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We included both random intercepts and random slopes at both the district and governorate level in the model.
We exclude the random slopes, β1 j, from the definition of mortality outliers due to a lack of consensus on how to identify performance outliers using both random intercepts and random slopes.
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We used the restricted maximum likelihood (REML) to determine the optimal random effects structure of our models (i.e. random intercept, random slope, or both random intercept and random slope), and maximum likelihood (ML) estimation to compare models with different fixed effects.
We ran models with both random intercept and random slope, and with random intercept only.
Both random intercept and random coefficient models were examined [ 12, 13].
A closer investigation showed that when both random intercept and random slope were statistically significant either the random slope or the random intercept was severely underestimated.
Despite considering the simplest structure for the residuals variance, our model with both random intercept and random slope on time ensures that correlation between repeated measurements is appropriately accounted for.
Mean rate of change of C-peptide AUC mean from 6 to 30 months (3-month assessment was not included to maintain a uniform contribution of scheduled assessments over time) was estimated using a mixed effects model with both random intercept and slope adjusting for age, sex, baseline C-peptide AUC mean, and treatment assignment.
In the most general case, a standard assumption for a GLMM with both random intercept and slope is that b i =(b0 i, b1 i ) Tfollows a bivariate normal distribution with mean zero and an unknown precision matrix Q= Q depending on parameters ϕ, i.e. b i | Q ∼ iid N 2 (0, Q − 1 ).
Both random intercepts (P < 0.0001) and random slopes (P < 0.0001) were deemed necessary.
The estimation procedure for both the random intercepts and the random intercepts and slopes models used Restricted Maximum Likelihood, as this is known to provide better estimates of standard errors than Maximum Likelihood [10].
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