Why Condition the Likelihood?
An interim estimate is often reported only after a rule has selected an arm: perhaps an arm exceeded a threshold, or it had the largest observed contrast. The sampling distribution of an estimate observed in that circumstance is not generally its original, unconditional distribution. In particular, the likelihood must account for how likely the observed selection event would have been under each possible treatment-effect value.
The introductory material on arm selection and the earlier material on selection events distinguish the estimand from the event that led to observing an arm. Here the focus is inferential: given an interim contrast and the selection event that occurred, how can the likelihood be formed and maximized? The event must be specified by the actual selection rule, not chosen after looking at the estimate.
For a vector of interim contrasts \(X=(X_1,\ldots,X_K)\), a rule might select arm \(j\) when \(X_j\) is the largest contrast, or when a specified subset of contrasts exceeds a threshold. The corresponding event \(A\) is a region in the vector's sample space. If the full vector is observed, its conditional density is the joint density restricted to that region and divided by \(P_\theta(A)\). If only part of the vector is retained, the numerator must instead be integrated over the unobserved values compatible with the selection event. Dividing a marginal density by the selection probability is not generally valid in that case.
Model and Conditional Score
A basic working model is that the interim contrast vector is multivariate normal with known covariance matrix: $$ X\sim N_K(\theta,\Sigma), $$ where \(\theta\) is the vector of true interim contrast means and \(\Sigma\) is positive definite. The selection rule is assumed known and applied to these contrasts as specified. The normal model, covariance structure, and rule must adequately represent the data and the actual selection process. The calculations below use known standard errors; estimating them or modelling additional stages requires an appropriate likelihood for those quantities as well.
The conditional likelihood follows directly from the definition of conditional density. Its score also has a useful interpretation. Write \(U_\theta(X)=\partial\log f_\theta(X)/\partial\theta\) for the ordinary score. Differentiating the selection probability gives $$ \frac{\partial}{\partial\theta}P_\theta(A) =\int_A \frac{\partial f_\theta(x)}{\partial\theta}\,dx =\int_A f_\theta(x)U_\theta(x)\,dx =P_\theta(A)E_\theta\{U_\theta(X)\mid A\}. $$ Consequently, $$ \frac{\partial}{\partial\theta}\log L_c(\theta;x,A) =U_\theta(x)-E_\theta\{U_\theta(X)\mid A\}. $$ At an interior conditional maximum, the observed ordinary score equals its expectation under the model conditional on the event that occurred. This is a score equation, not a claim that the estimate is unbiased.
For a normal vector with known \(\Sigma\), the ordinary score is \(\Sigma^{-1}(x-\theta)\). Thus the conditional score is $$ \Sigma^{-1}(x-\theta) -E_\theta\{\Sigma^{-1}(X-\theta)\mid A\}. $$ In general, computing the conditional expectation, or equivalently the event probability and its derivatives, can require a multivariate normal probability calculation. A one-dimensional truncation gives a transparent special case.
A Threshold-Selected Normal Contrast
Suppose \(X\sim N(\theta,\sigma^2)\), with known \(\sigma\), and an arm is retained only when \(X>c\). For an observed \(x>c\), the event probability is \(P_\theta(X>c)=1-\Phi((c-\theta)/\sigma)\), where \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution function. The conditional density is therefore $$ f_\theta(x\mid X>c) =\frac{\phi((x-\theta)/\sigma)} {\sigma\{1-\Phi((c-\theta)/\sigma)\}}, \qquad x>c. $$ Differentiating its log likelihood with respect to \(\theta\), and setting the score to zero, gives $$ x=\theta+\sigma\frac{\phi((c-\theta)/\sigma)} {1-\Phi((c-\theta)/\sigma)}. $$ The second term is the difference between the conditional mean of a lower-truncated normal variable and its untruncated mean. The equation says that the observed value is matched to its model-based mean conditional on passing the threshold.
Worked Example: A Contrast Selected Above Zero
Suppose a hypothetical interim contrast has standard error \(\sigma=1\) response unit, and the arm is retained only if \(X>0\). The observed contrast is \(x=1.5\) response units. Under the normal model, the conditional likelihood is proportional to $$ L_c(\theta)=\frac{\phi(1.5-\theta)}{\Phi(\theta)}, \qquad \theta<1.5\ \text{not required; the observation is supported whenever }1.5>0. $$ The score equation is \(1.5=\theta+\phi(\theta)/\Phi(\theta)\). Solving numerically gives \(\hat{\theta}_c\approx1.314\) response units. At this value, \(\phi(1.314)\approx0.1683\) and \(\Phi(1.314)\approx0.9056\), so $$ 1.314+\frac{0.1683}{0.9056}\approx1.500. $$ The unadjusted estimate would be \(1.5\) response units. Here the conditional maximum-likelihood estimate is lower because the estimate was observed after passing a threshold. The difference is not a universal fixed correction: it depends on the model, threshold, observed value, and selection event.
For a single lower-threshold event, the conditional mean \(m(\theta)=E_\theta(X\mid X>c)\) increases with \(\theta\). One justification is that its derivative is \(\operatorname{Var}_\theta(X\mid X>c)/\sigma^2>0\), obtained by differentiating the conditional expectation in the exponential-family normal model. Its range is from the threshold (as \(\theta\) becomes very negative) to infinity. Thus for any observed \(x>c\), the score equation \(m(\theta)=x\) has a unique finite solution. This useful existence and uniqueness fact is specific to this simple threshold setting; it should not be assumed automatically for more complicated selection regions or parameter vectors.
Winner Selection and Multiple Selected Arms
A winner rule can often be expressed through a contrast between estimates. Suppose two independent arm estimates have equal known variance \(1/2\), with means \(\theta_1\) and \(\theta_2\). Their difference \(D=X_1-X_2\) has distribution \(N(\delta,1)\), where \(\delta=\theta_1-\theta_2\). Selecting arm 1 as the winner is the event \(D>0\). Conditioning on that event gives a one-dimensional likelihood for the effect difference. This reduction is valid for inference on \(\delta\) because the average of the two estimates is independent of \(D\) under this equal-variance normal model.
Worked Example: Estimating the Difference for a Selected Winner
In hypothetical data, the interim estimates are \(X_1=1.2\) and \(X_2=0.2\) response units, so arm 1 is selected and \(d=1.0\) response unit. With the variances specified above, \(D\sim N(\delta,1)\), and the selection probability is \(P_\delta(D>0)=\Phi(\delta)\). The conditional likelihood for the observed difference is $$ L_c(\delta;d)=\frac{\phi(d-\delta)}{\Phi(\delta)}, \qquad d>0. $$ For \(d=1.0\), its score equation is \(1.0-\delta-\phi(\delta)/\Phi(\delta)=0\). At \(\delta=0.4805\), \(\phi(0.4805)\) is about \(0.3554\) and \(\Phi(0.4805)\) is about \(0.6846\); hence \(\delta+0.3554/0.6846\approx1.000\). The conditional MLE is approximately \(0.4805\) response units, whereas the observed difference is \(1.0\) response unit. The conditional estimate concerns the difference in true effects, not the identity of the truly best arm.
If several arms are retained by a threshold rule, the event probability must describe that joint rule. Even when the contrasts are independent, the likelihood denominator is the probability that all the required contrasts pass. The next example shows how this changes the estimating equation.
Worked Example: Two Contrasts Both Pass a Threshold
Suppose two hypothetical interim contrasts are independent, each distributed as \(N(\theta,1)\). Both are retained only when each exceeds zero. The observed values are \(1.2\) and \(0.8\) response units, so both pass. The event probability is \(\Phi(\theta)^2\), and the conditional likelihood is $$ L_c(\theta)=\frac{\phi(1.2-\theta)\phi(0.8-\theta)}{\Phi(\theta)^2}. $$ The log-likelihood score equation is $$ (1.2-\theta)+(0.8-\theta)-2\frac{\phi(\theta)}{\Phi(\theta)}=0, \quad\text{or}\quad 1.0=\theta+\frac{\phi(\theta)}{\Phi(\theta)}. $$ At \(\theta=0.4805\), the density and distribution values above give a right-hand side of approximately \(0.4805+0.3554/0.6846=1.000\). Thus the conditional MLE is about \(0.4805\) response units. Treating the two selected observations as ordinary unselected normal observations would give their mean, \(1.0\) response unit, and would omit the factor \(\Phi(\theta)^2\).
Computation, Diagnostics, and Alternatives
For a general MAMS selection rule, a practical calculation proceeds by first specifying the interim-data distribution, then encoding the observed event, and finally maximizing the conditional likelihood. The event probability must be evaluated under each candidate parameter value. If selection depends on correlated contrasts, the probability generally cannot be replaced by a product of marginal probabilities. Numerical integration, a validated multivariate normal probability routine, or carefully checked simulation may be needed.
State precisely which contrasts, thresholds, rankings, or ties determine the observed selection outcome.
Give the mean parameter and covariance matrix for the interim contrasts, including any shared-control covariance.
Use the joint density of the observed data and divide by the model probability of the specified event. Integrate over unobserved data if necessary.
Check the optimizer, event-probability calculation, score near the solution, and sensitivity to plausible covariance or model changes.
The main alternatives answer different inferential questions. The ordinary likelihood uses the sampling distribution without conditioning on selection; it can be appropriate when the data and analysis genuinely do not condition on selection, but it does not account for the observed selection event. Conditional maximum likelihood explicitly conditions on that event. A full likelihood for a complete selection path can condition on more than the final winner, but requires modelling the entire recorded path; it is not interchangeable with conditioning only on the final arm label. The choice depends on the information treated as fixed and the estimand of interest.
| Approach | What is conditioned on? | Key point |
|---|---|---|
| Ordinary likelihood | No selection event | Uses the unconditioned sampling model. |
| Conditional likelihood | The specified observed event | Includes the parameter-dependent event probability in its denominator. |
| Likelihood for a fuller path | Additional recorded stages or decisions | Requires the corresponding joint model and a precisely defined path. |
Diagnostics matter because an apparently plausible optimizer can still come from a misspecified selection probability. Verify that the observed data satisfy the event used in the likelihood; check that the event probability is between zero and one and changes sensibly with the parameter; compare an analytic score with a numerical derivative where possible; and use multiple starting values for multidimensional optimization. A very small event probability can make the conditional likelihood numerically unstable. A covariance model that ignores shared controls can distort the selection probability even if each marginal standard error looks reasonable.
Conditional maximum likelihood has important limits. It depends on a correctly specified sampling distribution and selection rule, and it does not guarantee an unbiased estimate in finite samples. The conditional likelihood may be difficult to compute, may have multiple local maxima in complex models, or may have no finite maximizer in some settings. Uncertainty assessment must also reflect the conditioning and model; the ordinary unselected standard error should not be assumed valid automatically. Finally, conditional inference on an interim contrast does not by itself incorporate later-stage data or answer every post-selection estimand.
Common Pitfall
Correction: It is constant with respect to the observed data \(x\), but not generally with respect to the unknown parameter \(\theta\). The conditional likelihood includes \(P_\theta(A)\), which changes as \(\theta\) changes. Omitting it maximizes the ordinary likelihood, not the likelihood conditional on the selection event.
A second frequent error is to condition on the event that is easiest to calculate rather than the event the rule actually used. If the rule selected the largest contrast, conditioning only on a threshold crossing is a different analysis. Likewise, if the full vector determined selection but only one contrast is analyzed, the unobserved components cannot simply be ignored without checking the required marginal conditional distribution.
Check Your Understanding
For each question, identify the relevant conditional-likelihood principle before checking the solutions.
- For \(X\sim N(\theta,1)\) observed only when \(X>c\), what parameter-dependent quantity must appear in the conditional likelihood denominator?
- Two independent standard-error-one contrasts have means \(\theta_1,\theta_2\). What are the mean and variance of their difference, and what event represents arm 1 winning?
- Why is the ordinary sample mean of two threshold-selected contrasts not generally the conditional MLE of their common mean?
- Name two checks that can reveal a faulty numerical conditional-likelihood calculation.
Solutions. (1) The denominator is \(P_\theta(X>c)=1-\Phi(c-\theta)\), since the standard deviation is one. (2) The difference has mean \(\theta_1-\theta_2\), variance \(1+1=2\), and arm 1 wins when \(X_1-X_2>0\). (3) The conditional likelihood divides the joint density by the parameter-dependent probability that both observations pass; the ordinary mean omits that factor. (4) Check that the observed data satisfy the encoded event and verify the event probability is between zero and one; also compare the analytic score with a numerical derivative or repeat optimization from multiple starting values.