# MGHT: Statistical Methodology for Missing-Data Imputation in Food Composition Databases
## 2 Materials and Methods
### Might--Missing Nutrient Value Imputation Using Null Hypothesis Testing
* Check \(p\)-value:
* If \(p\geq\alpha\) -- the null hypothesis \(H_{0}\) is not rejected, indicating that each country can borrow from any other;
* If \(p<\alpha\) -- the null hypothesis \(H_{0}\) is rejected, so we need to continue with the appropriate post-hoc procedure for the previously selected omnibus test. Figure 1: Flowchart of the methodology. * Apply appropriate post-hoc test [44], and depending on the type of post-hoc procedure applied, the methodology is split into two versions:
* MIGHTv1: Apply All vs. all type of post-hoc test (simultaneous) on the set and generate a matrix with the result values. All vs. all type of post-hoc test is a multiple comparison type of test, multiplicity or multiple testing problem, which occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values [45; 46]. * MIGHTv2: Apply One vs. all type of post-hoc test (step-down) \(n-2\) times and generate a matrix with the result values. One vs. all type of post-hoc test is a multiple test procedure of the sequentially rejective type, i.e., hypotheses are rejected one at a time until no further rejections can be done [47; 48; 49]. * Compare every value from the two generated matrices from the post-hoc testing with the selected significance level \(\alpha\) and generate new matrices with rules (separate matrix from the MIGHTv1 and separate matrix from the MIGHTv2 results). The matrices with rules are filled in according to:
* If \(H_{0}\)_is not rejected_, then put \"0\".This indicates that from the specific model we have gained information that country \(x\) can borrow from country \(y\). * If \(H_{0}\)_is rejected_, then put \"1\". This means that from the specific model we have gained information that country \(x\) cannot borrow from country \(y\). Table 3 gives a representation of these matrices, from which we can see that \(Country_{1}\) cannot borrow from \(Country_{2}\), and \(Country_{2}\) cannot borrow from \(Country_{m}\). 3. Generate two final matrices with rules (one for MIGHTv1 and MIGHTv2) from all the separate matrices with rules. These matrices are generated according to: * If all the \(n\)-2 matrices have \"0\" in a certain position, then in the final matrix we put \"0\". * If one or more of the \(n\)-2 matrices has \"1\" in a certain position, then in the final matrix we put \"1\". An example table of a sum from the rules from all the models is presented in Table 4. Here, \"0\" indicates that according to all models the countries from the specified row and column can borrow from each other, while everything else indicates the opposite, i.e., they cannot borrow from each other. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Country** & **Country\({}_{1}\)** & **Country\({}_{2}\)** &... & **Country\({}_{m}\)** \\ \hline Country\({}_{1}\) & / & 1 &... & 0 \\ Country\({}_{2}\) & 1 & / &... & 1 \\  � & � & � & � & � \\ Country\({}_{m}\) & 0 & 0 &... & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Example table of matrix with rules obtained from one model, where \(m\) is the number of countries, 0 indicates that according to this model the countries from the specified row and column can borrow from each other, while 1 indicates the opposite, i.e., they cannot borrow from each other, and / indicates “Not applicable”. 4. Test the rules on the separated, unseen instance. The final rules are incorporated in metrics for calculating the missing value of the selected nutrient in any food from the corresponding food group for any of the given countries (Equations (1) and (2)). \[Calculated_{average}=Average\Big{\{}\textit{values from FCDBs eligible for borrowing}\Big{\}}\] (1) \[Calculated_{median}=Median\Big{\{}\textit{values from FCDBs eligible for borrowing}\Big{\}}\] (2) 5. Compare the results from the rules with the actual value (Equation (3)). \[Error=\mid Actual_{value}-Calculated_{value}\mid\] (3)