Statistical models

The graded response model (GRM) was extended by incorporating variables (listed in Tables 1 and ​and2)2) to explain the latent construct – OHRQoL. The explanatory GRM with p explanatory variables, J items and K response options is formulated as

The frequency distribution of background variables

^aOnly collected in the baseline study

Regression coefficients with 95% credible intervals of explanatory GRM

Data in boldface: 95% credible interval excluded 0, considered as statistically significant at 0.05 level of significance

a_j = item discriminatory parameter for the j^th item; b_jk = item threshold parameter for the k^th response option in the j^th item; j = 1,2,3…J; k = 1,2,3…K-1; P⁺_jk is the probability of choosing the k + 1^th or higher response options in the j^th item. The latent score (θ) consists of a linear combination of explanatory variables and regression coefficients (x₁γ₁ + x₂γ₂ + … + x_pγ_p), plus an error term (ε) [17]. 14 items were fitted into the GRM model (8 items in RSF:8 plus 8 items in ISF:8 minus 2 overlapping items among the 2 short forms).

The explanatory GRM was used to investigate the baseline factors associated with OHRQoL at 12 years old. The explanatory variables included demographic variables (place of birth and year of residency in Hong Kong), oral health behaviors (snacking frequency between meals, brushing frequencies, use of fluoridated toothpaste, and previous participation of the School Dental Care Service (SDCS)), the family social economic status (parents’ place of birth, employment status, family income, education level and whether they have lived in Hong Kong for 7 years or more), DMFT and CPI. Each categorical variable was recoded as dummy variables and included into the models.

The Bayesian estimation method – Monte Carlo Markov Chain (MCMC) with Gibbs sampling was adopted for parameter estimation and implemented via WinBUGS (Additional file 1) [18]. Non-informative priors were used and the posterior distribution was constructed using 12,000 simulations after 8000 burn-ins. The parameter estimates resulted from the use of non-informative priors can resemble the maximum likelihood estimation in the classical (frequentist) approach [19]. The 95% credible intervals of the parameters (95% probability that the true parameters is within the interval) were obtained from the 12,000 simulations. Associations of explanatory variables with OHRQoL were established when the 95% credible interval excluded 0 or difference between each pair of regression coefficients within a factor excluded 0.

Longitudinal invariance of the questionnaire (Table 3) was investigated by the GRM with varying discriminatory (a_jt) and threshold parameters (b_jkt) across age 12 and 15 years old. The model is formulated as:

where t = 1 (at 12 years old) or 2 (at 15 years old). Meanwhile the scores (θ_t) of the same respondents across the 2 time points were allowed to be correlated. Since too few respondents chose “Every day/ almost every day”=4 in some CPQ_11–14 items, response options “Often” = 3 and “Every day/ almost every day” = 4 were combined in the longitudinal invariance analysis.

GRM for assessing longitudinal invariance

^aThe options “Often” = 3 and “Every day/ almost every day” = 4 were combined

Data in boldface: 95% credible interval excluded 0, considered as statistically significant at 0.05 level of significance

In order to separate the change in OHRQoL arise from the interpretation of items, 5 external anchor items (global self-rating items of OHRQoL and their perceived impact on OHRQoL) which reflect their own perceptions of OHRQoL, were used to equate the questionnaires administered to respondents at both time points onto the same scale [20]. The discriminatory and threshold parameters of the anchor items were set equal at both time points (a_j1 = a_j2; b_jk1 = b_jk2).

Using WinBUGS, the differences in the parameters across the 2 time points and their corresponding credible intervals were computed. Significant parameter drift were inferred when the 95% credible interval of change in item parameters excluded 0. Items with significant parameter drift were considered as biased (lack of longitudinal invariance).

For modeling the change in OHRQoL over the 3 years, δ is defined as the difference in OHRQoL at 15 years old (θ₂) and 12 years old (θ₁), i.e. δ = θ₂- θ₁. Three models relating the explanatory variables to OHRQoL in a longitudinal context are adopted:

Presuming the change in OHRQoL is attributed to their status at baseline: baseline explanatory variables (listed in Table 4) were used to explain the change in OHRQoL, i.e. δ = x₁₁γ₁₁ + x₂₁γ₂₁ + … + x_p1γ_p1+ ε.

Longitudinal explanatory GRM (Model 1): Regression coefficients with 95% credible intervals

Data in boldface: 95% credible interval excluded 0, considered as statistically significant at 0.05 level of significance

Presuming the change in OHRQoL is attributed to the change in status over 3 years: change in explanatory variables (listed in Table 5) were used to explain the change in OHRQoL, i.e. δ = x_1dγ₁ + x_2dγ₂ + … + x_pdγ_p + ε, where x_pd represents the change in the explanatory variables x_p over the 3 years.

Longitudinal explanatory GRM (Model 2): Regression coefficients with 95% credible intervals

Data in boldface: 95% credible interval excluded 0, considered as statistically significant at 0.05 level of significance

Presuming the effect of baseline and follow up factors on OHRQoL at their respective time point is of interest: baseline explanatory variables were used to explain the baseline OHRQoL; explanatory variables obtained in follow-up study were used to explain the follow-up OHRQoL (listed in Table 6). Static variables were used to explain both baseline and follow-up OHRQoL, i.e. θ_t = x_1tγ_1t + x_2tγ_2t + … + x_ptγ_pt + ε_t, where t = 1, 2.

Longitudinal explanatory GRM (Model 3): Regression coefficients with 95% credible intervals

Data in boldface: 95% credible interval excluded 0, considered as statistically significant at 0.05 level of significance

Again, WinBUGS was used to estimate the credible intervals (Additional file 1). Association of explanatory variables with OHRQoL at respective occasions were established when 95% credible intervals excluded 0 or differences in any pair of γ’s within a factor excluded 0.

To rule out the change in OHRQoL score due to change in interpretation of the items over 3 years, discriminatory and threshold parameters of items that were not invariant were allowed to vary across baseline and follow-up.