Anchor-based thresholds for a minimal clinically important difference and their use for responder shares and trial size
MID_between = d_M - d_S; c_star = argmin_c [(1 - Se_c) + (1 - Sp_c)]; c_95 = d_S + 1.645 * s_S; n = 2 * (z_a + z_b)^2 * sigma^2 / delta^2
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.
Between-group minimal important difference from mean changes in two anchor groups
MID_between = d_M - d_S
Subtracts the mean change of patients who rate themselves about the same from the mean change of patients who rate themselves a little better (or a little worse, for deterioration). The mean change of the minimally improved group on its own is the Jaeschke estimate of a within-person change; the difference between the two groups is, in the terms of Terwee and colleagues, a minimal important difference between groups, the quantity suited to a difference between trial arms.
Sensitivity, specificity and misclassification of one candidate change-score cut-off against an anchor
Se_c = a_c / N_I; Sp_c = (N_N - b_c) / N_N; M_c = (1 - Se_c) + (1 - Sp_c); J_c = Se_c + Sp_c - 1
Treats the change score as a diagnostic test and the anchor as the reference standard. At a candidate cut-off, sensitivity is the share of anchor-improved patients whose change reaches it and specificity the share of all other patients whose change falls below it; the ROC cut-off is the candidate with the smallest sum of the two misclassification proportions, the same as the largest Youden index.
Ninety-five per cent limit cut-off above the change scores of patients not importantly changed
c_95 = d_S + 1.645 * s_S
Places the threshold for improvement at the mean change of the patients the anchor classes as not importantly changed plus 1.645 standard deviations of their change scores, the one-sided 95 per cent point of a normal distribution. De Vet and colleagues note that it corresponds to 95 per cent specificity on the ROC curve, so it is stricter than the ROC cut-off.
Shares of each trial arm reaching a responder threshold when changes are normal with a common standard deviation
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
If changes in each arm are normally distributed with a common standard deviation, the share of an arm reaching a threshold is one minus the standard normal distribution function at the threshold's distance from the arm's mean in standard deviations, and the responder difference is the treated share minus the control share. The site's calculator has no normal distribution function, so Phi_C and Phi_T, its values at z_C and z_T, are entered as inputs (NORM.S.DIST(z, TRUE) in Excel, pnorm(z) in R).
Patients per arm for a two-arm trial with a continuous outcome and a minimal important difference as the target
n = 2 * (z_a + z_b)^2 * sigma^2 / delta^2
The usual normal approximation for two equal arms and a two-sided test sets the number per arm to twice the squared sum of the two standard normal quantiles, times the outcome variance, over the squared target difference. Because the target is a difference between arms, a between-group estimate such as MID_between is the relevant MCID; halving it quadruples the trial.