Deriving Power For Equal Allocation

Why Derive Power for Equal Allocation?

Power is the probability that a specified test rejects \(H_0\) when a particular alternative is true. It depends on the effect size, outcome variability, sample size, significance level, and test procedure. A calculation for equal allocation gives a clear reference point: with a fixed total sample size, the same number of participants is assigned to each of two treatments. This reference is useful when evaluating a design, but it represents a fair comparison only when the competing designs are assessed under matched trial conditions.

Here the outcome is continuous, and the target is the difference between two treatment means. The derivation uses a two-sample normal model and a two-sided test. It gives an exact power formula if the common population standard deviation is known and a useful normal approximation when that standard deviation is estimated.

These assumptions describe a fixed equal-allocation benchmark. In particular, the calculation does not by itself describe the power of a response-adaptive design, where assignments and group sizes may depend on accumulating trial data. The benchmark is informative only if the effect, outcome variability, total sample size, test, significance level, and outcome-measurement conditions are held comparable.

Deriving the Equal-Allocation Power Formula

The estimator of the treatment difference is \(\bar{Y}_A-\bar{Y}_B\). Under the stated model, the two sample means are independent, with variances \(\sigma^2/n\) each. Therefore,

Under a particular true difference \(\Delta\), the standardized statistic \(Z=(\bar{Y}_A-\bar{Y}_B)/(\sigma\sqrt{2/n})\) has a normal distribution with variance 1 and mean \(\lambda\), where \(\lambda\) is the effect measured in standard-error units. This mean is also called the noncentrality parameter.

Let \(\Phi\) be the standard normal cumulative distribution function, and let \(z_{1-\alpha/2}\) satisfy \(\Phi(z_{1-\alpha/2})=1-\alpha/2\). A two-sided z test rejects when \(|Z|>z_{1-\alpha/2}\). Under the alternative, rejection occurs either in the lower tail or the upper tail. Subtracting the mean \(\lambda\) from the two rejection boundaries gives the power formula:

The first term is the probability of rejection in the lower tail, and the second is the probability of rejection in the upper tail. For a negative \(\Delta\), \(\lambda\) is negative; the two tail probabilities exchange roles, leaving power unchanged for an effect of the same absolute size.

Planning a Sample Size for a Target Power

The same relationship can be rearranged to plan the number per treatment. For a positive effect, a common normal-approximation formula for target power \(1-\beta\) is

This formula comes from requiring the alternative mean of the standardized statistic, \(\lambda\), to exceed the upper rejection boundary by about \(z_{1-\beta}\). That is, set \(\lambda-z_{1-\alpha/2}\approx z_{1-\beta}\), substitute \(\lambda=\Delta/(\sigma\sqrt{2/n})\), and solve for \(n\). It focuses on the upper-tail rejection probability for a positive effect; the exact two-sided formula also includes the usually small probability of rejection in the opposite tail. Round the resulting \(n\) up to a whole participant per treatment, then check the resulting power using the two-tail formula.

Conditions for a Fair Design Comparison

A power number is not a property of an allocation rule alone. It is conditional on the assumptions and choices that produce it. For a fair comparison with another design, use the same total number of participants, treatment effect, outcome scale and variability, significance level, sidedness, analysis method, and follow-up conditions. If one calculation assumes complete outcomes and another includes loss to follow-up, their powers do not describe the same conditions.

For equal allocation, the sample size in the formula is the number in each group, not the total. If a trial has total sample size \(N\), then \(n=N/2\) when allocation is exactly equal. Confusing \(n\) with \(N\) changes the standard error and can substantially distort the power calculation.

Also distinguish a known-variance z test from the usual analysis with an estimated variance. Under the normal, equal-variance model, a pooled two-sample t test uses \(2n-2\) degrees of freedom and estimates the common variance from the data. Its critical value and power are not exactly the same as those of a z test. The normal formula is often a useful planning approximation, particularly when sample sizes are not small, but it should not be described as exact t-test power.

Diagnostics, Limitations, and a Common Pitfall

Before interpreting a calculated power, check whether its inputs and model describe the planned study. Confirm that the per-group sample size has been used, that the effect and standard deviation use the same outcome units, and that the intended test is two-sided if the two-tail formula is used. Ask whether outcomes are plausibly independent and whether a common variance and normal model are reasonable for the planned sample sizes. Strong skewness, outliers, clustering, unequal variances, or substantial missing outcomes can make this calculation a poor description of the test’s actual operating characteristics.

The calculation also treats \(\Delta\) and \(\sigma\) as known planning inputs, although in practice they are uncertain. Power can change sharply across plausible effect sizes, especially when the target effect is close to zero. It is therefore useful to show how power varies over a reasonable range of planning values rather than presenting one number as guaranteed performance. A normal-model calculation does not establish that an adaptive design has the same power: if allocation depends on observed outcomes or the group sizes are random, this fixed-allocation derivation is not that design’s power calculation.

Common mistake: substituting the total sample size \(N=2n\) for \(n\) in \(\sigma\sqrt{2/n}\), then reporting the resulting power as if it were for the planned equal-allocation trial. The standard error formula uses the number in each treatment group. For a total of 198 participants divided equally, the correct input is \(n=99\), giving \(SE=\sigma\sqrt{2/99}\), not \(\sigma\sqrt{2/198}\).

Another tempting shortcut is to report only the upper-tail probability under the alternative for a two-sided test. The correct calculation includes both rejection tails, as in the formula above. The omitted tail is often very small for a reasonably large positive effect, but the two-tail expression defines the planned test and should be used for the reported calculation.