Worked Sample-Size and Information Calculations

From a Planning Target to Group-Specific Enrollment

An information target becomes useful for trial planning when it can be translated into a number of participants in each group. The calculation depends on the maximum enrollment, the allocation ratio, and the outcome variability. An interim target may be stated as a fraction of maximum information or as an absolute information value; either way, the planned group sizes should be checked against the resulting information.

Consider a two-group trial comparing mean outcomes. Let \(n_T\) and \(n_C\) denote the treatment and control sample sizes, and let \(N=n_T+n_C\) be total enrollment. For independent participants, a difference in group means has variance

\operatorname{Var}(\bar{x}_T-\bar{x}_C) = \frac{\sigma_T^2}{n_T}+\frac{\sigma_C^2}{n_C}.

Here \(\sigma_T^2\) and \(\sigma_C^2\) are the planned outcome variances in the two groups. For this planning calculation, information is the reciprocal of that variance. It is greater when the estimated difference is more precise. If the outcome is measured in units, the variance is in squared units and information is in reciprocal squared units.

Definition: For a difference in two independent group means, the planned variance is \(V=\sigma_T^2/n_T+\sigma_C^2/n_C\), and the planned information is \(I=1/V\). The allocation ratio \(r\) is defined here as treatment enrollment divided by control enrollment: \(r=n_T/n_C\).

The calculation assumes independent observations within and between groups, the same outcome and analysis at the maximum and interim assessments, and planning variances that are appropriate for the population and measurement scale. It also assumes the allocation ratio is fixed as enrollment grows. Under these conditions, the variance formula gives the planning information directly. If the actual design uses a different variance model, that model must replace this one.

Allocation and Information at a Fixed Ratio

If the maximum total sample size is \(N_{\max}\) and the treatment-to-control ratio is \(r\), the corresponding group sizes before integer rounding are

n_{T,\max}=\frac{r}{1+r}N_{\max}, \qquad n_{C,\max}=\frac{1}{1+r}N_{\max}.

The same proportions apply at an interim target if enrollment follows the same ratio. Write \(q_T=r/(1+r)\) and \(q_C=1/(1+r)\), so that \(n_T=q_TN\) and \(n_C=q_CN\). Substituting these expressions into the variance formula shows why information fractions can be translated into sample-size fractions.

\begin{aligned} V(N) &=\frac{\sigma_T^2}{q_TN}+\frac{\sigma_C^2}{q_CN}\\ &=\frac{1}{N}\left(\frac{\sigma_T^2}{q_T}+\frac{\sigma_C^2}{q_C}\right). \end{aligned}

Define \(K=\sigma_T^2/q_T+\sigma_C^2/q_C\). Then \(V(N)=K/N\) and \(I(N)=N/K\). Thus, with fixed allocation and unchanged planning variances, information is proportional to total enrollment. This is a planning identity under the stated assumptions, not a guarantee that realized information will follow the same path.

Result: If the target is a fraction \(f\) of maximum information, the unrounded interim total is \(N_{\mathrm{int}}=fN_{\max}\), with group targets \(n_{T,\mathrm{int}}=fn_{T,\max}\) and \(n_{C,\mathrm{int}}=fn_{C,\max}\). Recalculate information using the final integer group sizes to check whether rounding still meets the target.

Worked Example: Equal Variances and a Half-Information Interim

Suppose a hypothetical trial plans a maximum total enrollment of 600 participants, with twice as many participants assigned to treatment as to control. The planned standard deviation is 12 outcome units in each group. Find the maximum group sizes and information, then calculate the group targets for an interim at 50% of maximum information.

The ratio is \(r=2\), so the maximum group sizes are \(n_{T,\max}=2(600)/3=400\) and \(n_{C,\max}=600/3=200\). The variance in both groups is \(12^2=144\) squared outcome units. Therefore,

V_{\max} = \frac{144}{400}+\frac{144}{200} = 0.3600+0.7200 = 1.080\ \text{units}^2, \qquad I_{\max}=\frac{1}{1.080}=0.9259\ \text{units}^{-2}.

At \(f=0.50\), the interim targets are \(n_{T,\mathrm{int}}=0.50(400)=200\) and \(n_{C,\mathrm{int}}=0.50(200)=100\), or 300 participants total. The variance at those counts is

V_{\mathrm{int}} = \frac{144}{200}+\frac{144}{100} = 0.7200+1.440 = 2.160\ \text{units}^2, \qquad I_{\mathrm{int}}=\frac{1}{2.160}=0.4630\ \text{units}^{-2}.

The information fraction is \(0.4630/0.9259=0.5000\), to the displayed precision. The group targets therefore achieve the planned half-information point under the assumed allocation and variances.

Allowing Different Outcome Variances

The same allocation calculations apply when the planned variances differ across groups. The allocation ratio determines group counts, while the two variances determine how much each group contributes to the variance of the difference. A smaller group can contribute substantially to uncertainty if its variance is large. The information should therefore be calculated from both terms rather than inferred from enrollment alone.

Worked Example: Unequal Variances and Integer Allocation Blocks

For a hypothetical trial, the maximum enrollment is 450, allocated in a 2:1 treatment-to-control ratio. The planned standard deviations are 10 outcome units for treatment and 14 outcome units for control. Find the maximum information and choose integer interim targets that meet or exceed 55% of that information while preserving the 2:1 ratio in whole allocation blocks.

The maximum group sizes are 300 treatment and 150 control. The planned maximum variance and information are

V_{\max} = \frac{10^2}{300}+\frac{14^2}{150} = \frac{1}{3}+\frac{98}{75} = 1.640\ \text{units}^2, \qquad I_{\max}=\frac{1}{1.640}=0.6098\ \text{units}^{-2}.

A 55% information target corresponds to an unrounded total enrollment of \(0.55(450)=247.5\). A 2:1 ratio in whole allocation blocks requires treatment and control counts of \(2b\) and \(b\), for an integer \(b\). The smallest block count that reaches the target total is \(b=83\), giving 166 treatment and 83 control participants, or 249 total.

Check the resulting information directly:

V_{\mathrm{int}} = \frac{100}{166}+\frac{196}{83} = \frac{50}{83}+\frac{196}{83} = \frac{246}{83} = 2.964\ \text{units}^2, \qquad I_{\mathrm{int}}=\frac{83}{246}=0.3374\ \text{units}^{-2}.

The achieved information fraction is \(0.3374/0.6098=0.5533\), or 55.33%, so the rounded targets meet the 55% goal. The next smaller whole block, 164 treatment and 82 control participants, would have variance \(100/164+196/82=3.000\) squared units and information \(1/3=0.3333\) reciprocal squared units. Its information fraction is \(0.3333/0.6098=0.5467\), which falls short. Rounding down would therefore miss the target.

Planning From an Absolute Information Target

A target may instead be supplied as an absolute information value. When allocation proportions are fixed, the information formula can be rearranged to solve for total enrollment. This is useful when the design specifies a precision or information requirement directly rather than as a fraction of maximum information.

Worked Example: Solving for Enrollment From Target Information

A hypothetical study plans a maximum of 500 participants, allocated 3:2 to treatment and control. The planned standard deviation is 8 outcome units in both groups. Find the group sizes corresponding to a target information of \(0.7500\) reciprocal squared units, and compare that target with maximum information.

At maximum enrollment, the group sizes are 300 treatment and 200 control. Thus,

V_{\max} = \frac{64}{300}+\frac{64}{200} = 0.5333\ \text{units}^2, \qquad I_{\max} = \frac{1}{0.5333} = 1.875\ \text{units}^{-2}.

The allocation proportions are \(q_T=3/5=0.60\) and \(q_C=2/5=0.40\). Consequently,

K=\frac{64}{0.60}+\frac{64}{0.40}=266.7, \qquad I(N)=\frac{N}{266.7}.

Solving \(0.7500=N/266.7\) gives \(N=200\). The group targets are \(n_T=0.60(200)=120\) and \(n_C=0.40(200)=80\). Checking those counts gives \(V=64/120+64/80=1.333\) squared units and \(I=1/1.333=0.7500\) reciprocal squared units. This target is \(0.7500/1.875=0.4000\), or 40% of maximum information, consistent with 200 being 40% of maximum enrollment.

Rounding, Diagnostics, and Limits

A planning calculation should end with a check using the actual integer enrollment targets, not only the unrounded result. If the target must not be undershot, round in a way that preserves the allocation rule and reaches at least the target information. Some ratios can be implemented only in blocks: a 2:1 ratio, for example, may require adding two treatment participants for every control participant. Calculate the variance and information at the selected counts, then compare the achieved information with the target.

The proportional conversion can fail when its assumptions do not hold. If treatment and control accrue at different rates, if the allocation ratio changes, or if missing outcomes reduce the usable sample sizes differently across groups, total enrollment alone does not determine information through the simple \(I(N)=N/K\) formula. Use the group-specific counts expected to contribute outcomes. Likewise, if the variances differ from the planning values, realized information can differ from planned information. These calculations describe expected planning information under specified assumptions; they do not establish that a particular interim analysis will have exactly that information.

The main alternative to using an information fraction is to specify the interim sample size directly. That is straightforward when a total enrollment target is the primary operational constraint. Information-based planning is more informative when comparing interim precision with maximum precision, especially when group variances or allocation ratios affect how much information a given enrollment provides. Neither approach removes the need to check the group-specific counts and variance model.

Mistake: Multiply the total interim enrollment by the allocation ratio and treat that product as the treatment sample size. Correction: Divide the total according to the ratio: for \(r:1\), treatment receives \(r/(1+r)\) of the total and control receives \(1/(1+r)\).
Mistake: Assume a rounded total sample size automatically achieves the desired information fraction. Correction: Recompute \(I=1/(\sigma_T^2/n_T+\sigma_C^2/n_C)\) using the rounded group counts and compare it with the target.
Mistake: Use enrollment fraction as information fraction even after the allocation, variance assumptions, or outcome availability changes. Correction: The equality follows only under the fixed-ratio and unchanged-variance assumptions; otherwise calculate information from the relevant group-specific variance model.

Key takeaway: Convert the allocation ratio into group proportions, calculate information from the group-specific variances and counts, and verify the target after integer rounding. Under a fixed allocation ratio and unchanged planning variances, information is proportional to total enrollment.

Check Your Understanding

Answer each question, then compare with the solutions below.

A hypothetical trial has a maximum of 360 participants and a 1:1 allocation. What are the maximum group sizes?
With 1:1 allocation, planned standard deviations of 5 units in both groups, and 100 participants per group, calculate the variance and information for the difference in means.
A design has maximum enrollment of 400 under a fixed 3:1 treatment-to-control ratio. What are the unrounded group targets at 25% of maximum information?
Explain why an interim enrollment fraction may not equal the information fraction if the allocation ratio changes.

Solutions. (1) The groups each receive \(360/2=180\) participants. (2) The variance is \(25/100+25/100=0.5000\) squared units, so information is \(1/0.5000=2.000\) reciprocal squared units. (3) At maximum, the group sizes are 300 treatment and 100 control; at 25%, the unrounded targets are 75 treatment and 25 control, 100 total. (4) With a changing ratio, the terms \(\sigma_T^2/n_T\) and \(\sigma_C^2/n_C\) do not both scale according to the same fixed proportions as total enrollment, so information need not be proportional to the enrollment fraction.