Phi(a)^\gamma}. \]
Using \(\bar\Phi(a)\sim\phi(a)/a\),
\[ \frac{\phi(a)}{\bar\Phi(a)^\gamma} \sim \underbrace{a^\gamma}_{\text{polynomial}} \underbrace{\phi(a)^{1-\gamma}}_{\text{remaining Gaussian factor}}. \]
\(\phi(a)^{1-\gamma}\) drives the objective to zero as \(a\to\infty\) and prevents escape to the boundary.
The Gaussian factor is cancelled exactly; the objective can support \(a\to\infty\) and \(\widehat\delta\to-\infty\).
The tempered score equation gives
\[ \widehat\delta_\gamma =y-\gamma\widehat\sigma_\gamma\sqrt\kappa\, \lambda(\widehat a). \]
When the fitted effect is far above \(c\),
\[ \widehat a\to-\infty, \qquad \lambda(\widehat a)\to0, \]
so
\[ \boxed{\widehat\delta_\gamma-y\longrightarrow0}. \]
If \(Y\overset{p}{\to}\delta\), then \(\widehat\delta_\gamma\overset{p}{\to}\delta\).
Instead of choosing \(\gamma\) directly, specify
\[ \boxed{ \bar\Phi(\widehat a) =P_{\widehat\delta,\widehat\sigma}(Y>c) \geq p_{\min} }. \]
Choose
\[ a_\gamma=\Phi^{-1}(1-p_{\min}), \qquad \boxed{ \gamma=\frac{a_\gamma}{\lambda(a_\gamma)} =\frac{a_\gamma p_{\min}}{\phi(a_\gamma)} }. \]
For Example 1, \(p_{\min}=0.05\) gives \(\gamma=0.797\) and a boundary-limit estimate of approximately \(0.003\) when \(s_p=1\).
At \(\delta=0.35\), tempered MCLE has mean bias \(0.017\), compared with \(0.042\) for LWB, \(0.077\) for CLWB, \(-0.023\) for percentile bias, and \(-0.063\) for Kirby MCLE.
Kirby MCLE corrects most strongly, but truncation at zero contributes to negative bias over much of the positive-effect range.
Tempered MCLE remains smooth and untruncated, with bias close to zero around and above \(c\).
Both strategies start with the same original Phase 2 Go population, \(Y>c\), and use the adjusted estimate \(d\) for Phase 3 planning.
Strategy B: retain the original \(Y>c\) decision; use the adjusted estimate for Phase 3 planning. At \(\delta=0.35\), achieved power is 71.1% versus 28.0% under Strategy A for tempered MCLE.
A very small adjusted effect may imply an economically infeasible trial—this does not force launch. Use feasibility review, a prespecified minimum clinically/market-relevant design effect, or Phase 3 sample-size reassessment. Strategy A instead risks discarding a genuinely large effect after an extreme downward adjustment.
Full MCLE can explain a barely positive result through a fitted Go probability approaching zero, producing an unbounded negative estimate.
Replacing \(-\log\bar\Phi(a)\) by \(-\gamma\log\bar\Phi(a)\) with \(\gamma<1\) removes the boundary singularity while preserving recovery of the unadjusted estimate far above \(c\).
Calibrating \(\gamma\) through \(p_{\min}\) makes the regularization interpretable and prespecifiable.
For development decisions, let observed \(Y\) determine Phase 2 Go/No-go; use the adjusted estimate for Phase 3 planning, with feasibility safeguards when the adjusted design effect is small.