Signature
z_C = (c - mu_C) / sigma; z_T = (c - mu_T) / sigma; P_C = 1 - Phi_C; P_T = 1 - Phi_T; RD = P_T - P_C
| Inputs | Definition | Unit |
|---|---|---|
c | For example the within-person ROC threshold | score points |
mu_C | Unit: score points | — |
sigma | Unit: score points | — |
mu_T | Unit: score points | — |
Phi_C | Entered from NORM.S.DIST(z_C, TRUE) or pnorm(z_C) | probability |
Phi_T | Entered from NORM.S.DIST(z_T, TRUE) or pnorm(z_T) | probability |
z_C | Unit: standard deviations | — |
|---|---|---|
z_T | Unit: standard deviations | — |
P_C | Unit: proportion | — |
P_T | Unit: proportion | — |
RD | Unit: proportion | — |
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.
Try this function
Implementations
Excel
Responder shares and their difference from named cells
With CutOff, MeanCtrl, MeanTrt and SDChange named, the formulas return the two standardised distances, the two responder shares and their difference, held in ZCtrl, ZTrt, PropCtrl, PropTrt and RespDiff.
=(CutOff-MeanCtrl)/SDChange; =(CutOff-MeanTrt)/SDChange; =1-NORM.S.DIST(ZCtrl,TRUE); =1-NORM.S.DIST(ZTrt,TRUE); =PropTrt-PropCtrl
Assumptions
Normal changes with a common standard deviation in both arms
Changes in each arm are normally distributed and differ only in their means, the simple model Snapinn and Jiang use to compare responder and mean-based analyses; with other shapes the shares have to come from the observed distribution of change.
Worked examples
Mean changes of 2 and 6 points with standard deviation 12 and a 6-point threshold
The control mean is a third of a standard deviation below the threshold, so 1 minus Phi(0.333), about 0.369, of control patients reach it; the treated mean equals the threshold, so half do, and the responder difference is about 0.131, as in the article, although the 4-point difference in means is below the between-group MID of 6.0.
c = 6; mu_C = 2; mu_T = 6; sigma = 12; Phi_C = 0.63056; Phi_T = 0.5; z_C = 0.3333; z_T = 0; P_C = 0.3694; P_T = 0.5; RD = 0.1306
Same trial with the 10.2-point limit as threshold
At a threshold of 10.2 points the shares fall to about 0.247 with control and 0.363 with treatment, a difference of about 0.116 (computed here for illustration), so the threshold chosen changes the responder shares a model would use.
c = 10.2; mu_C = 2; mu_T = 6; sigma = 12; Phi_C = 0.7528; Phi_T = 0.63683; z_C = 0.6833; z_T = 0.35; P_C = 0.2472; P_T = 0.3632; RD = 0.116
Common errors
Treating a mean difference below the MCID as no benefit
The Cochrane Handbook warns that a difference smaller than the MID may be read as trivial when a substantial proportion of patients achieved an important benefit; in the article's example a 4-point mean difference comes with about 13 percentage points more treated patients reaching the threshold.
Replacing the comparison of means by a responder analysis
Under the normal model of Snapinn and Jiang a responder analysis needed about 60 per cent more participants than a comparison of means even with the threshold midway between the arm means, and with a large enough sample any small difference in responder rates becomes statistically significant.
Sources
Snapinn and Jiang on responder analyses under a normal model
Snapinn SM, Jiang Q. Responder analyses and the assessment of a clinically relevant treatment effect. Trials. 2007;8:31. doi:10.1186/1745-6215-8-31 (full text read). Methods: with the measurement normally distributed with known variance and response defined as reaching a threshold x_0, power and sample size for a responder comparison depend on the mean difference and on x_0; even with the threshold midway between the group means the responder analysis needed approximately 60% more participants; with a large enough sample size any arbitrarily small difference in response rates can be statistically significant.
Guyatt and colleagues on the two steps from an MID to a proportion
Guyatt GH, Osoba D, Wu AW, Wyrwich KW, Norman GR. Methods to explain the clinical significance of health status measures. Mayo Clinic Proceedings. 2002;77(4):371-383. doi:10.4065/77.4.371 (abstract read). Abstract: the first step establishes the smallest change in score that patients consider, on average, to be important; the second estimates the proportion of patients who have achieved that minimum important difference.
Cochrane Handbook on results below the MID
Schunemann HJ, Vist GE, Higgins JPT, Santesso N, Deeks JJ, Glasziou P, Akl EA, Guyatt GH. Chapter 15: Interpreting results and drawing conclusions. In: Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024 (full text of the chapter read). Section 15.5.3.5: expressing results in MID units is risky in that a difference less than the MID may be interpreted as trivial when a substantial proportion of patients may have achieved an important benefit.
FDA 2009 on cumulative distributions of change in each arm
US Food and Drug Administration. Guidance for Industry. Patient-Reported Outcome Measures: Use in Medical Product Development to Support Labeling Claims. Silver Spring, MD: FDA; December 2009 (full text read). Section IV.E: the empiric evidence for any responder definition is derived using anchor-based methods; a display of the cumulative distribution of responses in each arm may be preferable to attempting to provide categorical definitions of responders.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0