Analyses were performed using data collected yearly when the child was enrolled in elementary school (aged 6–12 years; i.e., from 1999 through 2004 for two schools and from 2001 through 2006 for the third school). Children who had at least three visits between the ages of 6 and 12 years and met the criteria for one of the five refractive error groups were included in the analyses.
Growth curve models of AL, VCD, ACD, LT, and CR were developed using longitudinal data of the randomly selected eye. Fractional polynomial (FP) growth models were fitted for each refractive error group.
13 This method increases the flexibility afforded by the conventional family of polynomials models. The first-degree FP1 function is a power transformation model, β
1 Z P , where
Z is a single continuous covariate (age, in this analysis) and the powers
p are chosen from a restricted set,
S = {−2, −1, −0.5, 0, 0.5, 1, 2, 3} where
Z 0 denotes log
Z. The second-degree FP2 is an extension from the FP1 and FP2 functions, with powers
p 1,
p 2 including the formulations β
1 Z P 1 +β
2 Z P 2 and β
1 Z P 1 +β
2 Z P 2 ln Z, the latter being a so-called repeated-powers model. The best-fitting model is defined as the model with a
p (or combination of
p 1 and
p 2) that maximizes the likelihood among the FP1 and FP2 models.
To allow for possible nonlinearity in age at all levels of refractive error groups, we used the multivariate fractional polynomial interaction (MFPI) algorithm for investigating interactions between age and refractive error groups.
14 First, the relationship of ocular components and age in the children with persistent emmetropia or persistent hyperopia was modeled by a fractional polynomial (FP) function adjusted for sex, ethnicity, father's education level, and child's height (in centimeters), measured at each time point. The best transformation of age within the class of FP function was selected based on the maximum likelihood, with the constraint of the same powers for the persistent emmetropia and the persistent hyperopia groups. Second, the interaction between the age and the refractive error groups was examined by the likelihood ratio test, and the difference in deviance was compared by χ
50 test with 2
df. These procedures were repeated for pair-wise comparisons between children with persistent emmetropia or emmetropizing hyperopia, persistent emmetropia or newly developed myopia, and persistent emmetropia or persistent myopia for each ocular component. (Statistical analyses were performed with Stata, ver. 10.1; Stata Corp., College Station, TX.)