Blinded Pooled-Rate Estimation For Binary Outcomes

Why a Masked Pooled Rate Needs Careful Interpretation

For a binary outcome, each participant either experiences the event or does not. In a blinded sample-size re-estimation, the treatment labels may be hidden while the outcomes remain available. The total number of events can then inform planning, but it does not automatically estimate the event probability in either treatment arm. What it estimates depends on whether a common event probability is a reasonable model and, if the arm risks differ, on the allocation.

This distinction matters because the pooled rate is often used to estimate a nuisance quantity for planning. Under a common-risk model, the rate estimates that shared probability. Under unequal risks, it estimates an allocation-weighted average instead. If a planning calculation needs two arm-specific probabilities, the masked rate alone is insufficient; an assumed treatment contrast and the allocation are also needed.

Estimation Under a Common-Probability Model

Assume that outcomes are observed for all included participants, each outcome is Bernoulli, and outcomes are independent with the same event probability \(p\). The allocation counts are fixed and known, but the treatment labels attached to individual outcomes are masked. Under this common-probability model, the total event count satisfies \(X\sim\operatorname{Binomial}(N,p)\). The likelihood for \(p\), apart from a factor that does not depend on \(p\), is \(p^X(1-p)^{N-X}\).

For \(0<X<N\), differentiating the log likelihood gives \(X/p-(N-X)/(1-p)\). Setting this equal to zero yields the maximum likelihood estimate below. The same result applies at \(X=0\) or \(X=N\), where the likelihood is maximized at the boundary.

Under the model, \(E(\hat{p}_{\mathrm{pool}})=p\) and \(\operatorname{Var}(\hat{p}_{\mathrm{pool}})=p(1-p)/N\). Substituting the estimate for the unknown probability gives the usual estimated standard error, \(\sqrt{\hat{p}_{\mathrm{pool}}(1-\hat{p}_{\mathrm{pool}})/N}\). This standard error describes sampling variation under the common-probability model; it does not include uncertainty about whether that model is appropriate.

What the Pooled Rate Estimates When Risks Differ

If treatment 1 and treatment 0 have different event probabilities, write them as \(p_1\) and \(p_0\). With fixed arm sizes and independent Bernoulli outcomes within and between arms, the expected total event count is \(n_1p_1+n_0p_0\). Consequently,

So the pooled rate estimates the allocation-weighted average risk, not generally either arm's risk and not generally their simple average. With unequal allocation, the larger arm contributes more to the pooled rate. Masking prevents the analyst from assigning the observed events to arms, so the aggregate rate by itself cannot recover both \(p_1\) and \(p_0\).

For planning a risk-difference comparison, define the assumed contrast as \(\Delta=p_1-p_0\), with positive values indicating a higher event probability in treatment 1. If a pooled rate is used as a working estimate of the weighted average and a planning contrast \(\Delta\) is specified, the corresponding planning probabilities are

These expressions preserve the weighted average: \(w_1p_1+w_0p_0=\hat{p}_{\mathrm{pool}}\). They are a planning parameterization, not estimates of the hidden arm-specific event rates. The assumed contrast must be stated, the resulting probabilities must lie between 0 and 1, and the allocation fractions must match the design being planned.

Allocation Changes the Planning Variance

Even when the total sample size and pooled rate are fixed, allocation affects how precisely a risk difference can be estimated. The risk-difference variance depends on both arm sizes and both planning probabilities. Thus, the pooled rate is not enough to calculate that variance unless the allocation is known and the planning contrast (or another way to specify both arm probabilities) is supplied.

Assumptions, Diagnostics, and Alternatives

Before using a pooled rate in a re-estimation, check what information the masked data contain and what probability the calculation is intended to represent. A simple pooled estimate is most directly interpretable when each participant contributes one observed binary outcome, the denominator is the number of participants with outcomes, and the outcome probabilities are sufficiently stable across participants and relevant design strata. If outcomes are missing, \(X/N\) is not the event proportion among all enrolled participants unless the denominator and missingness handling have been defined accordingly.

Useful diagnostics include checking the event count against the analysis denominator, confirming the allocation fractions, and comparing the observed pooled rate with the range considered plausible at design. A rate near 0 or 1, very few events, or substantial variation across sites or strata can make a single pooled rate imprecise or unrepresentative. If follow-up differs between participants, the simple proportion may also fail to represent the intended time-specific risk. These problems should prompt review of the estimand and model rather than automatic use of the same pooled formula.

The main alternatives answer different questions. An unblinded analysis can estimate \(p_1\) and \(p_0\) separately from arm-specific event counts, but requires access to treatment labels and is not a blinded pooled-rate calculation. An external or design-stage event-rate assumption avoids using interim outcomes, but may not reflect the current trial population. A common-risk pooled estimate is simple and uses all masked outcomes efficiently under its model; it cannot reveal treatment-specific risks or verify that the common-risk assumption holds.

A pooled-rate estimate is a nuisance-parameter input, not a treatment-effect estimate. Blinding does not by itself guarantee that a sample-size update is appropriate: the update must use a defensible probability model, prespecified planning assumptions, and a re-estimation procedure compatible with the trial design. In particular, do not interpret the assumed contrast used to obtain planning probabilities as information learned from masked outcomes.