# Repeatability estimation using a mixed model with repeated measures by four raters

I am a medical doctor and definitely not an expert in statistics, although I think I do okay and I am familiar with R.

I have discussed the following issue with a statistician but I am still not sure whether we came up with the right solution. I hope one of you can help.

THE AIM:
I want to estimate the precision (repeatability) of a continuous variable measured on x-rays (PT) and evaluate the influence of different raters by specifying the variance between and within raters (inter- and intra-rater variance).

THE DESIGN:
I have had 4 raters measure the continuous variable on 67 x-rays twice, thus each x-ray has been measured 8 times in total.

THE DATA (Pre):

'data.frame':   536 obs. of  4 variables:
ID    : Factor w/ 67 levels
Rater : Factor w/ 4 levels
Time  : Factor w/ 2 levels
PT    : num  40.4 29.3 36 58.8 40.5 ...


THE MODEL we came up with was a linear mixed effect fit:

a <- lmer(PT~ (1|ID) + (1|Rater:ID), data=Pre)


From exploring the web I found that ":" is rarely used and that the model is equivalent to:

b <- lmer(PT~ (1|ID/Rater), data=Pre)


THE RESULT of either model a or b is:

Linear mixed model fit by REML ['lmerMod']
Formula: PT ~ (1 | ID/Rater)
Data: Pre

REML criterion at convergence: 2729.2

Scaled residuals:
Min      1Q  Median      3Q     Max
-5.7958 -0.2266 -0.0165  0.2214  4.8076

Random effects:
Groups   Name        Variance Std.Dev.
Rater:ID (Intercept)  4.630   2.152
ID       (Intercept) 97.178   9.858
Residual              2.695   1.642
Number of obs: 536, groups:  Rater:ID, 268; ID, 67

Fixed effects:
Estimate Std. Error t value
(Intercept)   18.611      1.214   15.34


THE INTERPRETATION according to the statistician was that:
The residual variance represents the intra-rater variance and the inter-rater variance is the sum of Rater:ID and residual variance.

THE QUESTION:
Are these interpretations valid?
My concern is especially if the residual variance is truly an expression of the intra-rater variance or if I should somehow include the "Time" variable in the model to specify the variance between the first and second measurements within raters within subjects.