SEM - Standard Error of Measurements

- Regarded as a “random variation in a measure when individual is tested many times“ (Hopkins, 2000, p.2)

- Measures the absoluet reliability

Formula:

\(SEM = SD\times \sqrt{(1-ICC)}\)

\(SD\) = Pooled standard deviation of measurements;

\(ICC\) = Reliability;

Note that according to the formula above, the calculation various by the form of ICC.

A generalized formula to calculate SEM suggested by Hopkins (2000):

\(SEM = \sqrt{S^2_{between trials} + S^2_{residual}}\)

Example in R

Untitled.utf8 
set.seed(1234)
library("rel")
data <- cbind(sample(50:100,200,replace=TRUE), sample(50:100,200,replace=TRUE))
head(data)
##      [,1] [,2]
## [1,]   77   53
## [2,]   65   92
## [3,]   71   64
## [4,]   86   66
## [5,]   93   92
## [6,]   96   88

Using ICC:

library("psych")
ICC(data)
## Call: ICC(x = data)
## 
## Intraclass correlation coefficients 
##                          type    ICC F df1 df2    p lower bound upper bound
## Single_raters_absolute   ICC1 0.0016 1 199 200 0.49       -0.11        0.12
## Single_random_raters     ICC2 0.0039 1 199 199 0.48       -0.11        0.12
## Single_fixed_raters      ICC3 0.0039 1 199 199 0.48       -0.11        0.12
## Average_raters_absolute ICC1k 0.0031 1 199 200 0.49       -0.26        0.21
## Average_random_raters   ICC2k 0.0078 1 199 199 0.48       -0.25        0.21
## Average_fixed_raters    ICC3k 0.0078 1 199 199 0.48       -0.25        0.21
## 
##  Number of subjects = 200     Number of Judges =  2
SEM = sd(data) * sqrt(1-0.0078)
SEM
## [1] 14.17431

Using Generalized Formula:

sem(data=data, type="mse", conf.level=0.95)
## Call:
## sem(data = data, type = "mse", conf.level = 0.95)
## 
##       Estimate LowerCB UpperCB
## Const   14.185  13.051   15.32
## 
## Confidence level = 95%
## Observations = 2
## Sample size = 200

Live Example: Try this yourself!

References:

Hopkins, W. G. (2000). Measures of Reliability in Sports Medicine and Science. Sports Medicine, 30(1), 1-15. http://dx.doi.org/10.2165/00007256-200030010-00001