Conditional Likelihood Estimation After Selection

Why Condition the Likelihood on Selection?

When a trial selects one treatment arm for further study because its interim estimate is larger, the observed stage-one data are not a random sample from their original distribution once we focus on trials making that selection. A likelihood that explicitly conditions on the selection event describes the stage-one data among trials with that event, together with the later data collected under the design. This is one way to make the selection rule part of the estimation model rather than treating the observed arm as if it had been chosen in advance.

Conditioning is a choice of likelihood, not a correction that automatically makes an estimate unbiased or preferable for every purpose. The conditional and unconditional likelihoods answer related but different questions. To construct the conditional likelihood correctly, the selection event, the sampling model, and the data retained for analysis must all be specified.

Model and Selection Event

Consider two candidate treatment arms, labelled 1 and 2, each compared with a common reference. Let \(\theta_j\) be the true treatment effect for arm \(j\), on the same scale and in the same units as its estimated effect. At stage one, let \(X_j\) be the estimated treatment effect for arm \(j\). Assume the following model:

X_1 \sim N(\theta_1,\sigma_1^2), \qquad X_2 \sim N(\theta_2,\sigma_1^2), \qquad X_1 \perp X_2.

The stage-one standard deviation \(\sigma_1\) is known and the same for both arms. The trial selects the arm with the larger stage-one estimate. Write \(J=1\) when \(X_1>X_2\), and \(J=2\) when \(X_2>X_1\). Under the continuous normal model, ties have probability zero. If arm 1 is selected, the trial also observes a stage-two estimate \(Y_1\), with

Y_1 \sim N(\theta_1,\sigma_2^2), \qquad Y_1 \perp (X_1,X_2),

where \(\sigma_2\) is known. If arm 2 is selected, the corresponding stage-two estimate is \(Y_2\sim N(\theta_2,\sigma_2^2)\), independent of the stage-one estimates. This model assumes independent stage-one contrasts, a correctly specified normal distribution, known variances, and stage-two data independent of stage one given the treatment effects. These assumptions need scrutiny in an actual trial: for example, contrasts sharing a control group may be correlated and would require a covariance model instead.

Definition: The conditional likelihood given the selection event \(J=1\) is the probability density of the observed data, conditional on the event \(X_1>X_2\), viewed as a function of the unknown treatment effects. It is the joint density of the observed data and the event divided by \(P_{\theta_1,\theta_2}(J=1)\).

The difference \(X_1-X_2\) is normal with mean \(\theta_1-\theta_2\) and variance \(2\sigma_1^2\). Therefore the probability of selecting arm 1 is

P_{\theta_1,\theta_2}(J=1) = P(X_1-X_2>0) = \Phi\left(\frac{\theta_1-\theta_2}{\sqrt{2}\sigma_1}\right),

where \(\Phi\) is the standard normal cumulative distribution function. This probability depends on both treatment effects. That dependence is why the selection probability must appear in the conditional likelihood.

Worked Example: Calculating the Selection Probability

Suppose the stage-one estimates have standard deviation \(\sigma_1=1\) treatment-effect unit, and consider the hypothetical parameter values \(\theta_1=0.5\) and \(\theta_2=0\). The probability that arm 1 is selected is

P(J=1) = \Phi\left(\frac{0.5-0}{\sqrt{2}(1)}\right) = \Phi(0.3536) \approx 0.6382.

Thus, at these parameter values, arm 1 is selected in about 63.8% of trials run under this model. This is a probability under the assumed parameters, not a probability that those parameters are true. If the effects were equal, the selection probability would be \(0.5\), by symmetry.

Constructing the Conditional Likelihood

Suppose arm 1 is selected and the observed estimates are \(X_1=x_1\), \(X_2=x_2\), and \(Y_1=y_1\), with \(x_1>x_2\). Write \(f_{\mu,v}(x)\) for the normal density at \(x\) with mean \(\mu\) and variance \(v\). The unconditional joint density of these observations is the product of the two stage-one densities and the stage-two density. To condition on selection, divide this joint density by the probability of the selection event:

L_c(\theta_1,\theta_2) = \frac{ f_{\theta_1,\sigma_1^2}(x_1) f_{\theta_2,\sigma_1^2}(x_2) f_{\theta_1,\sigma_2^2}(y_1) }{ \Phi\left((\theta_1-\theta_2)/(\sqrt{2}\sigma_1)\right) }, \qquad x_1>x_2.

The conditional likelihood is defined on observations satisfying the selection rule. The observed values remain in its numerator: conditioning on \(J=1\) does not mean discarding the stage-one measurements. Rather, their density is evaluated using the distribution truncated to the region \(x_1>x_2\). The stage-two factor is present because \(Y_1\) is observed after selecting arm 1; under the stated model it contributes no additional selection-probability term.

For maximizing the likelihood, constants that do not depend on \(\theta_1\) or \(\theta_2\) can be omitted. The conditional log-likelihood is then

\ell_c(\theta_1,\theta_2) = -\frac{(x_1-\theta_1)^2+(x_2-\theta_2)^2}{2\sigma_1^2} -\frac{(y_1-\theta_1)^2}{2\sigma_2^2} -\log\Phi(a), \qquad a=\frac{\theta_1-\theta_2}{\sqrt{2}\sigma_1}.

Score Equations and Numerical Estimation

A conditional likelihood estimator maximizes this log-likelihood over \(\theta_1\) and \(\theta_2\). To obtain the score equations, differentiate the normal terms in the usual way and differentiate the selection term using the chain rule. Define \(\lambda(a)=\phi(a)/\Phi(a)\), where \(\phi\) and \(\Phi\) are the standard normal density and distribution function. Since \(d\log\Phi(a)/da=\lambda(a)\), and \(da/d\theta_1=1/(\sqrt{2}\sigma_1)\), the two score equations are

\frac{\partial \ell_c}{\partial\theta_1} = \frac{x_1-\theta_1}{\sigma_1^2} +\frac{y_1-\theta_1}{\sigma_2^2} -\frac{\lambda(a)}{\sqrt{2}\sigma_1} =0,

\frac{\partial \ell_c}{\partial\theta_2} = \frac{x_2-\theta_2}{\sigma_1^2} +\frac{\lambda(a)}{\sqrt{2}\sigma_1} =0.

The opposite signs of the selection terms follow from the fact that increasing \(\theta_1-\theta_2\) increases the probability of selecting arm 1. The conditional likelihood divides by that probability, so its log contains \(-\log\Phi(a)\). Solving the equations generally requires numerical optimization: \(a\) depends on both unknown effects, and \(\lambda(a)\) is nonlinear.

Worked Example: Solving the Conditional Score Equations

In a hypothetical trial, let \(\sigma_1=\sigma_2=1\) treatment-effect unit. Suppose arm 1 is selected and the observations are \(x_1=2\), \(x_2=0\), and \(y_1=1.5\). The selection condition holds because \(2>0\). The score equations become

2-\theta_1+1.5-\theta_1-\frac{\lambda(a)}{\sqrt{2}}=0, \qquad 0-\theta_2+\frac{\lambda(a)}{\sqrt{2}}=0, \qquad a=\frac{\theta_1-\theta_2}{\sqrt{2}}.

Rearranging gives \(\theta_1=1.75-\lambda(a)/(2\sqrt{2})\) and \(\theta_2=\lambda(a)/\sqrt{2}\). Substituting these expressions into the definition of \(a\) gives a single nonlinear equation:

a=1.2374-0.75\lambda(a).

Numerical solution gives \(a\approx1.030\) and \(\lambda(a)\approx0.277\). Substitution into the two rearranged score equations yields

\hat{\theta}_{1,c} =1.75-\frac{0.277}{2\sqrt{2}} \approx1.652, \qquad \hat{\theta}_{2,c} =\frac{0.277}{\sqrt{2}} \approx0.196.

The conditional likelihood estimates the selected arm's effect as about 1.652 treatment-effect units and the other arm's effect as about 0.196 units. For comparison, maximizing the unconditional likelihood for these same observations gives \(\hat{\theta}_1=(2+1.5)/2=1.75\) and \(\hat{\theta}_2=0\). The difference arises from the selection-probability denominator, not from a change in the data or the model for the observations.

Evaluating the Likelihood and Checking the Model

The denominator is not an optional adjustment. It normalizes the joint density over the region consistent with the conditioning event. For example, at a proposed pair of effects, the conditional density can be evaluated by taking the ordinary joint density at the observed values and dividing by the probability of selecting the observed arm. Because a density has units and can exceed one, its numerical value alone is not a probability; comparisons of likelihood values are meaningful for the same observed data and model.

Worked Example: Evaluating Two Conditional Likelihood Values

Suppose a hypothetical trial selects arm 2, with observations \(x_2=0.8\), \(x_1=-0.2\), and \(y_2=0.6\). Let \(\sigma_1=1\) and \(\sigma_2=0.5\), so the stage-two variance is \(0.25\). For arm 2, the selection probability is

P(J=2) = \Phi\left(\frac{\theta_2-\theta_1}{\sqrt{2}}\right).

Ignoring normalizing constants that are the same for both parameter pairs, the conditional log-likelihood is

\ell_c(\theta_1,\theta_2) = -\frac{(x_1-\theta_1)^2+(x_2-\theta_2)^2}{2} -\frac{(y_2-\theta_2)^2}{2(0.25)} -\log\Phi\left(\frac{\theta_2-\theta_1}{\sqrt{2}}\right).

At \((\theta_1,\theta_2)=(0,0.5)\), the normal-term contribution is \(-0.02-0.045-0.02=-0.085\). The selection probability is \(\Phi(0.3536)\approx0.6382\), so

\ell_c(0,0.5)=-0.085-\log(0.6382)\approx0.364.

At \((\theta_1,\theta_2)=(-0.2,0.8)\), the normal-term contribution is \(0-0-0.08=-0.080\), and the selection probability is \(\Phi(0.7071)\approx0.7602\). Thus

\ell_c(-0.2,0.8)=-0.080-\log(0.7602)\approx0.194.

The first pair has the larger conditional log-likelihood of these two candidates. This comparison does not establish that it is the maximum; maximizing requires comparing against all parameter values, usually by numerical optimization.

A numerical fit should be checked rather than accepted solely because an optimizer returned a result. Useful checks include verifying that the observed data obey the stated selection rule, that the likelihood uses the correct selection probability, that the score is close to zero at an interior solution, and that the reported solution has a higher log-likelihood than nearby alternatives. Optimization from multiple starting values can reveal convergence to different stationary points. Extreme fitted effects can also signal weak information or a poorly conditioned likelihood.

What Conditioning Changes—and What It Does Not

The unconditional likelihood for this design uses the joint density of the observed stage-one estimates and the selected arm's stage-two estimate, without dividing by the probability of the observed selection event. That likelihood is appropriate when the intended analysis uses the full sampling model, including the fact that selection was a deterministic part of the design. The conditional likelihood instead uses the distribution of the observations given the selected arm. It is useful when inference is framed conditional on that event, but it is not simply the unconditional likelihood with an arbitrary penalty added.

A different analysis might condition only on the arm label while omitting stage-one values, or might use only stage-two data. Those analyses discard information used in the likelihood above and are not the same procedure. The conditioning event and the data included in the conditional density must be stated explicitly. In particular, conditioning on \(J=1\) does not mean that the numerical values \(x_1,x_2\) are fixed constants for the purpose of forming the likelihood; their truncated density still contributes.

This derivation also has important limits. It assumes known variances and independent normal contrasts. With correlated contrasts, unequal or estimated variances, more than two candidate arms, selection based on a threshold or a more complex rule, or a different stage-two allocation, the selection probability and joint density must be changed to match the design. A conditional likelihood estimate alone does not supply a confidence interval, guarantee unbiasedness, or account for model misspecification. Those require additional methods and arguments.

Common Mistakes

Omitting the selection denominator. Using only the ordinary product of normal densities gives the unconditional likelihood, not the likelihood conditional on selecting the observed arm.
Using the wrong selection probability. For the rule \(X_1>X_2\), the denominator is \(\Phi((\theta_1-\theta_2)/(\sqrt{2}\sigma_1))\) under the stated independent, equal-variance model. It is not \(\Phi(\theta_1/\sigma_1)\).
Discarding stage-one values after conditioning. The event determines the truncated region, but the observed stage-one values still appear in the conditional density.
Calling the conditional estimate unbiased by construction. Conditioning changes the likelihood; it does not guarantee unbiasedness or make the estimate universally preferable.
Applying this denominator to a different design. Shared controls, unequal variances, or a different selection rule change the probability of the event and therefore change the conditional likelihood.

Key takeaway: For a two-stage normal trial that selects the arm with the larger stage-one estimate, the conditional likelihood is the joint density of the observed stage-one and selected-arm stage-two data divided by the model-based probability of selecting that arm. The selection rule is part of the likelihood, and the assumptions used to calculate its probability must match the trial design.

Check Your Understanding

Answer each question, then compare with the solutions below.

Under the independent equal-variance stage-one model, derive \(P(J=2)\) in terms of \(\theta_1,\theta_2,\sigma_1\).
Why does the conditional likelihood for arm 1 include both stage-one densities even though the analysis conditions on \(J=1\)?
For arm 1 selected, what is the sign of the \(\lambda(a)\) term in the score for \(\theta_1\), and why?
If stage-one contrasts share a control group and are correlated, can the selection denominator above be used unchanged? Explain.
Does a larger conditional likelihood value at one parameter pair than at another prove that the first pair is the conditional maximum?

Solutions. (1) Since \(X_2-X_1\sim N(\theta_2-\theta_1,2\sigma_1^2)\), \(P(J=2)=P(X_2-X_1>0)=\Phi((\theta_2-\theta_1)/(\sqrt{2}\sigma_1))\). (2) Conditioning restricts the stage-one distribution to \(x_1>x_2\); the observed values still have a density within that region, so both stage-one density factors remain in the numerator. (3) The term is negative: the log-likelihood contains \(-\log\Phi(a)\), and \(a\) increases with \(\theta_1\), giving derivative \(-\lambda(a)/(\sqrt{2}\sigma_1)\). (4) No. Correlation changes the distribution of \(X_1-X_2\), so the selection probability must be recomputed under the appropriate joint model. (5) No. It establishes only a comparison between those two pairs; maximization requires checking the likelihood over the parameter space, including appropriate numerical diagnostics.