# MGHT: Statistical Methodology for Missing-Data Imputation in Food Composition Databases
## 2 Materials and Methods
### Might--Missing Nutrient Value Imputation Using Null Hypothesis Testing  
Working with FCDBs has its downsides. As analytical FCD is very valuable, relevant, difficult and expensive to obtain, and missing values are a very common problem in general in FCBDs, it is evident that in our case we will be working with small datasets. As we are working with food groups, i.e., selecting foods that belong to a specific food group, an important part here is to note that each set of values formed for one food group must consist of a minimum number of foods \(n\). This implies finding existing, non-null values for one nutrient in \(n\) foods that belong to a certain food group where all values are type \"A\" (Analytical result/s) or \"AG\" (Analytical, generic). To find the minimum number \(n\), we must go further with our methodology. The idea of hypothesis testing used with biological and chemical data is presented in [30; 31]. Modeling based on null hypothesis testing has been previously used in a lot of application in the area of economics and information systems [32]. Alongside effect size, confidence intervals, and \(p\)-values are part from frequentist statistics, albeit with increasing movements towards Bayesian approaches, for much of the health sciences. MIGHT follows the idea of frequentist statistics and involves statistically comparing values from different FCDBs, and based on the statistical significance between them creating rules for borrowing. One common approach for making a statistical comparison is null hypothesis testing. To perform null hypothesis testing, the number of instances, \(n\), in the datasets should satisfy a requirement for a minimum sample size, \(r\), for an appropriate statistical test. This implies that \(n\geq r\). Since we are working with small datasets, to build more robust models, we decided on leaving one instance out, then building models for the remaining instances, and at the end aggregating all the models to obtain the final rules for imputation. To achieve this, we require an additional instance: \(n\geq r+1\). Lastly, because this methodology needs to be tested on new, unseen instances, we require one more instance in our final set: \(n\geq r+2\). To generate rules for imputation between countries, the number of countries, \(m\), involved in the comparison should be \(m\geq 3\). For example, if we have only two countries, where one should borrow from the other, the user does not have any option for imputation, so we need at least three. This also comes from the fact that, using the traditional approaches, in most cases, users calculate either average or median from several countries. The starting point is a dataset such as the one presented in Table 2, with the restrictions of \(n\geq r+2\) and \(m\geq 3\). Before an appropriate statistical test is applied, some assumptions (i.e., conditions) must be checked. The most common assumptions checked are independence, normality of the data, and homoscedasticity of the variances [33]. In probability theory, two events are independent (i.e., statistically or stochastically independent) if the occurrence of one does not affect the probability of occurrence of the other. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. A random variable with Gaussian distribution is said to be normally distributed with some average value, \(\mu\), and standard deviation, \(\sigma\). Normality tests assess the likelihood that the given dataset comes from a normal distribution, which can be checked by using one of the following tests: Kolmogorov-Smirnov [34], Anderson-Darling [35], Shapiro-Wilk [36], and D'Agostino-Pearson [37] test. The assumption of homogeneity of variance is that the variance within each of the datasets is equal. To test for homogeneity of variance, several statistical tests exist, such as [PERSON]'s test [38], [PERSON]'s test [39], and [PERSON]'s test [40].