Signature
SEM = SD * sqrt(1 - R)
| Inputs | Definition | Unit |
|---|---|---|
SD | Standard deviation of the instrument's scores in the sample | instrument units |
R | Test-retest reliability in stable patients, or Cronbach's alpha | coefficient, no greater than 1 |
SEM | Typical size of the measurement error in one score | instrument units |
|---|
Function
Distribution-based yardsticks for a minimal important difference in patient-reported and utility scores
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.
Try this function
Implementations
Excel
Standard error of measurement from named cells
With ScoreSD and Reliab named, the formula returns the standard error of measurement, held in SEMVal.
=ScoreSD*SQRT(1-Reliab)
Assumptions
Reliability for the SEM estimated in a stable population
R is a reliability coefficient from the same population, preferably test-retest reliability in patients whose health has not changed; with Cronbach's alpha the SEM reflects only the variability of the instrument at one measurement.
Worked examples
SEM of an EQ-5D-5L index with SD 0.20 and reliability 0.80
With a standard deviation of 0.20 and test-retest reliability of 0.80, the SEM is 0.20 x 0.447 = 0.0894, shown as 0.089 in the article.
SD = 0.2; R = 0.8; SEM = 0.0894
SEM equal to half a standard deviation at reliability 0.75
At a reliability of 0.75 the SEM of a score with standard deviation 0.20 is 0.10, the same as half a standard deviation, as de Vet and colleagues note and the article repeats.
SD = 0.2; R = 0.75; SEM = 0.1
Common errors
Using one SEM as the minimal important change in every setting
De Vet and colleagues summarise studies in which one SEM matched anchor-based standards and a set of musculoskeletal studies using ROC methods in which the important change was 2.3 or 2.6 SEM, and conclude that one SEM equalling the minimally important change is not a universal truth.
Computing the SEM with Cronbach's alpha for change over time
Alpha comes from one measurement and describes internal consistency; de Vet and colleagues regard test-retest reliability in a stable population as more appropriate for change between two time points, and a higher alpha gives a smaller SEM that can understate the error in a change score.
Sources
De Vet and colleagues on the SEM formula and the reliability to use
de Vet HC, Terwee CB, Ostelo RW, Beckerman H, Knol DL, Bouter LM. Minimal changes in health status questionnaires: distinction between minimally detectable change and minimally important change. Health and Quality of Life Outcomes. 2006;4:54. doi:10.1186/1477-7525-4-54 (full text read). Comparison of SEM with anchor-based approaches: SEM = SD x sqrt(1 minus R), with test-retest reliability or Cronbach's alpha as the reliability parameter, test-retest reliability representing temporal stability and so more appropriate than alpha; for one set of studies the important change corresponded to 2.3 or 2.6 SEM, so one SEM equalling the important change is not a universal truth. Section on the distinction between minimally detectable and minimally important changes: using the formula, 1 SEM equals 0.5 SD when the reliability of the instrument is 0.75.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0