Sample-Size Calculations for Continuous Outcomes

Why Continuous-Outcome Sizing Needs Two Calculations

For a continuous outcome, the assumed treatment difference and outcome variance determine how much information is needed to distinguish treatment from control. In a multi-arm multi-stage (MAMS) design, that information must also be assigned to stages. The resulting maximum enrollment is not, by itself, a guarantee of the target power: power depends on the stopping boundaries and on the correlation among the repeated treatment-control comparisons.

This tutorial develops a transparent marginal calculation for a normally distributed outcome, then shows how to turn its cumulative sample-size target into stage sizes. Use that calculation as a planning building block. For the final MAMS design, the target power must be checked under the actual boundaries, shared-control structure, and interim stopping rules.

Definition: Let $j$ index treatment arms and $k$ index stages. Write $n_{0,k}$ for the number assigned to the shared control in stage $k$, and $n_{j,k}$ for the number assigned to treatment $j$ in that stage. The treatment-to-control allocation ratio is $r_j=n_{j,k}/n_{0,k}$, when that ratio is held constant across stages. Cumulative sample sizes through stage $k$ are $\sum_{\ell=1}^{k}n_{0,\ell}$ for control and $\sum_{\ell=1}^{k}n_{j,\ell}$ for treatment $j$. Maximum enrollment is the total planned enrollment if all arms continue through all stages.

The Marginal Sample-Size Formula

Assume independent participants, normally distributed outcomes with common variance $\sigma^2$ in each arm, and a planned treatment effect $\delta>0$, defined as the treatment mean minus the control mean. For now, treat $\sigma$ as known, or as a fixed planning value estimated from reliable external information. With cumulative sample sizes $n_j$ and $n_0$, the estimated treatment-control difference has standard error

SE=\sigma\sqrt{\frac{1}{n_j}+\frac{1}{n_0}}.

A positive treatment effect produces a positive standardized statistic when the difference is divided by this standard error. Suppose a single, one-sided comparison is assessed at a critical value $c$, and the desired power for detecting $\delta$ at that assessment is $1-\beta$. Under the assumed normal model, the statistic has mean $\delta/SE$ and variance 1 under the planned effect. Requiring its mean to exceed the critical value by $z_{1-\beta}$ standard errors gives the usual planning equation.

\frac{\delta}{SE}=c+z_{1-\beta}.

For a single test with one-sided significance level $\alpha$, $c=z_{1-\alpha}$. In a MAMS design, use the applicable design boundary for the comparison and look being planned; it need not equal the critical value for an unadjusted single test.

Result: If $n_j=r_j n_0$, the control sample size required for the marginal target is $$ n_0= \frac{\sigma^2(1+1/r_j)(c+z_{1-\beta})^2}{\delta^2}, \qquad n_j=r_jn_0. $$ For a one-sided single test, substitute $c=z_{1-\alpha}$. These are unrounded, cumulative sample-size targets for one treatment-control comparison.

The formula follows by substituting the standard error into the planning equation and squaring both sides. The ratio affects control allocation through $1+1/r_j$. If considering the total enrollment in a two-arm comparison, multiply the control requirement by $1+r_j$; the corresponding ratio factor is $(1+r_j)(1+1/r_j)=2+r_j+1/r_j$, which is minimized at equal allocation. Thus the control-size factor alone is not a measure of the total sample-size cost of an allocation ratio.

Worked Example: Equal-Allocation Marginal Target

Suppose a hypothetical continuous outcome has planning standard deviation $\sigma=10$ units. The design aims to detect a treatment-control difference of $\delta=4$ units with 80% marginal power at a one-sided single-test level of $\alpha=0.025$. Assume equal allocation, so $r=1$. The normal quantiles are $z_{0.975}=1.960$ and $z_{0.80}=0.8416$.

Using the marginal formula, $$ n_0= \frac{10^2(1+1)(1.960+0.8416)^2}{4^2} =12.5(2.8016)^2 =98.1120. $$

The unrounded target is 98.1120 participants in each arm. Rounding each arm up gives 99 control participants and 99 treatment participants for the single comparison. This calculation meets the stated marginal target under its assumptions; it does not yet establish the power of a multi-stage design with repeated looks.

From a Comparison to MAMS Enrollment

With $K$ treatment arms sharing a control and a common ratio $r$, a stage with $n_{0,k}$ control participants has $r n_{0,k}$ participants in each treatment arm before integer rounding. Its planned total is therefore $n_{0,k}(1+Kr)$, not $K$ separate control groups. For unequal ratios, the corresponding total is $n_{0,k}(1+\sum_{j=1}^{K}r_j)$. These expressions give maximum enrollment once summed across all planned stages.

A simple way to construct stage increments is to choose information fractions $f_k$ that increase to 1, with $f_0=0$, and allocate cumulative sample size in proportion to those fractions. With fixed allocation ratios and common variance, the information for one treatment-control contrast is proportional to its cumulative control size. Consequently, setting $n_{0,k}$ close to $(f_k-f_{k-1})n_{0,\mathrm{final}}$ gives the intended approximate information increments. Rounding and design constraints can alter the fractions slightly.

Worked Example: Allocating a Target Across Three Stages

Continue with the hypothetical target of 99 participants per arm from the previous example. Suppose a two-treatment design uses one shared control, equal allocation, and planned cumulative information fractions of 0.50, 0.80, and 1.00. For this illustration, treat 99 per arm as the marginal final target; do not interpret it as an already verified MAMS power calculation.

The control increments are approximately $0.50(99)=49.5$, $(0.80-0.50)(99)=29.7$, and $(1.00-0.80)(99)=19.8$. Choose integer control increments of 50, 30, and 19, giving cumulative control sizes of 50, 80, and 99. With equal allocation, plan the same increments in each treatment arm.

The stage enrollments across the two treatment arms and shared control are $3(50)=150$, $3(30)=90$, and $3(19)=57$. Maximum enrollment is $150+90+57=297$ participants. If all arms stop early according to the design rules, realized enrollment can be lower. The fractions after rounding are $50/99$, $80/99$, and $99/99$, close to but not exactly the nominal fractions.

The allocation is arithmetically consistent, but its actual power still needs to be calculated using the MAMS boundaries and the joint distribution of the stagewise statistics.

Shared Control, Ratios, and a Second Calculation

For several treatments, the shared control contributes to every treatment-control standard error. This saves participants relative to maintaining a separate control group for each comparison, but it also correlates the treatment-control statistics. Therefore, applying the one-comparison formula separately to every arm does not account for the joint chance of declaring one or more treatments effective, nor for the chance that a particular treatment crosses a boundary before stopping.

For an initial marginal calculation, use each arm's assumed $\delta_j$, the common variance, its allocation ratio $r_j$, and the relevant critical value. If the targets differ across treatments, each can imply a different control requirement. A design with one shared control must choose a control schedule that satisfies the planning needs of all arms, then evaluate the complete design. The arm with the largest marginal control requirement is a useful sizing constraint, but it does not replace the joint power calculation.

Worked Example: Unequal Allocation With a Shared Control

Suppose a hypothetical normally distributed outcome has planning standard deviation $\sigma=9$ units, and an assumed treatment effect is $\delta=3$ units. Use treatment-to-control ratio $r=1.5$, one-sided single-test $\alpha=0.025$, and marginal power 90%. Thus $z_{0.975}=1.960$ and $z_{0.90}=1.2816$.

The required control size for this one comparison is $$ n_0= \frac{9^2(1+1/1.5)(1.960+1.2816)^2}{3^2} =15(3.2416)^2 =157.6196. $$

The corresponding treatment target is $n_j=1.5(157.6196)=236.4294$. Rounding each arm upward gives 158 control participants and 237 participants in the treatment arm. Maximum enrollment for this two-arm comparison is $158+237=395$.

If three treatment arms use this same ratio and marginal target with one shared control, the illustrative rounded maximum is $158+3(237)=869$, not three times 395, because there is only one control group. This count is an enrollment calculation, not proof of 90% power for all arms or of a specified familywise error rate.

Checking the MAMS Power Claim

The marginal formula describes one comparison at one assessment, or a final assessment under a specified critical value. A MAMS design makes decisions at multiple stages and may stop arms early. A treatment can meet an efficacy boundary at an interim look or remain under consideration until a later look. The probability of efficacy therefore depends on the stagewise boundaries and on the dependence among statistics, especially where treatments share control observations.

In practice, the marginal target provides a starting scale for the cumulative sample sizes. The planned stage increments and integer allocations are then assessed under the actual MAMS decision rules. For each treatment, power is the probability, under its assumed effect configuration, of meeting the design's efficacy rule. The familywise type I error requirement concerns false efficacy declarations under the relevant null configuration; it is not automatically controlled by substituting the ordinary single-test $z_{1-\alpha}$ into every comparison. Use the critical boundaries specified for the design and evaluate the resulting stagewise probabilities.

A concise design check is:

Specify the outcome model.
State the outcome units, assumed effects, common or arm-specific variance assumptions, and allocation ratios.

Calculate marginal targets.
Use an appropriate critical value and desired marginal power to obtain unrounded cumulative sizes for each treatment-control contrast.

Choose stage increments.
Distribute the planned information across stages and calculate cumulative and maximum enrollment, accounting for shared control.

Verify the actual design.
Check power and error rates under the MAMS boundaries and dependence structure, then recalculate enrollment after any necessary changes.

Assumptions, Diagnostics, and Alternatives

The calculation assumes independent observations, a correctly specified treatment-control mean difference, a suitable common variance, and normally distributed outcomes (or a sufficiently accurate normal approximation for the sample means). It also assumes the allocation ratios and analysis contrast match the planned design. If outcomes are clustered, repeated within participants, strongly skewed in small samples, or have substantially different variances across arms, the standard error formula may not apply as written. An estimated variance used in planning also carries uncertainty that the plug-in calculation ignores.

Check the arithmetic and design before interpreting a target:

Units: $\sigma$ and $\delta$ must use the same outcome units, so the calculated sample size is dimensionless.
Scale: the target grows in proportion to $\sigma^2$ and inversely with $\delta^2$. A surprisingly small target can indicate a variance or effect entered on the wrong scale.
Allocation: verify that $r_j$ is treatment size divided by control size. For total two-arm enrollment, use the factor $(1+r_j)(1+1/r_j)$, not $1+1/r_j$ alone.
Stages and rounding: ensure stage increments sum to the stated cumulative sizes and that treatment and control counts are feasible integers. Recheck information fractions after rounding.
Boundaries: do not claim MAMS power from a single-look formula when repeated looks, shared controls, or multiplicity-adjusted boundaries change the decision probabilities.

A fixed, single-stage calculation is a useful benchmark and is simpler to interpret, but it does not represent the operating characteristics of an early-stopping design. Conversely, a MAMS calculation accounts for stopping and correlation but depends on the assumed effect configuration, variance, allocation, and boundaries. A sensible report gives both the assumptions and the maximum planned enrollment, and distinguishes the marginal benchmark from the verified MAMS power.

Common pitfall: “The formula gives 80% power, so the three-stage design has 80% power.” The formula gives the specified marginal power only for the comparison and critical value used in its derivation. Interim looks and shared-control dependence affect the probability of crossing the design's efficacy boundaries. Calculate power under those actual rules before making a MAMS power claim.

Check Your Understanding

Answer each question, then compare with the solutions below.

A one-sided single comparison has $\sigma=8$, $\delta=2$, equal allocation, $\alpha=0.05$, and target power 90%. Using $z_{0.95}=1.6449$ and $z_{0.90}=1.2816$, calculate the unrounded sample size per arm and the rounded size.
For $\sigma=6$, $\delta=3$, $r=2$, one-sided $\alpha=0.025$, and power 80%, calculate the unrounded control target. Use $z_{0.975}=1.960$ and $z_{0.80}=0.8416$.
A design has three treatment arms, one shared control, equal allocation, and control stage increments of 20, 15, 15, and 10. What is maximum enrollment if all arms continue through four stages?
Why is a marginal final-look power calculation insufficient to establish the power of a MAMS design with interim efficacy boundaries and a shared control?

Solutions. (1) The per-arm target is $8^2(2)(1.6449+1.2816)^2/2^2=32(2.9265)^2=274.06$. Round up to 275 participants per arm. (2) The control target is $6^2(1+1/2)(1.960+0.8416)^2/3^2=6(2.8016)^2=47.09$, or 48 control participants after rounding. The treatment target is twice the unrounded control target, $94.19$, so round up to 95 treatment participants. (3) Total control enrollment is $20+15+15+10=60$. Equal allocation gives 60 participants in each of three treatment arms, so maximum enrollment is $4(60)=240$. (4) Interim boundaries and early stopping determine the probability of declaring efficacy at each look, while shared control induces dependence among treatment-control statistics. The marginal formula represents neither the complete stopping rule nor that joint dependence; power must be evaluated under the actual design.