Signature
n = 2 * (z_a + z_b)^2 * sigma^2 / delta^2
| Inputs | Definition | Unit |
|---|---|---|
z_a | 1.96 for 5 per cent | none |
z_b | 1.2816 for 90 per cent power, 0.8416 for 80 per cent | none |
sigma | Unit: score points | — |
delta | For example the between-group MID | score points |
n | Unit: patients | — |
|---|
Function
Anchor-based thresholds for a minimal clinically important difference and their use for responder shares and trial size
Maps a meaningful step on an external anchor, usually a patient's global rating of change, onto the score scale of an outcome measure. The between-group estimate subtracts the mean change of patients reporting no change from that of patients reporting a small change; the ROC cut-off chooses the change score that best separates anchor-improved patients from the rest; the 95 per cent limit cut-off sits above almost all unchanged patients. A threshold then gives the share of patients reaching it in each arm, and a between-group difference sets the target difference of a sample size calculation. Distribution-based yardsticks are HE-FM-DMID-001 to HE-FM-DMID-003 on the Distribution-Based MID page. Notation follows the Minimal Clinically Important Difference article.
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Implementations
Excel
Patients per arm from the significance level, power, standard deviation and target difference
With AlphaLevel, PowerLevel, SDOutcome and TargetDiff named, the formulas return the two normal quantiles, the unrounded number per arm and the number rounded up, held in ZAlpha, ZBeta, NPerArmRaw and NPerArm.
=NORM.S.INV(1-AlphaLevel/2); =NORM.S.INV(PowerLevel); =2*(ZAlpha+ZBeta)^2*SDOutcome^2/TargetDiff^2; =ROUNDUP(NPerArmRaw,0)
Assumptions
Two equal arms, a continuous outcome and a two-sided test
The outcome is approximately normal with the same standard deviation in both arms, the arms are of equal size and the analysis compares means with a two-sided test; other designs need other formulas.
Target difference important to a stakeholder group
DELTA2 asks for a target difference that is important to at least one key stakeholder group and realistic; it need not be the minimum important value if a larger difference is realistic.
Worked examples
Between-group MID of 6.0 points with standard deviation 12, 5 per cent significance and 90 per cent power
Twice (1.96 + 1.2816) squared times 144 over 36 gives 84.1, so 85 patients per arm are needed, as in the article.
z_a = 1.96; z_b = 1.2816; sigma = 12; delta = 6; n = 84.1
Target difference of 4 points in the same trial
A 4-point target gives 189.1, or 190 patients per arm, as in the article, so the chosen difference drives trial size and cost.
z_a = 1.96; z_b = 1.2816; sigma = 12; delta = 4; n = 189.1
Standardised difference of 0.78 with 80 per cent power
With the outcome measured in standard deviations, a difference of 0.78 at 5 per cent significance and 80 per cent power needs 25.8, or 26 per arm, the figure Whitley and Ball give.
z_a = 1.96; z_b = 0.8416; sigma = 1; delta = 0.78; n = 25.8
Common errors
Using a within-person threshold as the target difference between arms
Taking the 8.0-point mean change of the minimally improved group as delta would give 47.3, or 48 per arm (computed here for illustration), against 85 for the between-group MID of 6.0, so the trial would be underpowered for the difference that matters between arms.
Setting the target difference from a distribution-based value
DELTA2 favours anchor and opinion-seeking methods and says the distribution method should not be used; a half standard deviation target of 6 points would coincide with the MID here only by chance.
Sources
Whitley and Ball on the sample size formula for comparing two means
Whitley E, Ball J. Statistics review 4: sample size calculations. Critical Care. 2002;6(4):335-341. doi:10.1186/cc1521 (full text read; the sample size equation is printed as an image, and the article's stated result was recomputed here). Section on sample size formulae: for comparing means in two groups of equal size the number per group is 2 c_(p,power) divided by the squared standardised difference d, the target difference in standard deviations; Table 2 gives c of 7.9 for 80% power and 10.5 for 90% power at P = 0.05, and a standardised difference of 0.78 with 80% power needs 26 per arm.
DELTA2 guidance on the target difference
Cook JA, Julious SA, Sones W, Hampson LV, Hewitt C, Berlin JA, et al. DELTA2 guidance on choosing the target difference and undertaking and reporting the sample size calculation for a randomised controlled trial. BMJ. 2018;363:k3750. doi:10.1136/bmj.k3750 (full text read). Recommendations: the target difference for a definitive trial should be one considered important to at least one key stakeholder group and does not necessarily have to be the minimum value; the anchor and opinion-seeking methods are to be favoured and the distribution method should not be used; under the conventional approach, halving the target difference quadruples the sample size for a two-arm parallel-group trial with a continuous outcome.
Canonical Identity
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