# Design of an imputation methodology by random selection using regression trees

Lelly Useche, Jean Pérez Parra, Carlos Garcia-Mendoza, Ana Ides Chacón

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Lelly Useche, Jean Pérez Parra, Carlos Garcia-Mendoza, Ana Ides Chacón. Design of an imputation methodology by random selection using regression trees. Bull. Comput. Appl. Math., 2021, 9 (2), pp.97-121. hal-03579181

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# Design of an imputation methodology by random selection using regression trees

Lelly Useche<fnref n="1" />, Jean Pérez Parra<fnref n="2" />, Carlos Garcia-Mendoza<fnref n="3" /> <fnref n="4" />, Ana Ides Chacón<fnref n="5" />

CompAMa Vol.9, No.2, pp.97-121, 2021 - Received April 20, 2021 - Accepted December 21, 2021

Special Issue II JEM 2020, Eds. Carmen Judith Vanegas and Saúl Buitrago Boret

## Abstract

One of the biggest issues in the information collection stage is the absence of data, this research focuses specifically on the scenario when the loss is partial, completely random and the data is quantitative. There are classic techniques to impute data, however, these have not been able to accurately impute the real data. A design of an imputation methodology by random selection is proposed through the use of regression trees, comparing theoretically and empirically with and without the use of the tree for different data loss percentages. Unbiased estimators of variances and biases are obtained by evaluating their properties, which improves the estimates. As a disadvantage of the proposed design, it does not solve the alteration of the distribution of the data and the relationship between the variables.

<footnote n="1">¹Departamento de Matemáticas y Estadística, Instituto de Ciencias Básicas, UTM, Ecuador. (lelly.useche@utm.edu.ec). (ORCID iD: https://orcid.org/0000-0002-4294-9009).</footnote>

<footnote n="2">²Departamento de Química, Instituto de Ciencias Básicas, UTM, Ecuador. (jean.perez@utm.edu.ec). (ORCID iD: https://orcid.org/0000-0002-7971-1782).</footnote>

<footnote n="3">³Departamento de Matemáticas y Estadística, Instituto de Ciencias Básicas, UTM, Ecuador. (cdgarcia1034@utm.edu.ec). (ORCID iD: https://orcid.org/0000-0002-6580-2560).</footnote>

<footnote n="4">⁴Corresponding author.</footnote>

<footnote n="5">⁵Departamento de Matemáticas y Estadística, Instituto de Ciencias Básicas, UTM, Ecuador. (anaidesch@gmail.com). (ORCID iD: https://orcid.org/0000-0003-3382-5407).</footnote>
**Keywords:** Absence of data; imputation; regression trees; random selection.

## Resumen

La gran problemática en la etapa de recolección de la información es la ausencia de datos, esta investigación se basa específicamente cuando la pérdida es parcial, completamente aleatoria y los datos son cuantitativos. Existen técnicas clásicas para imputar los datos, sin embargo, no se ha logrado imputar con precisión el dato real. Se propone un diseño de una metodología de imputación por selección aleatoria mediante el uso de árboles de regresión, comparando de manera teórica y empírica con y sin el uso del árbol para diferentes porcentajes de pérdida. Se obtienen estimadores insesgados de las varianzas y sesgos evaluándose las propiedades de los mismos, lo cual mejora las estimaciones. Como desventaja del diseño propuesto, se tiene que este no solventa la alteración de la distribución de los datos y relación entre las variables.

**Palabras claves:** Ausencia de datos; imputación, árboles de regresión; selección aleatoria.

# 1 Introduction

Most statistical methods and various data analysis methodologies are designed on the assumption of a complete database. However, when collecting information either through surveys or direct measurements to study phenomena, it is common to find patterns of missing or lost data. Either due to prejudices of the respondent or physical losses of the experimental units involved in the study. That is why it is extremely important to have methodologies that allow us to correct incomplete data matrices. The main problems of not having a complete database are: 1) loss of efficiency; 2) complexity in handling information; 3) flaws in bias estimates and; 4) inefficient estimates [1].

At present there are several methods of data imputation which have tried to solve the main problems of the absence of information, however, a way to improve the imputations achieved with these classic techniques is still being sought, since until now, there is still underestimation or overestimation of the variance, change in the relationships of variables, change in their distributions, among others. This research shows an imputation methodology through the use of Regression Trees and the classical technique of Imputation by Random Selection for quantitative data, in order to optimize the existing
classical imputation.

The research began with a bibliographic review of the state of the art of imputation techniques, loss patterns, sampling and non-sampling errors, types of non-response, description of the data imputation model, regression trees and the importance of classification in data imputation. Then the proposed technique is detailed algebraically, that is, the imputation by classical random selection together with the regression tree, determining unbiased estimators for the mean, totals and variance. Then, an empirical evaluation is made with quantitative data and compared with the use and without the use of regression trees for different percentages of missing data between 5% and 30%. Finally, the results and their respective analysis are shown.

## 2 Preliminaries

### Imputation:

Imputation is filling the empty spaces of the incomplete database with plausible values, and thus obtaining a complete file to analyze it. It is used as a treatment for partial lack of response [2]. In [3], imputation techniques are defined as those that generate a specific value for a missing real value in a data set.

### Description of the imputation model:

In [4] the imputation model is described as follows: Let $U$ be a finite population of $N$ units $U = \{U_i; i = 1, 2, ..., N\}$. Let $Y = (y_i)$ be the vector of the response variable where $y_i$ represents the i-th element. Let $X = (x_{ik})$ be a $(N \times K)$-matrix of auxiliary variables with $x_{ik}$ as the k-th variable for the i-th element. $X$ is represented as $X = (\vec{x}_1, \vec{x}_2, ..., \vec{x}_k, ..., \vec{x}_K)$ where $\vec{x}_k = (x_{1k}, x_{2k}, ..., x_{NK})^t$ is a vector with $N$ values of $x_{ik}$. It is assumed that the variable $y_i$ is subjected to non-response and $x_{ik}$ is completely observed. $R = (r_i)$ is defined as a $(N \times 1)$-vector of variable indicators for $Y$, which identifies whether $y_i$ is lost or not. That is;

$$r_i = \begin{cases} 1 & \text{if } y_i \text{ is observed} \\ 0 & \text{if } y_i \text{ is not observed} \end{cases}$$
In matrix notation the population is defined as:

$$ \begin{bmatrix} x_{11} & \dots & x_{1k} & \dots & x_{1K} \\ x_{21} & \dots & x_{2k} & \dots & x_{2K} \\ \vdots & & \vdots & & \vdots \\ x_{m1} & \dots & x_{mk} & \dots & x_{mK} \\ x_{m+1,1} & \dots & x_{m+1,k} & \dots & x_{m+1,K} \\ \vdots & & \vdots & & \vdots \\ x_{N1} & \dots & x_{Nk} & \dots & x_{NK} \end{bmatrix} \begin{pmatrix} \vec{Y} \\ y_1 \\ y_2 \\ \vdots \\ y_m \\ 0 \\ \vdots \\ 0 \end{pmatrix} \begin{pmatrix} \vec{R} \\ 1 \\ 1 \\ \vdots \\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, $$

where *m* is the amount of records when *Y* is observed (measured) and the zeros (0) represent the missing values.

In [4] it is stated that under the model-based approach, $x_{ik}$ and $y_i$ are considered random variables with distribution $f(X, Y/\theta)$ indexed by the parameter $\theta$. The response $R$ is also included as a random variable with distribution $f(R/X, Y/\phi)$. $X$ is fully observed and $Y$ is subject to non-response, and the full form of the distribution can be written as $f(X, Y, R/\theta, \phi)$ indexed by the parameter $\theta$ and $\phi$ with $R$ as the response indicator.

The joint distribution of $X$, $Y$ and $f(X, Y, R/\theta, \phi)$ can be decomposed as the product of the probability distribution of $X$ and $Y$ indexed by the parameter $\theta$ and the conditional distribution of $R$ given $X$ and $Y$ (the distribution for the missing data mechanism) indexed by the parameter $\phi$, that is:

$$ f(X, Y, R/\theta, \phi) = f(X, Y/\theta)f(R/X, Y, \phi) \quad (1) $$

If $Y$ is subjected to non response, we can write $Y = (Y_{obs}, Y_{per})$, where $Y_{obs}$, which is a vector of size $n \times 1$, represents the observed values of $Y$ and $Y_{per}$ is a vector of size $(N-m) \times 1$ which represents the missing values of $Y$. Furthermore, the distribution $f(X, Y, R/\theta, \phi)$ can be written as $f(X, Y_{obs}, Y_{per}, R/\theta, \phi)$, and the Equation 1 can be written as:

$$ f(X, Y_{obs}, Y_{per}, R/\theta, \phi) = f(X, Y_{obs}, Y_{per}/\theta) f(R/X, Y_{obs}, Y_{per}, \phi) $$

The distribution of the observed data can be obtained by integrating $Y_{per}$ outside the joint distribution of $X$, $Y$ and $R$. That is, $f(X, Y_{obs}, R) = \int f(X, Y, R) dY_{per}$, more specifically:

$$ f(X, Y_{obs}, R/\theta, \phi) = \int f(X, Y_{obs}, Y_{per}/\theta) f(R/X, Y_{obs}, Y_{per}, \phi) dY_{per} $$
Assumptions about the model are normally made in order to obtain a valid estimate. One of the most common assumptions is Missing at Random (MAR). That is, $f(R/X, Y_{obs}, Y_{per}, \phi) = f(R/X, Y_{obs}, \phi)$. Therefore assuming MAR and given the current observed data is $(X, Y_{obs}, R)$ we have:

$$f(X, Y_{obs}, R/\theta, \phi) = f(X, Y_{obs}/\theta)f(R/X, Y_{obs}, \phi)$$

In this case the maximum common likelihood procedure used for the complete observed data can be used to estimate the required parameter $\theta$ when the data is incomplete. That is, $\theta$ can be estimated by maximizing $f(X, Y_{obs}/\theta)$. Thus, the missing data mechanism is ignorable, that is, the second part of the right hand side of the last equation can be ignored in the estimation of $\theta$. If MAR holds, the inference of $\theta$ is based on the likelihood function $L(\theta/x, Y_{obs})$ which is a function of $\theta$ proportional to $f(X, Y_{obs}, \theta)$.

## Classification and Regression Trees (CART):

Classification and Regression Trees (CART) were developed by Breiman, Freidman, Olshen and Stone in the book Classification and Regression Trees, published in 1984. For classification trees, the goodness of a division is quantified by a measure of impurity; a division is said to be pure if, after division, all instances of the choice of a branch belong to the same class [5].

In [6], under the model-based assumption, the probability function of Y is given, which is subject to non-response, given the terminal node (classification group) defined by $X_t$ as $f(Y/\vec{x}_i \in X_t)$, that is, the probability function of Y given a set of the values of the explanatory variables identifying the terminal node. To simplify the notation we write:

$$f(Y/\vec{x}_i \in X_t) = f(Y/t)$$

The construction of the regression tree is carried out using recursive division of nodes. The splits will be stopped according to the researcher's specifications as a minimum number of items per node, although the information contained in the learning data is probably over-fitted. That is why we then proceed to a “pruning” stage, obtaining simpler or more general nodes. Finally, the most optimal tree is selected, where the information is adjusted to the learning data, this selection is based on certain optimization criteria, such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).

## Previous works:
Initially, the lack of response was treated by replacing the missing data with the mean of the present data of the variable, a method proposed in [7]. During the 1970s, data imputation meant identifying and replacing records without information. In this context, Hot-deck procedures, in their different variants, were widely applied to supply information in censuses and surveys, and the methods of adjustment by averages and Cold-deck were frequently used [8]. In the 70s, the methods were studied according to the patterns of information loss being [9] who distinguished between values lost or missing in a random way (MAR), and lost or missing values in a completely random way MCAR. In [10] the differences between the weighted adjustment techniques were established.

In the 90s, [11] proposed the k-nearest neighbor of missing values as an estimation method. Other research has looked for ways to improve imputation techniques or create new ones, such as those based on neural networks [12], factor analysis [13], using response probability [14], Hot fractional deck [15], in principal component analysis [16], or based on GH-Biplot decomposition [17]. Since the year 2000, the use of classification trees has been implemented as an improvement of the imputation procedures, in which the benefits of regression trees as a methodology prior to imputation with the mean have been studied, classical technique and commonly used, as is the case of [18] with classification trees and [2] with regression trees, in both cases unbiased estimators were obtained.

However, in recent years the classical imputation techniques have continued to be evaluated in combination with other estimation techniques. Among these investigations we have: [19] carried out an imputation procedure based on the maximum likelihood algorithm in longitudinal models with Poisson and Negative Binomial response variables, obtaining good estimates. In [20] the closest k-neighbor imputation algorithm to different loss mechanisms with different loss percentages and databases of different sizes is evaluated. Also, [21] proposed a review of the imputation methods applied to the National Household Expenditure Survey (ENGHO).

# 3 Methodology

As in all statistical research, the evaluation of the results of the imputation technique to be used must be an essential requirement to determine the reliability of the base obtained after the lack of response has been treated. There
are many ways to carry out the evaluation and validation of one or more
imputation techniques that have been applied to a particular situation, the
most common of which is the use of simulation. The simulation in this case
consists of applying artificial losses in some variables of an original database
that contains all the data, (obtaining a database that we will call the available
database). Subsequently, all the records that have at least one missing data
will be deleted from this database, finally obtaining a smaller, but complete
database. Assuming that the relationships and characteristics of the missing
data are the same as that of the present data (completely random loss), the
same missing pattern obtained from the original database is applied to the
complete database.

Faced with such losses, the proposed imputation technique is applied,
and this new database obtained is evaluated by comparing it with the com-
plete database by using univariate techniques (contingency table, frequency
histogram), bivariate (matrix comparison correlations, ANOVAS), joint and
marginal frequency distributions, individual values, among other theoretical
aspects. On the other hand, the reaction to this evaluation of the database
will be studied with respect to the percentage of data loss, which will be
arbitrarily originated, (5%, 10%, 20% and 30%) to know up to what per-
centage the database "is attributable", observing certain criteria, such as, the
influence of the percentage of loss on the results obtained with the same im-
putation technique, among others. Then proceed in the same way but under
the proposed methodology. It should be noted that the regression tree has
already been created in previous research, which was subjected to a cross val-
idation (80:20) obtaining acceptable AIC and BIC indices in the validation
sample [2]. The process is applied for each variable to be imputed and for
each percentage of loss that has been considered. For the statistical analysis,
the software SPSS 25 Spanish version was used.

Imputation by Random selection:

For the imputation for each case with one or more missing cells, a donor is
selected randomly, from where the data to be used to impute the missing data
of the database comes. Among its advantages we have is that a possible/real
value is assigned for a lack of response, if the selection is without replacement,
the distribution of the imputed data will resemble the respondents. The
disadvantage is the possible non-conservation of the relationship between the
variables and the assumption that respondents and non-respondents have the
same probability of response and similar characteristics.
**Imputation technique by random selection under the use of regression trees:**

Let $Q$ be the total number of nodes, $m_q$ the number of observed records and $(N_q - m_q)$ the number of non-observed records in each node $q$. Establishing a random selection without replacement and letting $w_j$ be the randomly taken observation in the j-th trial of $m_q$ in $j - 1$ observations, we have:

**Mean estimator:**

$$ \bar{Y} = \frac{1}{Q} \sum_{q=1}^{Q} \left[ \frac{1}{N_q} \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) \right] \quad (2) $$

**Expectancy of the mean estimator:**

In order to find the unbiased estimator for the mean we calculate the expectancy of the estimator in Equation 2 for each node:

$$ E(\bar{Y}) = \frac{1}{N_q} \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) = \frac{1}{N_q} E \left[ \sum_{i=1}^{N_q} y_i r_i \right] + \frac{1}{N_q} E \left[ \sum_{j=1}^{N_q - m_q} w_j \right] $$

Then, making use of the assumption Missing Completely at Random (MCAR), where the distribution of the donors is equal to that of the missing data:

$$ E(\bar{Y}) = \frac{1}{N_q} \left[ \sum_{i=1}^{N_q} \mu \varphi \right] + \frac{1}{N_q} \left[ \sum_{j=1}^{N_q - m_q} E(w_j) \right] $$

where $\varphi$ is a parameter of the mean of the $r_i$, and $w_j$ being equally distributed as the $y_i$ as mentioned earlier when MCAR is assumed:

$$ \mu = \left[ \frac{N_q \varphi + (N_q - m_q)}{N_q} \right] \quad (3) $$

Now, obtaining an unbiased estimator for each node, in Equation 3 we have:

$$ \bar{Y}' = \frac{N_q}{N_q \varphi + N_q - m_q} \bar{Y} = \frac{N_q}{N_q \varphi + N_q - m_q} \frac{1}{N_q} \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) $$

Therefore, the expectancy of the mean estimator through random selection, using regression trees is:

$$ \bar{Y}' = \frac{1}{Q} \sum_{q=1}^{Q} \left[ \frac{1}{N_q \varphi + N_q - m_q} \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) \right] \quad (4) $$
This will be the unbiased mean estimator by the technique of imputation with random selection.

**Bias of the mean estimator:**

It is defined as the difference between the parameter $\mu$ and the expectation of the mean, but because an unbiased estimator was obtained, it means that its bias is equal to zero.

**Variance of the mean estimator:**

$$V(\bar{Y}') = V \left[ \frac{1}{N_q\varphi + N_q - m_q} \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) \right] \\ = \frac{1}{(N_q\varphi + N_q - m_q)^2} V \left[ \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) \right]$$

Therefore the variance of the mean estimator is:

$$V(\bar{Y}') = \frac{1}{(N_q\varphi + N_q - m_q)^2} \left[ E \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) - E^2 \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) \right] \quad (5)$$

Separating the previous equation in its two terms and expanding the first term we get:

$$E \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) = E \left[ \left( \sum_{i=1}^{N_q} y_i r_i \right)^2 + 2 \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q - m_q} w_j \right) + \left( \sum_{j=1}^{N_q - m_q} w_j \right)^2 \right] \quad (6)$$

Further separating this expression in Equations 7, 8, 9 (first, second and third term respectively) and expanding the first term we get:

$$N_q(\mu^2\sigma_r^2 + \varphi\sigma^2 + \sigma^2\sigma_r^2) + N_q^2(\mu^2\varphi^2) \quad (7)$$

Expanding the second term in Equation 6 we get:
$$2E\left(\sum_{i=1}^{N_q} y_i r_i\right) \left(\sum_{j=1}^{N_q-m_q} w_j\right) = 2E\left(\sum_{i=1}^{N_q} y_i r_i\right) E\left(\sum_{j=1}^{N_q-m_q} w_j\right) \quad (8)$$

Given that independence between $y_i$ and $w_j$ is assumed, we get:

$$2E\left(\sum_{i=1}^{N_q} y_i r_i\right) \left(\sum_{j=1}^{N_q-m_q} w_j\right) = 2\left(\sum_{i=1}^{N_q} E(y_i)E(r_i)\right) E\left(\sum_{j=1}^{N_q-m_q} w_j\right)$$

Assuming independence between $y_i$ and $r_i$ we get a simplified version of Equation 8:

$$2E\left(\sum_{i=1}^{N_q} y_i r_i\right) \left(\sum_{j=1}^{N_q-m_q} w_j\right) = 2N_q(N_q - m_q)\mu^2\varphi$$

Expanding the third term in Equation 6 we obtain:

$$E\left(\sum_{j=1}^{N_q-m_q} w_j\right)^2 = (N_q - m_q) [V(w_j) + E^2(w_j)] + 2\left(\frac{N_q - m_q}{2}\right) E(w_j w_i) \quad (9)$$

Assuming independence between $w_i$ and $w_j$ we get:

$$(N_q - m_q)(\sigma^2 + \mu^2) + (N_q - m_q)(N_q - m_q - 1)\mu^2$$

Therefore the simplified version of 9 is:

$$E\left(\sum_{j=1}^{N_q-m_q} w_j\right)^2 = (N_q - m_q) \left[ (\sigma^2 + \mu^2) + (N_q - m_q - 1)\mu^2 \right]$$

Joining the simplified versions of Equations 7, 8, 9, we obtain:

$$\begin{aligned} N_q \mu^2 \sigma_r^2 + N_q \varphi \sigma^2 + N_q \sigma^2 \sigma_r^2 + N_q^2 \mu^2 \varphi^2 + 2 N_q^2 \mu^2 \varphi \\ - 2 N_q m_q \mu^2 \varphi + N_q \sigma^2 + N_q \mu^2 + N_q^2 \mu^2 - N_q m_q \mu^2 - \\ N_q \mu^2 - m_q \sigma^2 - m_q \mu^2 - m_q N_q \mu^2 + m_q^2 \mu^2 + m_q \mu^2 \end{aligned}$$

Therefore the simplified version of Equation 6 is:
$$E \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) = N_q \mu^2 \sigma_r^2 + (N_q \varphi + N_q \sigma_r^2 + N_q - m_q) \sigma^2 \\ + (N_q^2 \varphi^2 + 2N_q^2 \varphi - 2N_q m_q \varphi + N_q^2 - 2N_q m_q + m_q^2) \mu^2$$

Expanding the second term of Equation 5 we get:

$$E^2 \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) = [N_q \mu \varphi + (N_q - m_q) \mu]^2 \quad (10)$$

Which can be further expanded as follows:

$$E^2 \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) = N_q^2 \mu^2 \varphi^2 + 2N_q \mu \varphi (N_q - m_q) \mu \\ + (N_q^2 - 2N_q m_q + m_q^2) \mu^2$$

Obtaining therefore the fully expanded version of Equation 10 as:

$$E^2 \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) = N_q^2 \mu^2 \varphi^2 + 2N_q^2 \mu^2 \varphi \\ - 2N_q \mu^2 \varphi m_q + N_q^2 \mu^2 - 2N_q m_q \mu^2 + m_q^2 \mu^2$$

Joining the simplified versions of Equations 6 and 10 we obtain:

$$V(\bar{Y}') = \frac{1}{(N_q \varphi + N_q - m_q)^2} \left[ N_q \mu^2 \sigma_r^2 + (N_q \varphi + N_q \sigma_r^2 + N_q - m_q) \sigma^2 \right. \\ + (N_q^2 \varphi^2 + 2N_q^2 \varphi - 2N_q m_q \varphi + N_q^2 - 2N_q m_q + m_q^2) \mu^2 \\ \left. - (N_q^2 \mu^2 \varphi^2 + 2N_q^2 \mu^2 \varphi - 2N_q \mu^2 \varphi m_q + N_q^2 \mu^2 - 2N_q m_q \mu^2 + m_q^2 \mu^2) \right]$$

Which can be written as:

$$V(\bar{Y}') = \frac{1}{(N_q \varphi + N_q - m_q)^2} \left[ N_q \mu^2 \sigma_r^2 + (N_q \varphi + N_q \sigma_r^2 + N_q - m_q) \sigma^2 \right] \quad (11)$$

Obtaining in this way the variance of the mean estimator for the technique of imputation by random selection.

**Consistency:**
In order to evaluate the consistency of the estimator we calculate the
limit of the variance when $N$ tends to infinity, that is:

$$ \lim_{N \to \infty} \left( \frac{1}{(N_q \varphi + N_q - m_q)^2} \left[ N_q \mu^2 \sigma_r^2 + (N_q \varphi + N_q \sigma_r^2 + N_q - m_q) \sigma^2 \right] \right) $$

This calculation was carried out in Matlab. Since the result is zero it can
be said that the estimator is consistent.

## Total estimator

$$ \hat{T} = \sum_{q=1}^{Q} \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) \quad (12) $$

Finding an unbiased estimator for the total, the following is calculated:

## Expectancy of the total estimator

$$ E(\hat{T}) = \sum_{q=1}^{Q} \left( E \sum_{i=1}^{N_q} y_i r_i + E \sum_{j=1}^{N_q - m_q} w_j \right) \quad (13) $$

Applying a similar approach to that used for Equation 3:

$$ E(\hat{T}) = N_q \mu \varphi + (N_q - m_q) \mu $$

Therefore we obtain:

$$ E(\hat{T}) = (N_q \varphi + N_q - m_q) \mu \quad (14) $$

Now, obtaining an unbiased estimator for Equation 14

$$ \hat{T}' = \frac{1}{N_q \varphi + N_q - m_q} \hat{T} $$

Therefore the unbiased estimator is:

$$ \hat{T}' = \frac{1}{Q} \sum_{q=1}^{Q} \frac{1}{N_q \varphi + N_q - m_q} \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) \quad (15) $$

## Variance of the total estimator

$$ V(\hat{T}') = \frac{1}{(N_q \varphi + N_q - m_q)^2} V \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) $$
Making use of the expression:

$$V(\hat{T}') = E(x^2) - [E(x)]^2$$

And based on the approach followed for the mean estimator in Equation 5 we get:

$$V(\hat{T}') = \frac{1}{(N_q\varphi + N_q - m_q)^2} [N_q\mu\sigma_r^2 + (N_q\varphi + N_q\sigma_r^2 + N_q - m_q)\sigma^2] \quad (16)$$

And thus the variance of the estimator of the total is obtained for the imputation technique by random selection. Note that no biases were calculated for the means and totals, since unbiased estimators were obtained in both cases.

**Consistency:**

In order to evaluate the consistency of the estimator we calculate the limit of the variance when N tends to infinity, that is:

$$\lim_{N \to \infty} \left( \frac{1}{(N_q\varphi + N_q - m_q)^2} \left[ N_q\mu\sigma_r^2 + (N_q\varphi + N_q\sigma_r^2 + N_q - m_q)\sigma^2 \right] \right)$$

This calculation was carried out in Matlab. Since the resulting value is zero, then it is concluded that the estimator is consistent.

**Estimator of the variance:** Based on the general formula for the variance estimation inside each node:

$$S^2 = \frac{\sum_{i=1}^{N_q} y_i^2 - N_q \bar{y}^2}{N_q - 1} \quad (17)$$

For the imputation method by random selection we get:

$$\sum_{i=1}^{N_q} y_i^2 = \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right)^2 = \left( \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q - m_q} w_j^2 \right)$$

$$N_q \bar{y}^2 = N_q \left[ \frac{1}{N_q} \left( \sum_{i=1}^{N_q} y_i r_i + \sum_{j=1}^{N_q - m_q} w_j \right) \right]^2 \\ = \frac{1}{N_q} \left[ \sum_{i=1}^{N_q} y_i^2 r_i + 2 \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q - m_q} w_j \right) + \sum_{j=1}^{N_q - m_q} w_j^2 \right]$$
Substituting this result into Equation 17:

$$S^2 = \frac{1}{N_q(N_q - 1)} \left[ N_q \sum_{i=1}^{N_q} y_i^2 r_i + N_q \sum_{j=1}^{N_q-m_q} w_j^2 - \sum_{i=1}^{N_q} y_i^2 r_i - 2 \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q-m_q} w_j \right) - \sum_{j=1}^{N_q-m_q} w_j^2 \right]$$

Therefore, we obtain:

$$S^2 = \frac{1}{N_q} \left( \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q-m_q} w_j^2 \right) - \frac{2}{N_q(N_q - 1)} \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q-m_q} w_j \right) \quad (18)$$

And thus the estimator of the variance is obtained using the imputation technique by random selection inside each note.

## Expectancy of the variance estimator

$$E(S^2) = \frac{1}{N_q} E \left( \sum_{i=1}^{N_q} y_i^2 r_i \right) + \frac{1}{N_q} E \left( \sum_{j=1}^{N_q-m_q} w_j^2 \right) - \frac{2}{N_q(N_q-1)} E \left( \sum_{i=1}^{N_q} y_i r_i \right) E \left( \sum_{j=1}^{N_q-m_q} w_j \right)$$

Since:

$$\begin{align*} E \left( \sum_{i=1}^{N_q} y_i^2 r_i \right) &= N_q(\sigma^2 + \mu^2)\varphi \\ E \left( \sum_{j=1}^{N_q-m_q} w_j^2 \right) &= (N_q - m_q)\mu E \left( \sum_{i=1}^{N_q} y_i r_i \right) = N_q\mu\varphi \\ E \left( \sum_{j=1}^{N_q-m_q} w_j \right) &= (N_q - m_q)(\sigma^2 - \mu^2) \end{align*}$$

Then we obtain:

$$E(S^2) = \frac{1}{N_q} (N_q(\sigma^2 + \mu^2)\varphi) + \frac{1}{N_q} (N_q - m_q)(\sigma^2 + \mu^2)$$
$$ - \frac{2}{N_q(N_q - 1)} N_q \mu \varphi(N_q - m_q) \mu $$

Therefore the estimator is:

$$ E(S^2) = \varphi(\sigma^2 + \mu^2) + (N_q - m_q) \left[ \frac{1}{N_q}(\sigma^2 + \mu^2) - \frac{2}{N_q - 1}\mu^2\varphi \right] \quad (19) $$

And thus the expectancy of the estimator of the variance is obtained with the imputation technique by random selection.

## Variance of the variance estimator

$$ V(S^2) = V \left[ \frac{(N_q - 1) \left[ \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q - m_q} w_j^2 \right]}{N_q(N_q - 1)} - \frac{2 \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q - m_q} w_j \right)}{N_q(N_q - 1)} \right] $$

$$ V(S^2) = \frac{1}{N_q^2(N_q - 1)^2} \left[ E \left\{ (N_q - 1) \left[ \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q - m_q} w_j^2 \right] \right. \right. \\ \left. \left. - 2 \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q - m_q} w_j \right) \right\} - E^2 \left\{ (N_q - 1) \left[ \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q - m_q} w_j^2 \right] \right. \right. \\ \left. \left. - 2 \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q - m_q} w_j \right) \right\} \right] \quad (20) $$

Separating this expression into its two main terms, and expanding the first term we get:

$$ E \left[ (N_q - 1)^2 \left[ \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q - m_q} w_j^2 \right]^2 \right. \\ \left. - 4(N_q - 1)^2 \left[ \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q - m_q} w_j^2 \right] \left[ \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q - m_q} w_j \right) \right] \right] $$
$$ +4 \left( \sum_{i=1}^{N_q} y_i r_i \right)^2 \left( \sum_{j=1}^{N_q - m_q} w_j \right)^2 \quad (21) $$

Expanding the second term of Equation 20 we get:

$$ N_q(\sigma^2 + \mu^2)\varphi + (N_q - m_q)(\sigma^2 + \mu^2) - 2N_q\mu^2\varphi(N_q - m_q) \quad (22) $$

Joining Equations 21 and 22 we get:

$$ \begin{aligned} V(S^2) = E \left[ (N_q - 1)^2 \left[ \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q - m_q} w_j^2 \right]^2 \right. \\ \left. - 4(N_q - 1)^2 \left[ \sum_{i=1}^{N_q} y_i^2 r_i + \sum_{j=1}^{N_q - m_q} w_j^2 \right] \left[ \left( \sum_{i=1}^{N_q} y_i r_i \right) \left( \sum_{j=1}^{N_q - m_q} w_j \right) \right] \right. \\ \left. + 4 \left( \sum_{i=1}^{N_q} y_i r_i \right)^2 \left( \sum_{j=1}^{N_q - m_q} w_j \right)^2 \right] - \left[ N_q(\sigma^2 + \mu^2)\varphi \right. \\ \left. + (N_q - m_q)(\sigma^2 + \mu^2) - 2N_q\mu^2\varphi(N_q - m_q) \right] \end{aligned} \quad (23) $$

The fully expanded expression has a fourth degree term, therefore we leave the expression as is.

# 4 Empirical development of the proposal

## Analysis of the imputed data base:

For the empirical development of this research, the Agricultural Census of Venezuela for 1998 is used, made up of a quantitative database. The creation of the trees and the exploratory analysis of the data was carried out in previous investigations [2]. Some of the characteristics of the database are:

* In the 984 parishes of Venezuela and taking only the variables to be imputed into account, it can be said that approximately, on average, per parish there are 1153 calving cows, 1687 milking cows, 927 steers, 4136 breeding hens and 183 slime.
* The variables present high variability, which indicates that it would be advisable to previously classify the data to find donors more similar to the absent registries.

* Regarding their marginal distributions, uni-modal, non-symmetric distributions with a bias to the right can be seen, which indicates that the data do not have normality and therefore it will be necessary to transform them so that they comply with the assumptions of some techniques that will be necessary to apply to compare the results.

* All the correlations are low, except for the correlation of 0.7131 between the variables "milking cows" and "steers".

In cases like these (low correlations) it would not be of great importance to maintain the relationship between variables, therefore the technique of classifying one variable at a time with CART could yield good results.

## Tree construction

The tree to be used was the one optimized by [2], the design of the tree that was selected was: Tree (33,30,10,33) which indicates; 33% of the records make up the learning sample, 30 minimum items to segment, 10 levels maximum, and 33% of the records make up the test sample (validation). The regression trees were carried out for each variable that had at least one missing data, but in the case of the variables "breeding hens" and "slime" they could not be classified, therefore, it will be imputed only for the variables: "calved cows", "milking cows" and "steers".

# 5 Comparison between the original data base and the bases with different loss percentages

## Checking assumptions for analysis of variance:

For this, the verification of the ANOVA assumptions is carried out through histograms and Q-Q plot, normality was tested, which required transformations of the data at a logarithmic scale. For the fulfillment of the independence assumption the graphs of residuals against the time sequence and the assumption of randomness was verified by means of the graph of residuals
against the adjusted value of the data, not observing any dependency pattern. Therefore, the resulting ANOVAS will be adequate and the conclusions would be reliable.

**Comparison of the marginal distributions:**

For this technique, changes in the distribution of kurtosis and asymmetry are observed for the variables milking cows, calving cows and reproductive hens in losses of 5%, 10% and 20%, that is, this technique alters the distribution of the variables.

**Mean comparisons:**

It begins with the comparison of the most common statistic used which is the Average.

**ANOVAS of comparisons of transformed data (log) by variable according to percentage of loss:**

It will be tested whether the means between the percentages of loss of the databases, the original complete with those subjected to the proposed imputation process, are significantly different or not for each of the lost variables. This process begins with the comparison of the mean of the different loss percentages and including the original database, that is, the statistical hypotheses raised are:

$H_0 : \mu_{CODB} = \mu_{5\%} = \mu_{10\%} = \mu_{20\%} = \mu_{30\%}$

$H_1 :$ At least one of the means differs from the rest,

where CODB means “complete original database”.

<table><caption>Tab. 1: P-values of one way ANOVAS between loss percentages. Imputation by random selection.</caption><thead><tr><td>Variables</td><td>P-values</td></tr></thead><tbody><tr><td>Calved cows</td><td>0.993</td></tr><tr><td>Milking cows</td><td>0.994</td></tr><tr><td>Steer</td><td>0.898</td></tr><tr><td>Reproductive Chickens</td><td>0.800</td></tr></tbody></table>

The imputations of missing data are not significantly different for the different loss percentages. The F statistic of the analysis of variance that allows
to establish the equality of the effect of the loss percentages on the imputations, yielded statistically non-significant values ($P \ge 0.05$), this occurred for each variable that was imputed and in doing previous use of CART as seen in Table 1. Therefore, the data confirm the hypothesis of equality of means for the four loss percentages (5%, 10%, 20%, 30%) and the Original Database.

**Comparison at the microdata level by variable. Analysis of the variance between biases according to the loss percentages:**

The averages of the difference between the real value and the imputed value are compared.

$$H_0: \mu_{\text{bias } 5\%} = \mu_{\text{bias } 10\%} = \mu_{\text{bias } 20\%} = \mu_{\text{bias } 30\%}$$

$H_1$: At least one of the means differs from the rest.

<table><caption>Tab. 2: Bias comparison. P-values of one way ANOVAS between loss percentages. Imputation by random selection.</caption><thead><tr><td>Variables</td><td>P-values</td></tr></thead><tbody><tr><td>Calved cows</td><td>0.788</td></tr><tr><td>Milking cows</td><td>0.818</td></tr><tr><td>Steer</td><td>0.097</td></tr></tbody></table>

In general, it can be seen in Table 2, for the rest of the study variables, with an error of 5%, there are no significant differences between the biases of the different loss percentages for the imputed variables.

**6 Comparison of the variance and co-variance matrices to identify the relationships between variables**

In order to check whether the imputation of the data maintains the relationship between the variables, a comparison is made between the variance and covariance matrices which are made up of the variables that had at least one data to be imputed with respect to the complete database.

According to [22], the multivariate proofs for co-variance matrices equality are as follows: For $k$ multivariate populations, the hypothesis of variance matrices equality is

$$H_0: \sum_1 = \sum_2 = \cdots = \sum_k$$
and for $k = 2$ we have:

$$H_0: \sum_1 = \sum_2 \qquad (24)$$

This is not exactly a proof of $H_0: \sum_1 = \sum_2$ as being analogous to the univariate case. It is assumed that the independent samples of sizes $n_1, n_2, \dots, m_k$ of normal multivariate distributions. In order to prove this, the following is calculated:

$$M = \frac{|S_1|^{v_1/2} |S_2|^{v_2/2} \dots |S_k|^{v_k/2}}{|S_{pl}|^{\sum_i v_i/2}} = \left( \frac{|S_1|^{v_1/2}}{|S_{pl}|} \right) \left( \frac{|S_2|^{v_2/2}}{|S_{pl}|} \right) \dots \left( \frac{|S_k|^{v_k/2}}{|S_{pl}|} \right).$$

If $S_1 = S_2 = S_{pl}$ then $M = 1$, and as the disparity between $S_1, S_2, \dots, S_k$ is incremented, $M$ tends to zero. For $k = 5, p = 5$ variables and $\nu = 268$ where $k$ is the number of matrices to compare, $\nu$ is the degree of freedom ($n - 1$). Table 3 shows the theoretical and obtained results:

<table><caption>Tab. 3: Theoretical and obtained T values. Imputation technique by random selection.</caption><thead><tr><th>Obtained value</th><th>Theoretical value</th></tr></thead><tbody><tr><td>T<sub>2</sub> 0.4569</td><td>M<sub>T<sub>2</sub></sub> 0.4546</td></tr><tr><td>T<sub>3</sub> 14.1450</td><td>M<sub>T<sub>3</sub></sub> 14.0743</td></tr><tr><td>T<sub>4</sub> 42.4048</td><td>M<sub>T<sub>4</sub></sub> 42.1929</td></tr><tr><td>T<sub>5</sub> 31.6364</td><td>M<sub>T<sub>5</sub></sub> 31.4783</td></tr></tbody></table>

Seeking an approximation to

$$X^2 \left( \frac{1}{2}(k-1)p(p+1) \right)$$

which is rejected if $MT \ge X_{\alpha}^2$.

Since the calculated value is greater than the theoretical value, the covariance matrices equality hypothesis is rejected, that is, $H_0: \sum_{original} = \sum_{imputed}$, therefore, as corroborated in the uni-variant correlations, the technique using random selection does not preserve the relations between variables.
# 7 Comparison of the imputed data base with and without the use of the regression trees.

The aim is to determine whether there are significant differences in the use of the Regression Tree in the data estimates, using the random selection imputation technique. For this, the comparison is done with the greatest loss of data, that is, 30%, the results are observed in Table 4.

<table><caption>Tab. 4: Means and standard deviations of the imputed data base with and without the use of the regression tree</caption><thead><tr><th rowspan="2">Variable</th><th colspan="2">Calved cows</th><th colspan="2">Milking cows</th><th colspan="2">Steer</th></tr><tr><th>Mean</th><th>Stand. dev.</th><th>Mean</th><th>Stand. dev.</th><th>Mean</th><th>Stand. dev.</th></tr></thead><tbody><tr><td>CODB</td><td>1152,8</td><td>3239,3</td><td>1686,3</td><td>4207,2</td><td>926,9</td><td>2186,2</td></tr><tr><td>Without tree</td><td>928,8</td><td>2538,9</td><td>1382,7</td><td>3497,5</td><td>832,5</td><td>1830,5</td></tr><tr><td>With tree</td><td>1145,8</td><td>3149,8</td><td>1683,9</td><td>4224,6</td><td>905,4</td><td>1633,2</td></tr></tbody></table>

To make a more effective comparison, an ANOVA is applied comparing the complete databases, without the use of the regression tree and with the use of the regression tree, that is,

$H_0: \mu_{CODB} = \mu_{\text{mean without tree}} = \mu_{\text{mean with tree}}$

$H_1$: At least one of the means differs from the rest

<table><caption>Tab. 5: P-values of one way ANOVAS between imputed data with and without the use of regression tree and the complete data base.</caption><thead><tr><th>Variables</th><th>P-value</th></tr></thead><tbody><tr><td>Calved cows</td><td>0.18*</td></tr><tr><td>Milking cows</td><td>0.08**</td></tr><tr><td>Steer</td><td>0.09**</td></tr></tbody></table>

* No significant difference

** Significant difference

As all the P-values are greater than 0.05, as seen in Table 5, it is concluded that there are no significant differences in the estimates of the data, since
the technique preserves good estimates, however, at P-values lower than 0.1
there are significant differences in the variables Milking Cows and Steers,
which in Tukey's posterior tests corresponds to the difference to the mean
without tree.

8 Conclusions

The proposed methodology is based on the elaboration of a Regression Tree
as a previous step to the classical imputation by Random Selection, whose
imputation is carried out in the final nodes where there are donors as similar
as possible to those absent due to the fact of having passed by the classifica-
tion process. Therefore, this allows more efficient estimators to be obtained
than those generated by the classic Random Selection Method. In addition,
the proposed methodology identifies the atypical data, additionally allowing
a data editing phase to review inconsistencies in the information, and the
possibility of eliminating the atypical data and replacing it with an imputed
one.

In the empirical analysis, making use of quantitative data with completely
random losses of 5%, 10%, 15%, 20% and 30%, applying the imputation
techniques by Random Selection, with previous classification of the use of re-
gression trees, relationships between variables are preserved and sometimes
improved when compared with techniques without the use of regression trees.
Furthermore, the regression trees remain robust, since in the ANOVAS no
significant differences were found between the loss percentages and the orig-
inal database.

The estimators for the Mean and Totals of the random selection impu-
tation technique under the use of regression trees have properties that allow
obtaining unbiased data. On the other hand, as an added benefit, no as-
sumption testing, or data transformation is needed to build the data and
impute. In this study, it was necessary to do so to make use of the Analysis
of Variance technique.

The proposal does not solve one of the problems of the classical tech-
nique, which is the non-preservation of the marginal distributions. Another
drawback in the proposed methodology is that the regression tree cannot
always be created for variables with missing data, if the variable is highly
unrelated to the complete variables, the tree is not formed and the proposed
methodology cannot be built.
References

[1] R. Rindfuss, M. Choe, N. Tsuya, L. Bumpass, and E. Tamaki. Do low survey response rates bias results? evidence from Japan. *Demographic Research*, 32(Article 26):797-828, 2015. doi: 10.4054/DemRes.2015.32.26.

[2] D. Mesa, V. Márquez, L. Useche, and A. Chacón. Estrategia de imputación con la media bajo el uso de árboles de regresión. *Comunicaciones en Estadística*, 10:9-40, 2017. https://dialnet.unirioja.es/servlet/articulo?codigo=6765744.

[3] S. Khan and A. Latiful. Sice: an improved missing data imputation technique. *Journal of Big Data*, 7(Article number: 37):1-21, June 2020. doi: 10.1186/s40537-020-00313-w.

[4] D.B. Rubin and R.J. Little. Statistical analysis with missing data. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., 2014. ISBN 978-1-119-01356-3.

[5] R. Medina-Merino and C. Nique-Chacón. Bosques aleatorios como extensión de los árboles de clasificación con los programas R y Python. *Interfases*, (010):165-189, 2017. doi: 10.26439/interfases2017.n10.1775.

[6] L. Brieman, J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Wadsworth and Brooks, Monterry, CA, 1984. https://www.bibsonomy.org/bibtex/27f293aa2bdfd10960ef36928f2795f1d/flashspys, ISBN 9780412048418.

[7] S.S. Wilks. Moments and distributions of estimates of population parameters from fragmentary samples. *The Annals of Mathematical Statistics*, 3(3):163-195, August 1932. doi: 10.1214/aoms/1177732885.

[8] F. Medina and M. Galván. Imputación de datos: teoría y práctica, volume 54 of Estudios estadísticos y prospectivos. Comisión Económica para América Latina y el Caribe (CEPAL), 2007. ISBN 978-92-1-323101-2.

[9] D.B. Rubin. Inference and missing data. *Biometrika*, 63(3):581-592, December 1976. doi: 10.1093/biomet/63.3.581.
[10] I.G. Sande. Imputation in surveys: Coping with reality. *The American Statistician*, 36(3a):145-152, March 1982. doi: 10.1080/00031305.1982.10482816.

[11] R. Todeschini. Weighted k-nearest neighbour method for the calculation of missing values. *Chemometrics and Intelligent Laboratory Systems*, 9(2):201-205, September 1990. doi: 10.1016/0169-7439(90)80098-Q.

[12] P. Koikkalainen. Neural networks for editing and imputation. In *DataClean 2002 Conference, Jyväskylä (Finland)*, pages 1013–1035, 2002. http://erin.it.jyu.fi/dataclean/abstracts/node33.html#SECTION0009300000000000000.

[13] Z. Geng and K. Li. Factorization of posteriors and partial imputation algorithm for graphical models with missing data. *Statistics & Probability Letters*, 64(4):369-379, Octuber 2003. doi: 10.1016/S0167-7152(03)00181-0.

[14] J.K. Kim and H. Park. Imputation using response probability. *Canadian Journal of Statistics*, 34(1):171-182, 2006. doi: 10.1002/cjs.5550340112.

[15] J. Kim and W. Fuller. Fractional hot deck imputation. *Biometrika*, 91(3):559-578, 2004. http://www.jstor.org/stable/20441123.

[16] T. Gleason and R. Staelin. A proposal for handling missing data. *Psychometrika*, 40(2):229-252, June 1975. doi: 10.1007/BF02291569.

[17] M. Vásquez. *Aportación al análisis Biplot: Un enfoque algebraico*. PhD thesis, Universidad de Salamanca, Spain, January 1995. https://www.deberes.net/tesis-doctorales/espana/matematicas/aportaciones-al-analisis-biplot-un-enfoque-algebraico/.

[18] R. Chambers, T. Crespo, Seppo Laaksonen, P. Piela, P. Tsai, and T. De Waal. The AUTIMP-project: Evaluation of WAID. pages 1-10, January 2001. https://www.researchgate.net/publication/255565338_The_AUTIMP-project_Evaluation_of_WAID.

[19] D. Martínez. Estimación e imputación de datos faltantes en modelos longitudinales con variable respuesta tipo Poisson y binomial negativa
con exceso de ceros. Master’s thesis, Universidad Nacional de Colombia,
Bogotá D.C., Colombia, 2018.

[20] A. Polo Serrano. *Evaluación del algoritmo kNN para imputación de datos*. PhD thesis, Universitat Politecnica de Valencia, Valencia, Spain, March 2017. https://riunet.upv.es/handle/10251/78412.

[21] J. Bussi, L. Hernández, G. Marí, F. Méndez, and G. Mitas. Visualización y métodos de imputación de datos faltantes en la encuesta de gastos en los hogares. In *Vigesimoterceras Jornadas “Investigaciones en la Facultad” de Ciencias Económicas y Estadística*, volume 133. Universidad Nacional del Rosario, December 2018. http://hdl.handle.net/2133/14011.

[22] A.C. Rencher. *Methods of Multivariate Analysis*. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., 2002. ISBN 9780471418894, doi: 10.1002/0471271357.