# Correlation of Bone Textural Parameters with Age in the Context of Orthopedic X-ray Studies
## 2 Materials and Methods
### Statistical Analysis  
Statistical analysis was aimed at showing which textural features correlate significantly with age (age distribution presented in Figure 2).  
In the first step, the Shapiro-Wilk test [20] and Q-Q plot were used to evaluate the distribution of continuous variables. Continuous data were presented as mean (standard deviation, SD) or median (interquartile range, IQR) and compared using Student's \(t\)-test and Mann-Whitney-Wilcoxon U test, respectively. When one of the subgroups in the study groups had a non-normal type distribution, data were presented as median (IQR) and Mann-Whitney-Wilcoxon U test. Categorical data were presented as numbers (percentages), and [PERSON]'s exact test was used to compare them. This determined the steps for further analysis, which was the calculation of [PERSON]'s Rank correlation or Pearson's coefficient for each ROI. Regression models were used as quantitative methods to assess whether and how the predictor; AGE correlates with the selected textures. Further, mixed-effects regression models were used as an extension of fixed-effects regression models to account for hierarchical data structures (Tables 2 and 3 show comparison results of generated models). Such analysis takes into account intra-individual association between variables because we had a mixed data set; some data were collected for participants at several different time points. LME4 R packages were used to develop mixed-effect models.  
Figure 2: Age distribution, including SEX differences.  
The idea behind regression analysis is expressed in the equation below where f(x) is the y-value we want to predict, \(\alpha\) is the intercept (the point where the regression line crosses the y-axis at x = 0), \(\beta\) is the coefficient (the slope of the regression line).  
\[\text{f(x)}=\alpha+\beta\text{ix}+\epsilon\]  
SE is a measure that tells how much the coefficients would vary if the same regression were applied to multiple samples from the same population. Thus, a relatively small SE value indicates that the coefficients will remain very stable if the same regression model is fitted to many different samples with identical parameters. A large SE value, on the other hand, says that the model is variable and not very stable or reliable because the coefficients change substantively if the model is applied to many samples.  
\[StandardError(SE)=\frac{\frac{\sum(\overline{x}-\overline{x}_{i})^{2}}{N-1}}{ \sqrt{N}}=\frac{SD}{\sqrt{N}}\]  
The equation below represents a formal representation of a mixed-effects regression with varying:  
\[\text{f(x)}=\alpha\text{i}+\beta\text{ix}+\epsilon\]  
In this model, each level of the random variable has a different intercept and a different slope. So, to predict the value of a data point, it takes the appropriate intercept (model intercept + random effect intercept) and adds the level factor of that random effect multiplied by the value of x.  
The inclusion of a random effect structure with random intercepts is justified as the AIC (Akaike Information Criterion) of the model with random intercepts is substantially lower than the AIC of the model without random intercepts. The Akaike Information Criterion is a mathematical method for evaluating how well a model fits the data it was generated from. In statistics, AIC is used to compare different possible models and determine which one is the best fit for the data. The lower the AIC score, the better.