Introduction

The standard of education in a country has a significant impact on the growth of any community or nation. It is widely accepted that the development of human resources must be the first step towards any real progress. A country’s level and quality of education are directly related to its development. Social, demographic, dietary, and environmental factors have all been suggested to have a significant impact on academic ability in elementary school and higher education. Children are regarded as a nation’s most valuable resource and help shape its future1. More than 200 million children in poor nations suffer from developmental deficiencies, according to reports in2,3. It is crucial to address inadequate child development since these children will likely have lower educational attainment, worse health in later life stages, a decreased chance of finding work, and lower incomes3.

According to recent studies, cognitive performance is better predicted by the mother’s educational background4. Research on the Senna project in Brazil has demonstrated how socio-familial factors affect kids’ motivation for learning5,6,7. The author8 concentrates only on children aged 4–10 (preschool and early primary education). It doesn’t explore how family and school impacts might change as children grow older. And the study focused on the school environment, which is measured mainly by community and type of school. But other important school factors like place of residence, distance from their school, regional differences, and parents’ educational background were not examined.

According to some authors, children from families with lower cultural and socioeconomic advantages struggle more in school and have fewer language and numeracy skills. Children from the most privileged social classes typically continue to face these challenges and inequalities throughout later school years4,6, and8. It should be noted that community (rural vs. urban) and type of school (public vs. private) are also relevant in cognitive performance and academic achievement having better results. From the evidence of8 the family and school are two fundamental institutions that, at present, share educational functions, contributing to and influencing the development of children. Furthermore, authors9 examined that there is a link in developing countries where educational systems and access to them vary widely.

It is commonly known that risk factors for poor cognitive development include viral diseases, poverty, malnutrition, poorer parental socioeconomic status, and family-environmental adversity10,11. Individual-level influences, particularly in developing nations, have been the focus of certain authors’ studies on cognitive development12. The authors13 studies concerning academic results were taken from report cards, but other indicators like standardized test scores could provide a broader view of achievement.

Various studies have provided evidence of the importance of the cognitive development of individuals, which also affects their educational achievements. Research indicates that health problems due to miserable nutritional status in primary school-age children, distance from school, place of residence, and unsatisfactory classroom performance are among the most common causes of low school enrollment14. Approximately 200 million children worldwide do not reach their developmental potential as a result of under nutrition and poverty14. Of these, more than 170 million children are stunted (i.e., height-for-age Z-score (HAZ) more than 2 standard deviations below the reference for sex and age). Therefore, less height-for-age and weight-for-age in childhood is one of the reasons behind the high child mortality rates observed in developing countries14,15,16. And it is a chronic childhood is linked to slower cognitive development and serious health impairments later in life and reduces the quality of life of individuals.

As previously indicated, some studies have presented students’ cognitive achievement in different ways. Most of the authors used some measures of cognitive achievement, like science and mathematics, separately. Additionally, they made an effort to demonstrate some descriptive statistics and classical regression models. This indicates that there are limitations in the literature regarding the cognitive achievement of primary school students. The authors in study17 examined the physical growth and cognitive development of children, indicating a relationship between cognitive development and nutritional status of children.

This paper makes several important contributions to the examination of the determinants of children’s cognitive development. Rather than focusing on a single dimension of socioeconomic status, we take a broader approach by examining the resources that parents provide for their children. We utilize measures of parental schooling, household size, and children’s nutritional status, allowing us to observe the relationships between these various household and other factors.

In this study, we will investigate both individual and community-level factors that concurrently impact cognitive achievement. We will employ multivariate multilevel modeling on the three measures of student cognitive achievement, which will allow us to jointly regress multiple outcomes on multilevel explanatory variables. This approach will enable us to investigate students’ cognitive achievements and explore its association with their multidimensional factors in Ethiopia using the Young Lives data.

Data description and sample design

Since 2002, Ethiopia, India, Peru, and Vietnam have been the sites of the groundbreaking longitudinal study on children poverty known as Young Lives. Addis Ababa, Amhara, Oromia, Southern Nations Nationalities and Peoples (SNNP), and Tigray are the five areas in Ethiopia where sites were selected, but other regions were excluded because the number of child participants was too small9,18. We used round 5 data from the old cohort in Ethiopia, which was gathered in 2016–17, for this investigation18. Children with missing data on outcomes and predictors at the individual level were dropped. Outliers in outcomes based on the WHO standards19 were also dropped. Details on sample variable construction and other information related to the data are available at www.younglives.org.uk18. Thus, a total sample of 1726 children was used in this study.

Outcome variables: The study focus on three main indicators of cognitive achievement: mathematics skill score, reading skill score, and Peabody skill score. The Peabody Picture Vocabulary skill score (PPVT), a popular assessment of a child’s receptive language abilities that is used to gauge vocabulary growth in both high-income and low-income nations, was used to measure receptive vocabulary4,5. A mathematics skills score including counting, number discrimination, number awareness, basic operations with numbers, and brief problems pertaining to everyday math application was used to evaluate students’ mathematical ability. A test that included simple activities like matching words to images to more difficult ones like reading short texts and answering comprehension questions was used to evaluate reading skills score. All tests were designed by educational experts and adapted to the formal curricula of the country and in the local language as appropriate9. Therefore the three indicators of cognitive achievement were used as the dependent variables, and the values were directly converted to 100%.

Predictor variables

The predictor variables were the sex of child, the time taken from home to school, the transportation used from home to school, the household size of the family, the parents educational level for both mother and father, age of child in month, height-for-age, weight-for-age for child, and access to services. The variables at the school level we have used were place of residence region, and school type.

Statistical models

In this study we used a multivariate regression model and multivariate multilevel models.

Multivariate multiple linear regression model

A popular statistical technique for examining the relationship between two or more variables, including the response variable and the predictor variable20. In this study, our interest is to predict/explain several responses of \(\:P\) response variables \(\:{(\text{Y}}_{1},{\text{Y}}_{2}\dots\:,{\text{Y}}_{\text{p}})\) from a set of predictors\(\:{(\text{X}}_{1}\), \(\:{\text{X}}_{2},\dots\:,{\text{X}}_{\text{q}})\)24,25. Therefore, the multivariate multiple linear model is given by:

$$\:{\text{Y}}_{\text{i}1}={{{\upbeta\:}}_{01}+{\text{X}}_{\text{i}1}{\upbeta\:}}_{11}+{{\text{X}}_{\text{i}2}{\upbeta\:}}_{21}+\dots\:+{{\text{X}}_{\text{i}\text{p}}{\upbeta\:}}_{\text{p}1}+{\epsilon}_{\text{i}1}$$
$$\:{\text{Y}}_{\text{i}2}={{{\upbeta\:}}_{02}+{\text{X}}_{\text{i}1}{\upbeta\:}}_{12}+{{\text{X}}_{\text{i}2}{\upbeta\:}}_{22}+\dots\:+{{\text{X}}_{\text{i}\text{p}}{\upbeta\:}}_{\text{p}2}+{\epsilon}_{\text{i}2}$$
(1)
$$\:{\text{Y}}_{\text{i}3}={{{\upbeta\:}}_{03}+{\text{X}}_{\text{i}1}{\upbeta\:}}_{13}+{\text{X}}_{\text{i}2}{{\upbeta\:}}_{23}+\dots\:+{\text{X}}_{\text{i}\text{p}}{{\upbeta\:}}_{\text{p}3}+{\epsilon}_{\text{i}3}$$

.

Equivalently in matrix form, it can be written as follows:

$$\:\mathbf{Y}=\mathbf{X}\varvec{\upbeta\:}+\mathbf{\epsilon}$$
(2)
$$\begin{aligned} & \:\left(\begin{array}{ccc}{\text{Y}}_{11}&\:{\text{Y}}_{12}&\:\begin{array}{cc}\cdots\:&\:{\text{Y}}_{1\text{m}}\end{array}\\\:{\text{Y}}_{21}&\:{\text{Y}}_{22}&\:\begin{array}{cc}\cdots\:&\:{\text{Y}}_{2m}\end{array}\\\:\begin{array}{c}\vdots\\\:{\text{Y}}_{\text{n}1}\end{array}&\:\begin{array}{c}\vdots\\\:{\text{Y}}_{\text{n}2}\end{array}&\:\begin{array}{cc}\begin{array}{c}\cdots\:\\\:\cdots\:\end{array}&\:\begin{array}{c}\vdots\\\:{\text{Y}}_{\text{n}\text{m}}\end{array}\end{array}\end{array}\right)=\left(\begin{array}{ccc}\begin{array}{cc}1&\:{\text{X}}_{11}\end{array}&\:{\text{X}}_{12}&\:\begin{array}{cc}\cdots\:&\:{\text{X}}_{1\text{p}}\end{array}\\\:\begin{array}{cc}1&\:{\text{X}}_{21}\end{array}&\:{\text{X}}_{22}&\:\begin{array}{cc}\cdots\:&\:{\text{X}}_{1\text{p}}\end{array}\\\:\begin{array}{cc}\begin{array}{c}\vdots\\\:1\end{array}&\:\begin{array}{c}\vdots\\\:{\text{X}}_{\text{n}1}\end{array}\end{array}&\:\begin{array}{c}\vdots\\\:{\text{X}}_{\text{n}2}\end{array}&\:\begin{array}{c}\begin{array}{cc}\vdots&\:\vdots\end{array}\\\:\begin{array}{cc}\cdots\:&\:{\text{X}}_{np}\end{array}\end{array}\end{array}\right)\\& \left(\begin{array}{ccc}\begin{array}{cc}{{\upbeta\:}}_{01}&\:{{\upbeta\:}}_{02}\end{array}&\:\cdots\:&\:{{\upbeta\:}}_{0\text{m}}\\\:\begin{array}{cc}{{\upbeta\:}}_{11}&\:{{\upbeta\:}}_{12}\end{array}&\:\cdots\:&\:{{\upbeta\:}}_{1\text{m}}\\\:\begin{array}{cc}\begin{array}{c}\vdots\\\:{{\upbeta\:}}_{\text{p}1}\end{array}&\:\begin{array}{c}\vdots\\\:{{\upbeta\:}}_{\text{p}2}\end{array}\end{array}&\:\begin{array}{c}\cdots\:\\\:\cdots\:\end{array}&\:\begin{array}{c}\vdots\\\:{{\upbeta\:}}_{\text{p}\text{m}}\end{array}\end{array}\right)+\left(\begin{array}{ccc}\begin{array}{cc}{\epsilon}_{11}&\:{\epsilon}_{12}\end{array}&\:\cdots\:&\:{{\:\epsilon}}_{1\text{m}}\\\:\begin{array}{cc}{\epsilon}_{21}&\:{\epsilon}_{22}\end{array}&\:\cdots\:&\:{\epsilon}_{2m}\\\:\begin{array}{cc}\begin{array}{c}\vdots\\\:{\epsilon}_{\text{n}1}\end{array}&\:\begin{array}{c}\vdots\\\:{\epsilon}_{\text{n}2}\end{array}\end{array}&\:\begin{array}{c}\cdots\:\\\:\cdots\:\end{array}&\:\begin{array}{c}\vdots\\\:{\epsilon}_{nm}\end{array}\end{array}\right)\end{aligned}$$

.

Where \(\:\text{Y}\) is the \(\:(n\times\:m)\) matrix of dependent variable, \(\:\text{X}\) is the \(\:n\times\:(p+1)\) matrix of independent variables, \(\:{\upbeta\:}\) is \(\:(p+1)\times\:m\) matrix of regression coefficient (parameters), and \(\:\epsilon\) is the \(\:(n\times\:m)\) matrix of residuals (errors).

In this study, principal component analysis (PCA) was analyzed for a reduction in data set by finding linear combinations of the original independent variables23,27.

Multivariate multilevel model

Multilevel modeling, or hierarchical linear models, are the superior statistical approaches for hierarchical data, which are generated by sampling procedures like the Young Live survey mentioned above. Several levels are used to study the multilevel modeling, which shows that lower levels are nested within higher levels21. The assumption violation that occurred in multiple linear regression as a result of the hierarchical data structure can also be accommodated by multilevel modeling20,21,22,23.The models assume that the dependent variable is measured at the lowest level and the independent (explanatory) variables are measured at all available levels23,24,25,28.

Since the data are clustered in nature—specifically, the joint dependent variables at level 1 are nested within students (level 2), who are themselves nested within schools (level 3)—we relied on the multivariate multilevel model as advocated in24,25,26. To address our aims, we followed a step-up model-building approach28,29. First, a null model (M1) with no predictors was estimated to jointly assess the variances and covariances of response variables. In the second model (M2), all student-level predictors were included. In the final model (M3), school contextual predictors were added.

Therefore, this study applies MVMM to identify the schools and individual characteristics that influence Ethiopian children’s cognitive achievement from three perspectives and its relationship with their multidimensional factors: the children performance in reading skill score, Peabody vocabulary skill score, and mathematics skill score.

Model 1: null model (Variance component model)

The variance component model a model without explanatory variables and has the form of29:

$$\:{Y}_{ijk}={\beta\:}_{0}+{U}_{0j}+{\epsilon}_{ij}$$
(3)

.

The model breaks down the total variance into two components: school and student levels. These represent the variations in children’s cognitive achievement both within and between schools29,30. The interclass correlation measures correlation between observations within a cluster as:

$$\:ICC=\frac{{{\sigma\:}_{u}}^{2}}{{{\sigma\:}_{u}}^{2}+{{\sigma\:}_{\epsilon}}^{2}}$$
(4)

.

Where \(\:\text{I}\text{C}\text{C}\) is the intraclass correlation coefficient, \(\:{{\sigma\:}_{u}}^{2}\) is variance of random intercept at the school level, and \(\:{{\sigma\:}_{\epsilon}}^{2}\) is variance intercept at the individual level.

Model 2: random intercept and fixed slope model

This model has a random intercept and fixed slopes for the variables at students and school levels. The model is defined as32:

$$\:{Y}_{ijk}={\beta\:}_{0}+{\beta\:}_{1}{X}_{1ij}+\dots\:+{\beta\:}_{m}{X}_{mij}+{U}_{oj}+{\epsilon}_{ij}$$
(5)

.

Model 3: random intercept and random slope models

This model, also known as random slopes models, is an extension of multilevel or hierarchical linear modeling. Finally, the general form of a random coefficient multilevel model can be written as29,30:

$$\:{Y}_{ijk}={\beta\:}_{0}+{\beta\:}_{1}{X}_{1ij}+\dots\:+{\beta\:}_{m}{X}_{mij}+{u}_{oj}+{u}_{1j}{X}_{1ij}+\dots\:+{u}_{1m}{X}_{mij}+{\epsilon}_{ij}$$
(6)

.

\(\:{Y}_{ijk}\)​: represent the i cognitive outcome for \(\:j\:\)student in \(\:k\:\)school, \(\:{\beta\:}_{0}\)is overall intercept (fixed effect), \(\:{\beta\:}_{1}\)​: Average slope of the predictor for predictor \(\:{x}_{ij}\) across all groups (fixed effect), \(\:{u}_{0j}\)​: Random intercept for group \(\:j\:\)(group-level variation) capturing school-specific deviations from the overall intercept (assumed \(\:{u}_{0j}\sim\:N(0,{{\sigma\:}_{u0}}^{2})\), \(\:{u}_{1j},\dots\:,{u}_{1m}\): random slope for group j, capturing group specific deviation from the overall slope (\(\:{u}_{1j}\sim\:N(0,{{\sigma\:}_{u1}}^{2})\)), \(\:{x}_{ij}\): Predictor variable for individual \(\:i\) in group \(\:j\) and \(\:{\epsilon}_{ij}\)​: Individual-level residual error (assumed \(\:{\epsilon}_{ij}\)​ \(\:\sim\:N(0,{{\sigma\:}_{{\epsilon}}}^{2})\)).

Parameter Estimation of the models

The multivariate multiple and multilevel linear regression model parameters can be estimated using maximum likelihood method31. The data analysis was conducted using SPSS and R version 4.2.3 software.

Results and discussion

Based on Young live data in 2016/17, the percentage of male students in the study was more than female students, which was around 53.1%. Descriptions of each response variable and predictor variable are summarized as descriptive statistics in Tables 1 and 2. The highest average score of student achievement was found in Peabody picture vocabulary skill score with 73.8 and a standard deviation of 15.8. The average score of cognitive achievement in mathematics skill score was 37, and the standard deviation was 14.9. And the average score of students in reading skill score was 51.7 with a standard deviation of 21.

The average height-for-age z-score and weight-for-age z-score were found to be −1.33 and − 1.64, respectively, with standard deviations of 1.1 and 0.97. About 95.6% of the students used transportation by walking, and around 4.4% of other students used public buses and other ways of transportation. On average, the distance that a student traveled from his/her home to school was approximately 22.3 min with a standard deviation of 15.2 min. The average number of members in a household was found to be 5.8, which implies that on average there were 5 members in a household.

With regard to the educational levels of mothers, 48.5%, 27.2%, 13.1%, and 11.2% had no formal education, primary school, secondary school, and diploma and above levels, respectively. This implies that most of them have no education. The educational levels of fathers 33.9%, 21.3%, 19.5%, and 25.3% for no formal education, primary, secondary, and diploma and above level, respectively. Regarding access to some services, it was observed that around 64.1% and 53.8% have access to electricity and drinking water respectively.

The average age of student participants in the study was approximately 181 months. Similarly, the results also show that five regions participated in the study: SNNP, Oromia, Tigray, Amhara, and Addis Ababa. The SNNP Region has the largest representation, with 25% of the students coming from this region. Oromiya follows with 22.4%, representing a large proportion of the students as well. Tigray and Amhara, at 19.8% and 18.4% respectively, have relatively similar proportions. Addis Ababa has the smallest representation, with 14.5% of students from this region.

Concerning place of residence, 59.4% of the students were from rural areas, and 40.6% of students were from urban areas. About 26.4% and 73.6% of the students who participated in the study came from private and public schools, respectively.

Table 1 Summary of descriptive statistics for the response Variables.
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Table 2 Summary of descriptive statistics for predictor Variables.
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Results of multivariate regression modeling

A multivariable linear regression was conducted, incorporating various predictor variables for three indicators of student cognitive achievement, as presented in Appendix A1. For mathematics skill scores, the statistically significant predictors at the 5% level included: type of transportation used by the student, height-for-age, school type, mother’s educational level, household size, place of residence, region, and access to electricity. However, the sex of the children, age in months, time taken from home to school, weight-for-age, father’s educational level, and access to drinking water were not statistically significant at the 5% level.

For cognitive achievement in reading skills scores, the results indicate that the following factors are statistically significant at the 5% level: age in months, type of transportation used by the student, height-for-age, school type, mother’s educational level, time taken from school to home, household size, place of residence, region, and access to electricity. Conversely, sex, weight-for-age, father’s educational level, and access to drinking water are not statistically significant at the 5% level.

For the Peabody Picture Vocabulary Test (PPVT) scores, the statistically significant predictors at the 5% level include: age in months, type of transportation from home to school, time taken in minutes, height-for-age, weight-for-age, school type, mother’s educational level, region, household size, place of residence, access to drinking water, and access to electricity. However, sex and father’s educational level were not statistically significant at the 5% level.

By comparing the performance of the three selection techniques we opted for stepwise regression techniques in order to select the variables for the case of three indicators of cognitive achievement of students. Therefore, predictor variables presented in appendix B1-B3 were selected.

Section I: results of principal component

Table 3 presents the results of a principal component analysis with five components. And the result shows that the first principal component is the most significant because it represents a comprehensive measure of student cognitive achievement across various skills. It combines high contributions from mathematics, reading, and PPVT skill scores, making it an all-encompassing indicator of student performance.

Table 3 Results of principal Components.
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Results of the multilevel models analysis

In this section we want to see the results of the three multilevel models analysis for the three cognitive achievement skill score.

Section II: results of the cognitive achievement in mathematics skill score variable

The results from Table 4 below indicate that the overall average of cognitive achievements in mathematics score was found to be significant, and it is estimated to be 37.75% with a standard error of 1.33.The variance of school-level mean cognitive achievement in mathematics skill score around the grand mean is 39.33. The standard deviation is 6.27, indicating that most schools have a mean on cognitive achievement in mathematics skill score within \(\:\pm\:6.27\) points of the overall mean. The results indicate that 17.6% of the variability in mathematics scores is due to differences between schools, while 82.4% is due to individual differences within schools.

Table 4 Result of null model for mathematics cognitive achievement Score.
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The results presented in Table 5 below show that the type of transportation student used, the number of the household size, access to electricity, the place of residence, the region, the father’s educational level, the mother’s educational level, the height for age, and the school type were statistically significant at the 5% level of significance. The cognitive achievement in mathematics skill score increased by 0.74% if students used the mode of transportation of public buses as compared to the reference category of walking. But there is a negative relationship between the household size and cognitive achievement of mathematics score; it implies that increasing the number of household size decreases the cognitive achievement of students in mathematics skill score by 0.47%.

The results show that the students whose mothers have a primary education score 1.95% higher than those whose mothers have no education, on average. Students whose mothers have a secondary education score 5.13% higher, and those with mothers who have a diploma or higher score 1.49% higher than students whose mothers have no education respectively, and these differences are statistically significant at the 5% level of significance, and similar interpretation for father educational level. These indicates that higher parental education is strongly associated with better student cognitive achievement of mathematics skill score. A 1-unit increase in height for age is associated with a 3.92% increase in cognitive achievement of mathematics skill scores. This suggests that chronic malnutrition (stunting) is negatively associated with cognitive achievement. The results concerning the regional effects show that students in the Oromiya and SNNPR regions score − 2.81% and − 2.13% lower than those in Tigray, and these differences are not statistically significant. But students in Addis Ababa score 12.17% higher compared to those in Tigray. This difference is statistically significant, suggesting regional disparities in mathematics skill scores.

Students in private schools score 2.56% higher than those in public schools, and the difference is statistically significant at the 5% level of significance. And students with access to electricity score 2.82% higher than those without access, suggesting the importance of electric power infrastructure for learning. Students in urban areas score 1.06% higher than those in rural areas, and the differences is statistically significant at the 5% level of significance. The variance intercept of school-level mean cognitive achievement in mathematics skill score around the grand mean is 2.43 with a standard deviation of 1.56, indicating that most schools have a mean on cognitive achievement in mathematics skill score within \(\:\pm\:1.56\) points of the overall mean.

Table 5 Result of random intercept for achievement in mathematics Score.
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The results in Table 6 suggest that the place of residence, age of students in month, household size,,access to electricity, school type, height-for-age, mother educational level, father educational level, and region have statistically significant results at a 5% level of significance. But the other variables were not statistically significant. The variance intercept of the random slope is 4.11, which indicates that the variability in the relationship between a cognitive achievement in mathematics skill score and the predictor variables, such as school type, and mother educational level, across the schools is significant. This indicates that there is significant variation in mathematics skill scores across schools, even after accounting for all the included predictors.

The random slope variance for school type, and mother educational level were 10.60, 11.30, 0.12, and 4.04, respectively. The values indicate that the effect of mother educational level, and school type on the cognitive achievement in mathematics skill score varies across schools with standard deviations of 3.26, 3.36, 0.34, and 2.02, respectively. After accounting for both fixed and random effects, the variance (181.04) and standard deviation (13.45) show that individual mathematics skill scores within schools vary by ± 13.45 points. The correlations between various factors and mathematics cognitive achievement scores indicate varying relationships. There is a weak correlation (0.09) between the intercept and mother’s primary education and the same interpretation for other variables. The effect of attending a private school on mathematics cognitive achievement has moderate correlations (0.57 with the intercept, 0.89 with diploma education, 0.36 with secondary education, and 0.48 with primary education).

Table 6 Result of random slope for mathematics cognitive achievement Score.
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Section III: results of the multilevel models for cognitive achievement in reading score

The results shown in Table 7 below show that the mean score of cognitive achievements in reading skill was found to be significant, and it is estimated to be 53.44% with a standard error of 2.43. The variance of school-level mean cognitive achievement in reading skill score around the grand mean is 135.50 with a standard deviation of 11.60, indicating that most schools have a mean cognitive achievement in reading skill score within \(\:\pm\:11.60\) points of the overall mean.

The intraclass correlation coefficient shows that 30.70% of the variation in the cognitive achievement of reading skills score of the students is accounted to the between schools, while 69.30% is accounted to the within students. It indicates that pupils’ scores in reading skills vary notably across students.

Table 7 Result of null model for reading cognitive achievement Score.
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From the results of Table 8 below, the amount of time it took to get to school, the student’s age in months, the size of the household, access to electricity, the student’s height for age, the type of school, the place of residence, the student’s region, and the mother’s educational attainment were found to be significant at the 5% level of significance. According to the results, there is a negative correlation between the cognitive achievement of reading skills and some variables, such as the amount of time spent traveling from home to school, and the size of the household. However, there is a positive correlation between other predictor variables and the student cognitive achievement in reading skills.

The regional effects: students in the SNNP region score − 6.67% lower as compared to those in the Tigray region (reference category). The student in the Addis Ababa region scores 23.26% higher as compare it with Tigray region (reference category). Students whose mothers have a primary and secondary education level score were 2.18% and 2.9% respectively and higher than those whose mothers have no formal education. And other variables can be interpreted in similar ways. The results also indicate that the variance of school-level mean cognitive achievement in reading skill score around the grand mean is 31.62 with a standard deviation of 5.62, indicating that most schools have a mean cognitive achievement in reading skill score within \(\:5.62\pm\:\) points of the overall mean.

Table 8 Result of random intercept for reading cognitive achievement Score.
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From the result of Table 9 below if the household size member increases by one person, the average of cognitive achievement in reading skill score is decreased by 1.13%. The students with access to electricity in reading skill score was 2.91% higher as we compare with those without access of electricity. The results presented show that the cognitive achievement in urban areas was 6.45% higher than that in rural areas, and the private schools had a score 3.35% higher than public schools in the cognitive achievement of reading skill score. The results also indicate that the cognitive achievement skill score of reading in the Addis Ababa region was 23.50% higher than that in the Tigray region. But, in SNNPR, it is 6.87% lower than in the Tigray region. The cognitive achievement in reading skill score for secondary educational level of mother’s was 3.08% higher than that of no formal educational levels. If the height-for-age score of the students increases by a unit, the cognitive achievement of reading skills of the students increased by 0.89%.

The variance intercept of the random slope is 27.90, which indicates that the variability in the relationship between a cognitive achievement in reading score and school type, and mother’s educational level across the schools. The random coefficient variance for school type, and mother educational level was 5.91, 4.12, 2.93, and 8.35 respectively. These indicate that the effect of school type, and mother educational level on the cognitive achievement in reading skill varies across schools with standard deviations of 2.43, 2.03, 1.71, and 2.89, respectively. The results presented as the effect of attending a private school, the effect of mothers having primary education, the effect of mothers having secondary education, the effect of mothers with diplomas or higher education on reading skill scores vary across schools by \(\:\pm\:2.43\), \(\:\pm\:2.03\) \(\:\pm\:1.71,\:\pm\:2.89\) points, respectively. After accounting for both fixed and random effects, individual reading skill scores within schools vary by \(\:\pm\:17.67\) points.

Table 9 Result of the intercept and random slope for reading achievement Score.
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Section IV: results of the cognitive achievement in PPVT skill score

The results displayed in Table 10 below show that the average score of cognitive achievements in Peabody picture vocabulary skill score was found to be significant, and it is estimated to be 75.48% with a standard error of 1.99. The variance of school-level mean cognitive achievement in PPVT skill score around the grand mean is 92.70 with a standard deviation of 9.60, indicating that most schools have a mean on cognitive achievement in PPVT skill score within \(\:\pm\:9.60\) points of the overall mean.

Approximately 37.30% of the variation in the cognitive achievement of Peabody picture vocabulary skills of the students is accounted to the between schools, while 62.70% is accounted to the within students. It indicates that pupils’ scores in Peabody picture vocabulary skills vary notably across students.

Table 10 Result of null model for PPVT cognitive achievement Score.
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The results in Table 11 below show that the time taken from home to school, household size, access to electricity, height-for-age, weight-for-age of student, place of residence, school type, region, and mother’s educational level are found significant at the 5% significance level. From the results, the predictors like time taken from home to school, and household size have a negative relationship with the cognitive achievement in Peabody vocabulary skill score. The other predictors with the response variable have a positive relationship.

With regard to random intercept for cognitive achievement in PPVT, if the time taken from home school increases, the average score in Peabody Vocabulary score decreased by 0.06%. If the household size members increased by a person, the cognitive achievement in PPVT decreased by 0.69%. The students with access to electricity score 2.33% higher than those without access, suggesting the importance of infrastructure for learning (better lighting for studying). The results presented show that the cognitive achievement in urban areas was 1.69% higher than that in rural areas, and the private schools had a score 2.15% higher than public schools in the cognitive achievement of PPVT skill score. The results also indicate that the cognitive achievement skill score of PPVT in the Addis Ababa region was 13.59% higher than that in the Tigray region. But, in SNNPR, it is 7.13% lower than in the Tigray region. The cognitive achievement in PPVT skill score for primary educational level of mother’s was 1.88% higher than that of no formal educational levels.

As shown in Table 11, there was a statistically significant positive relationship between height and weight-for-age Z score with cognitive achievement in Peabody picture vocabulary among students (p-value = 0.004 and p-value = 0.023), respectively. This implies that a single unit change in height and weight-for-age Z score resulted in 0.92% and 0.74% unit changes in PPVT skill score, respectively.

From the results of the random intercept in Table 11 below, it indicates that the variance of school-level mean cognitive achievement in PPVT skill score around the grand mean is 44.36 with a standard deviation of 6.66, indicating that most schools have a mean on cognitive achievement in PPVT skill score within \(\:\pm\:6.66\) points of the overall mean. And the results also show that the variance of individuals’ cognitive achievement in PPVT skill scores around their school’s mean is 158.7 and standard deviation 12.6, indicating typical variability within each school.

Table 11 Result of random intercept for PPVT cognitive achievement Score.
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The results displayed in Table 12 indicates that the time taken from home to school, household size, access to electricity, mothers educational level, height-for-age of students, weight-for-age of students, school type, place of residence, and the region of the students have statistically significant at the 5% level of significance. The result also shows that time of student spent from home to school and household size have a negative relationship with a cognitive achievement in PPVT skill score.

The average of cognitive achievement in PPVT skill score vary across schools with a variance of 35.7, and the PPVT scores differ by approximately \(\:\pm\:5.98\) points from the overall average score. This indicates that there is significant variation in the baseline PPVT skill score levels across schools, even after accounting for the predictors. The results of random slope were also stated in the Table 12. The variance (3.92) and standard deviation (1.98) of school type (private) implies that the effect of attending a private school on the cognitive achievement in PPVT skill score varies across schools by \(\:\pm\:1.98\) points. Therefore, the benefit of being in a private school is not consistent across schools. In some schools, being private is associated with a larger improvement in PPVT skill scores, while in others, the benefit is smaller.

Table 12 Result of intercept and random slope for PPVT cognitive achievement Score.
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Model comparisons

The model comparisons for the three measures of cognitive achievement are shown in Table 13. AIC, BIC, and DIC were used to compare the performance of the models. According to the available data, the model with the lowest AIC, BIC, and DIC values is the best model. When compared to alternative null, random intercept models, the multivariate multilevel random coefficient models for the three measures of students’ cognitive achievement—mathematics, reading, and Peabody Vocabulary skill score—have the lowest values of AIC, BIC, and DIC.

Table 13 Comparisons for the three Models.
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Remarks

Model 1 is null model, Mode 2 is random intercept and fixed model, and Model 3 is random intercept and random slope model.

Discussion

This study aimed to examine the variation in children’s cognitive achievement and explore its association with their multidimensional factors in Ethiopia, using data from the Young Lives data. Accordingly, the average cognitive achievement in reading, mathematics, and Peabody vocabulary skill score were 37%, 51.7%, and 73.8%, respectively. The results of this study are comparable to those of20, which was focused on reading skill, was lower (48.12%) than the current study.

Students from urban areas attending private schools have better cognitive achievement in the three indicators, and this finding was similar to the study done by8. The cognitive achievement in mathematics, reading, and Peabody vocabulary skill score is better predicted by the mother’s educational background. It reveals that mothers having primary, secondary, diploma, and above educational levels have better cognitive achievement skill score compared to mothers with no formal educational. These results is supported by4,6, and8 authors who basically focused on reading and science skill scores.

For the three indicators, the weight-for-age and height-for-age were statistically significant in relation to the cognitive achievement skill score. The results show that lower HAZ and WAZ have lower cognitive achievement skill scores in reading, mathematics, and PPVT. These results were nearly identical to authors15,16, who considered only Peabody vocabulary skill score and mathematics skill score. The time taken from school to home was statistically significant for the cognitive achievement in mathematics score, and supported by the authors9,10. There are some limitation in the study, there are different measures of cognitive achievement used in each region because of missing data. The fact that similar patterns persist despite these differences suggests that each type of resource matters across different contexts.

Conclusions

Cognitive achievement among children can be influenced by various factors including biological factors (health condition, nutritional status), socio economic factors (parental educational level, income level, and home environment), and educational.

The study suggested that the multilevel random coefficients model best fits the data for the three indicators of cognitive achievement skills. We can conclude that cognitive achievement in mathematics skills score is significantly influenced by factors such as place of residence, age, household size, access to electricity, school type, height-for-age, mother’s educational level, father’s educational level, and region, all at the 5% significance level. Additionally, the relationship between mathematics skill scores and factors like school type and maternal education varies significantly across schools, with individual scores differing by ± 13.45 points. Correlations between various factors and mathematics cognitive achievement show varying degrees of relationship, with strong correlations observed for mother’s secondary education and moderate correlations for mother’s diploma or higher education and attending private schools.

The results indicate that cognitive achievement in reading skills score is significantly influenced by factors such as age, household size, access to electricity, height-for-age, school type, place of residence, mother’s educational level, and region at the 5% level. Notably, larger household sizes and longer travel times to school are associated with lower reading skill scores, while access to electricity, urban residence, attending private schools, and higher maternal education levels have positive effects. There is significant variability in reading skill scores across schools, with differences observed in the impact of private school attendance and maternal education. Strong correlations suggest that maternal education plays a substantial role in students’ cognitive achievement in reading skills.

The results for cognitive achievement in PPVT skill scores show that factors such as time taken from home to school, household size, access to electricity, mother’s educational level, height-for-age, weight-for-age, school type, place of residence, and region are significant. The analysis reveals a negative relationship between PPVT scores and both time taken from home to school and household size, while other predictors show positive relationships. There is significant variation in PPVT scores across schools, with an average variance of 35.70 and a standard deviation of ± 5.98 points. The impact of attending private schools and the educational level of mothers on PPVT scores varies considerably, indicating inconsistent benefits. Strong correlations were found between having a highly educated mother and various factors influencing cognitive achievement, particularly the positive effects of attending private schools and higher maternal education levels.

According to the study’s findings ICC indicates that variations among schools account for roughly 17.60%, 30.70%, and 37.30% of the variation in cognitive accomplishment in mathematics skill score, reading skill score, and Peabody vocabulary skill scores, respectively. The within-students component accounts for about 82.40%, 69.30%, and 62.70% of the variation in the three indicators of cognitive achievement of the above, respectively.