Converting Target Information to Sample Size

From a Precision Target to a Sample Size

In a group sequential design, information is a measure of how precisely an accumulating estimate determines the treatment contrast. This tutorial addresses a practical planning step: if a design requires a specified information level, how many participants are needed to attain it? The answer depends on the estimator's variance, the allocation between groups, and the assumptions used to specify that variance.

The introductory material on statistical information and precision defines information for an estimator through its variance. Here, write the target as \(I^*\). For a scalar estimator \(\hat{\theta}\), the precision information used below is \(I=1/\operatorname{Var}(\hat{\theta})\), so the target is equivalent to requiring \(\operatorname{Var}(\hat{\theta})\leq 1/I^*\). Information has units reciprocal to variance; it is not a count of participants and does not, by itself, describe the probability of rejecting a null hypothesis.

This conversion requires the variance model to be appropriate for the planned data. For normal outcomes, the formulas below assume independent observations, the stated allocation, and fixed known variances. For binary outcomes, they assume independent Bernoulli outcomes within and between groups and specified planning probabilities. If variances or probabilities are uncertain, the calculation is conditional on the values selected for planning; it does not remove that uncertainty.

One-Sample and Two-Sample Normal Means

Suppose \(X_1,\ldots,X_n\) are independent with \(X_i\sim N(\mu,\sigma^2)\), where \(\sigma^2\) is fixed and known. Then \(\operatorname{Var}(\bar{x})=\sigma^2/n\), so the information about \(\mu\) in the sample mean is \(I(n)=n/\sigma^2\). Therefore a target \(I^*\) requires \(n\geq I^*\sigma^2\). Equivalently, a target standard error \(SE^*\) corresponds to \(I^*=1/(SE^*)^2\).

For two independent groups, let \(n_1\) and \(n_0\) be the group sample sizes, and let the contrast be \(\bar{x}_1-\bar{x}_0\). With fixed group variances \(\sigma_1^2\) and \(\sigma_0^2\), its variance is \(\sigma_1^2/n_1+\sigma_0^2/n_0\). Define total sample size \(n=n_1+n_0\) and allocation ratio \(r=n_1/n_0\), so \(n_1=rn/(1+r)\) and \(n_0=n/(1+r)\). Substitution gives the total sample-size formula.

When the variances are equal to \(\sigma^2\), the coefficient simplifies to \(\sigma^2(1+r)^2/r\). Equal allocation (\(r=1\)) then gives \(n\geq 4I^*\sigma^2\), with half of the total in each group before rounding. This is the total sample size, not the sample size in each arm.

Binary Outcomes: Planning for a Risk Difference

For independent binary outcomes, let \(p_1\) and \(p_0\) be the planned event probabilities in groups 1 and 0. Let \(\hat{p}_1-\hat{p}_0\) estimate the risk difference, with \(v_1=p_1(1-p_1)\) and \(v_0=p_0(1-p_0)\). Under fixed group sizes and independent Bernoulli outcomes, the variance of this difference is exactly \(v_1/n_1+v_0/n_0\). This variance identity does not require a large-sample approximation.

Using the allocation ratio \(r=n_1/n_0\), the variance becomes \((1+r)(v_1/r+v_0)/n\). Defining planning information as the reciprocal of this variance yields the following sample-size calculation. The probabilities must be stated because they determine the variance coefficient.

The formula gives precision information for the risk-difference estimator under the specified Bernoulli model. If normal-based inference is planned, its adequacy is a separate issue: the normal approximation generally requires sufficiently large expected event and non-event counts in each group. Adequate expected counts are not needed for the exact variance formula itself.

Why the Conversion Works—and What It Does Not Do

The central algebra is simple but useful to make explicit. Suppose the planned variance has the form \(C/n\), where \(C\) contains the variance assumptions and allocation ratio but not the total sample size. Then \(I(n)=1/(C/n)=n/C\). Requiring \(I(n)\geq I^*\) gives \(n/C\geq I^*\), hence \(n\geq I^*C\). The normal and binary formulas above differ in the coefficient \(C\); both follow from this same step.

An information target is not, on its own, a power calculation. Power also depends on the effect size under an alternative, the significance level, and the test or stopping rules. A design can attain a specified variance yet have inadequate power for a clinically important effect. Conversely, power-based planning can be expressed in terms of information after the effect and testing requirements have been specified. These are related planning approaches, but they answer different questions.

The calculation is also only as credible as its inputs. A poor variance estimate, misspecified event probabilities, incorrect allocation, or departures from independence can make planned and achieved information differ. Inspect the assumed inputs and the final integer allocation; for a binary design using normal inference, inspect expected event and non-event counts as well. More complex features such as clustering, missing outcomes, overdispersion, or estimated variances require variance adjustments beyond the formulas used here.

Common Pitfalls and Planning Checks

• Using the arm size where the total is required. In a two-group design, the displayed formulas solve for \(n=n_1+n_0\). Calculate \(n_1\) and \(n_0\) from the allocation ratio afterward.
• Ignoring allocation in the variance. The same total \(n\) can yield different contrast variances when the allocation differs, particularly with unequal group variances or event probabilities.
• Treating assumed binary probabilities as observed facts. They are planning inputs. Recalculate the variance under alternative plausible probabilities to see how sensitive the required total is.
• Calling the Bernoulli variance formula approximate. With independent Bernoulli observations and fixed group sizes, the variance of the difference in sample proportions is exact. A normal approximation may be used for inference, but that is a separate assumption.
• Rounding down or failing to verify. The continuous solution is a lower bound; integer allocation can change information. Substitute the selected group sizes into the variance formula to check the result.
• Equating target information with adequate power. Information describes precision under a variance model. It does not encode an effect size or the decision procedure.

A useful final report states the target information, the estimator and its units, the variance assumptions, the allocation ratio, the continuous sample-size result, the rounded group sizes, and the achieved information after rounding. This makes the conversion transparent and exposes whether the design depends heavily on uncertain planning inputs.