Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Distribution-based yardsticks for a minimal important difference in patient-reported and utility scores

MID_SD = k_SD * SD_0; SEM = SD * sqrt(1 - R); MDC = 1.96 * sqrt(2) * SEM; RC = (x_2 - x_1) / (sqrt(2) * SEM)

Expresses an important or detectable change in a questionnaire or utility score as a multiple of the spread of scores or of the instrument's measurement error: half a baseline standard deviation, a benchmark effect size, one standard error of measurement, the minimally detectable change and the reliable change index for one person. These yardsticks describe precision, not what patients value, and methodological and regulatory guidance ranks them behind anchor-based estimates. Standardised mean differences themselves are HE-FM-CD-001 on the Cohen's d page. Notation follows the Distribution-Based MID article.

  • Standard error of measurement from the score standard deviation and reliability

    SEM = SD * sqrt(1 - R)

    The standard error of measurement (SEM) is the standard deviation of scores times the square root of one minus a reliability coefficient, and estimates in the instrument's units how far an observed score is likely to stray from a person's true score. Test-retest reliability in a stable population suits change between two time points better than Cronbach's alpha. One SEM equals half a standard deviation when reliability is 0.75.

  • Minimally detectable change and reliable change index from the standard error of measurement

    S_diff = sqrt(2) * SEM; MDC = 1.96 * S_diff; RC = (x_2 - x_1) / S_diff

    A change score involves two measurements, so its standard error is the square root of 2 times the SEM. The minimally detectable change (MDC) is 1.96 of these standard errors, the 95 per cent limit for no change, and the reliable change index (RC) of Jacobson and Truax divides one person's change by the same standard error; a value beyond 1.96 in either direction would arise from measurement error alone with probability of 5 per cent or less. Both thresholds apply to one person's change, not to a difference in group means.