1. Random-effects model based on Early vs
Random-effects model based on Early vs. Late Infantile Strabismus Surgery Study
H.J. Simonsz 1, M.J.C. Eijkemans 2
1 Department of Ophthalmology, Erasmus Medical Center Rotterdam
2 Department of Public Health, Erasmus Medical Center Rotterdam

2. Statistical procedure to estimate the Random-Effects Model
(dr. ir. M.J.C. Eijkemans, Dept. Public Health, Erasmus MC Rotterdam)
Fixed effects: hoek ~ months + sfeq1
Value Std.Error DF t-value p-value
(Intercept) <.0001
months
sfeq
Correlation:
(Intr) months
months
sfeq
Random effects:
Formula: ~ months + sfeq1 | patnr
Structure: General positive-definite
StdDev Corr
(Intercept) (Intr) months
months
sfeq
Residual

3. Prediction of angle of individual child in future, given angle and spherical equivalent at given age.
The random effect Zb for the individual child is defined as the deviation of average, the fixed effect Xb.  Define Z as the vector {1, months, sfeq}, and the matrix D as the covariance-matrix of the random-effects estimations.
The model formula for the real angle of strabismus y at time h for child i
1st term: fixed-effects = average angle for a child with covariates xi,h
2nd term: random-effects = deviation for the individual child relative to average
child (regression-coefficient b with index i)
3rd term: residual variance (3.84)2=14.75 degrees.
The variance around average is equal to the random-effects variance plus the residual variance and depends on covariates (z). The matrix D is filled with ELSSS values, I(n) is the unit matrix of n, the number of orthoptic measurements per child.

4. The variance around the prediction:
Consists of uncertainty in the estimations, random effects and the residuals. The total variance equals the variance of the difference between the predicted and the real value.
The variance matrix of the fixed effects is obtained by combining the SE’s and the correlations of the fixed effects estimations.
A 68% (SD) and 95% (2 SD) prediction interval around the estimated angle is given by, based on the variance formula with the estimated values filled in the matrices

5. The uncertainty about the angle grows rapidly with time because of the random effect of the slope of the lines (SD +/- 0.26).
An angle of 15° is immediately corrected towards the average, 19°, because the model takes variability and imprecision of the measurement into account (‘Shrinkage’).

8. The model adjusts the slope according to the tendency in successive measurements of the angle of strabismus.
The uncertainty about the slope decreases with additional measurements because the random effect of the slope of the lines decreases.
Even if we would know the slope precisely, there is additional variation of the angle around this slope for an individual child.

10. Refractive error does have some influence, with an optimum at S + 4, possibly some very early accommodative esotropia cases were among the ELISSS study population, that responded well to early glasses.