# Techniques for Robust Imputation in Incomplete Two-Way Tables
## 2 Material and Methods
### Numerical Example 1: Simulation Study  
To evaluate the performance of the five imputation systems (SVD88, M5 RobSVD, rSVD84, GOKimputation, and M5 COKimputation), we simulated eight incomplete and contaminated 100 \(\times\) 8 size matrices (Table 1) that had the structure of an AMMI model, using the steps given by [20; 26] as follows:  
1. Create a matrix \(\mathbf{X}\) with \(n=100\) rows and \(p=8\) columns with observations drawn from a uniform distribution \([-0.5;0.5]\). 2. Compute the SVD of \(\mathbf{X}\) to obtain the matrices \(\mathbf{U}\), \(\mathbf{V}\) and \(\mathbf{D}\), containing, respectively, the left and right singular vectors and the singular values of \(\mathbf{X}\). 3. Simulate the grand mean \((\mu)\), the row \((\alpha)\) and column \((\beta)\) main effects, where \(\mu\sim N(15,3)\), \(\alpha\sim N(5,1)\) and \(\beta\sim N(8,2)\). 4. Simulate a matrix \(\mathbf{Y}\) with AMMI2 structure (i.e., AMMI with the first two singular values/vectors): \[\mathbf{Y}=\mathbf{1}_{n}\mathbf{1}_{p}^{\mathsf{T}}\mu+\mathbf{\alpha}_{n}\mathbf{1}_{p}^{T}+\bm {1}_{n}\mathbf{B}_{p}^{\mathsf{T}}+28\times\mathbf{U}[,1]\mathbf{D}[1,1]\mathbf{V}[,1]^{T}+15 \times\mathbf{U}[,2]\mathbf{D}[2,2]\mathbf{V}[,2]^{T}\] where \(\mathbf{A}[,i]\) represents \(i\)-th column of the matrix \(\mathbf{A}\)\((i=1,2)\), \(\mathbf{A}[i,i]\) represents the element in the row \(i\) and column \(i\) of the matrix \(\mathbf{A}\), \(\mathbf{1}_{n}\mathbf{1}_{p}^{\mathsf{T}}\mu\) is a matrix \((n\times p)\) with the grand mean \(\mu\) in all positions, \(\mathbf{\alpha}_{n}\mathbf{1}_{p}^{\mathsf{T}}\) is a matrix \((n\times p)\) of rows main effects (all matrix rows equal), \(\mathbf{1}_{n}\mathbf{\beta}_{p}^{\mathsf{T}}\) is a matrix \((n\times p)\) of columns main effects (all matrix columns equal). For each complete \(\mathbf{Y}\) matrix, observations were then randomly deleted at the specified percentages shown in Table 1 and outliers were produced in the remaining data at the companion percentages in Table 1. The positions of the outliers were chosen randomly, and the outliers were generated from \(N\Big{(}\mu_{j}+5\sigma_{j}^{2},\sigma_{j}^{2}\Big{)}\), where \(\mu_{j}\) and \(\sigma_{j}^{2}\) represent the mean and variance of the \(j\)-th column (or \(j\)-th environment) of the non-missing values. The five imputation procedures were applied to the eight incomplete and contaminated matrices, the five imputation procedures were applied, having previously recorded the true data that had been randomly removed. This made it possible to assess the performance of the imputations by calculating the prediction error \(\mathrm{P_{e}}\) defined by [4] as the square root of the mean squared error (MSE) between the true values and the corresponding imputations. Numerical Example 2: Cross-Validation on Real Data from Experiments with Genotype-by-Environment Interaction  
The Ontario winter wheat dataset with 18 genotypes in nine environments was published by [27] to show an application of biplot analysis, and recently these data were used to illustrate new proposals for distribution-free multiple imputation [9] and new bootstrap methods to determine the optimal number of multiplicative components in AMMI models [28]. A difficulty in evaluating the behavior of robust imputation systems in complete real data is the lack of a priori knowledge of what \"good behavior\" should be. For this reason and following the recommendation of [19], we consider adding a small contamination of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Matrices** & **\% of Missing** & **\% of Outliers** \\ \hline Matrix1 & 10 & 0 \\ Matrix2 & 20 & 0 \\ Matrix3 & 10 & 5 \\ Matrix4 & 10 & 10 \\ Matrix5 & 10 & 15 \\ Matrix6 & 20 & 5 \\ Matrix7 & 20 & 10 \\ Matrix8 & 20 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the simulated matrices. the matrix elements and removing some elements. Initially, 10% of the elements of each environment were removed, but unlike the previous simulation study, we used an MNAR mechanism.