Journal of Current Researches on Business and Economics

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doi: 10.26579/jocrebe-9.1.4

Journal of Current Researches

on Business and Economics

(JoCReBE)

ISSN: 2547-9628 http://www.jocrebe.com

Hierarchical Linear Model Analysis to Explore the Impact of

Contextual Realities on Eighth Grade Students’ Mathematics

Performances

Cahit POLAT1 Keywords Mathematics performance, hierarchical linear model, contextual realities. Abstract

Young students’ mathematics performance is an important indication of their future academic success and career. Researchers have been searching for the factors that impact the learning and achievement of mathematics. This study seeks to explore the associations between math achievement of eighth grade students and their socio-economic statuses, amount of time they spend on homework, and their family sizes. Moreover, the effect of schools’ regions and minority rates have also been investigated in terms of students’ math success. A hierarchical linear model was used to run the related analyses.

Article History Received 12 June, 2019 Accepted 30 June, 2019 1. Introduction

Student achievement is a subject of debate and interest in many forums ranging from educators, policy makers and researchers (Ireland, 2016; Rodgers, 2010). Most discussions have targeted issues relating to student achievement gap among different student groups (McGraw Lubienski & Strutchens, 2006). These issues range from school and home contextual factors, to individual student issues such as socio-economic status and ethnic identity (Ireland, 2016; Klinger et al., 2006). While these issues are crucial and important, studying them separately does not provide the holistic picture present within the learning environment (Aitkin & Longford, 1986). Particularly, students’ mathematics achievement plays an important role on students’ education and many educational institutions work on increasing their students’ math performances. It is also well-known that academic achievement may be related to many factors such as parents’ involvement in their child’s education (Fan & Chen, 2001; Keith et al, 1986), school neighborhood (Wang, & Holcombe, 2010) and families’ socio-economic status (White, 1982). On the other hand, newer statistical techniques such as Hierarchical Linear Modeling (HLM) presents an opportunity to examine several interactions and relationships between these variables in attempt to answer most of the questions with greater details and accuracy (Burstein, 1980).

1_{Corresponding Author. ORCID: 0000-0002-1423-5084. Öğr. Gör. Dr., Harran University, Faculty of}

Economics and Administrative Sciences, Econometrics, cahitpolat@harran.edu.tr

Year: 2019 Volume: 9 Issue: 1

Research Article/Araştırma Makalesi

For cited: Polat, C. (2019). Hierarchical Linear Model Analysis to Explore the Impact of Contextual Realities on Eighth Grade Students’ Mathematics Performances. Journal of Current Researches on Business and Economics, 9 (1), 49-60.

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50 Polat, C. (2019). Hierarchical Linear Model Analysis to Explore the Impact of Contextual Realities on Eighth Grade Students’ Mathematics Performances

2. Problem Statement and Purpose of Study

The learning environment presents a complex picture that cannot be understood by simple analysis methods (Ireland, 2016; Klinger et al., 2006). When discussing inequality in education, some studies address issues ranging from socio-economic status to different types of schools, ethnicity, “a zip code as a predictor of success”, and how they impact the achievement gap (Bankston and Caldas, 1998; Bowey, 1995; Crane, 1991; Duncan, Brooks-Gunn & Klebanov, 1994; Felner et al., 1995; Jencks and Meyer, 1990; Levine & Levine, 1996; Peng, Wright, & Hill, 1995; Simon & Hocevar, 1998; Thompson, 2002). To fully understand student achievement, it is important to use advanced methods to explore the issues that impact it while exploring the possible existing inter-relationships (McCormac, 2013; Rodgers, 2010). On the other side, large scale educational studies provide numerous sets of secondary data on ranging topics and on websites such as National Education Longitudinal Study (NELS) (National Center for Educational Statistics, 1996). These datasets further have several sets of useful grouping variables that can be used to explore how several contextual factors impact student achievement among other important variables. Moreover, Hierarchical Linear Modelling (HLM) is an advanced statistical technique to investigate relationships between variables particularly for data structures like school-student and organization-person type and it has been increasingly applied in organizational research (Hofmann & Gavin, 1998). Furthermore, HLM makes predictions at higher levels for coefficients and errors of lower level models so that contribution of multilevel data may provide more accurate results (Thum,1997). In other words, parameters of lower level models are associated with higher level models’ coefficients where coefficients of higher level models are regressed for lower level models. Therefore, it is called regression of regression by many researchers (De Leeuw and Kreft, 1986).

Contextual analysis can be defined as studying the effects group variables on individuals’ behavior (Iversen & Gudmund, 1991) and HLM is a suggested technique to create an effective contextual analysis plan (Boyd and Iversen,1979; Mason, Wong, & Entwisle, 1983). Thus, contextual variables are group related variables, for example region or minority rate of a school. Even some studies explored relationship between student achievement and variety of factors such as students time spending on studying (Lammers, Onweugbuzie, & Slate, 2001), family size (Poole & Kuhn, 1973), socio-economic status (Caro, 2009) and schools’ environmental conditions (Catsambis & Beveridge, 2001), an advanced method such as HLM is needed to explore more detailed relationship between these variables when school effect (level-2) takes into consideration to obtain more accurate results.

Consequently, this study seeks to explore relationships among student related variables (Math achievement score, weekly amount of time spending on homework, family size, and socio-economic status) and some contextual, school setting variables (region, minority percentage), and how they influence student achievement using HLM. The purpose of this study was investigating whether amount of time spending on homework, family size and socio-economic status predict students’ math achievement scores. Due to the complexity in understanding student achievement based on only these individual factors, it is

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Journal of Current Researches on Business and Economics, 2019, 9 (1), 49-60. 51

important to examine how school level predictors (region and minority rate) interrelate with the other variables to impact student achievement. In this case, it was considered that the level one (student) level variables are nested within the second (school) level variables.

3. Research Question and Hypothesis

This study attempted to answer the following research question: Are there any significant moderation effects of a school region and its minority percentage with students’ weekly amount of time spending on homework, family size, and socio-economic status on student’s math scores? Furthermore, the following hypothesis was investigated:

H: There is at least one significant interaction between school variables (region, minority percentage) and the student level variables (weekly amount of time spending on homework, family size, and socio-economic status) to predict students’ math achievement scores.

To predict math scores using any or all the potential three one and two level-two predictors, it is highly likely that most of the students’ scores within and between schools’ variation may remain unexplained. This expectation posits the use of random effects in the models tested at all levels (De Leeuw & Kreft, 1986). 4. Methods

4.1. Data Collection and Sampling

This study utilized secondary data comprising over 27.394 of 8th_{grade students}

from 1052 schools in the U.S.A. This free data was data accesses via the National Education Longitudinal Study (NELS). Surveys were administered to students, their parents, teachers, school administrators, and school counselors, while student assessments targeting their science, mathematics, reading and other school related topics’ talents were administered for 8th grades. After organizing and cleaning missing data, there were 21695 of 8th_{grade students from 981}

schools to be used into this HLM analysis. 4.2. Definition of Variables

The outcome variable for this analysis was based on students’ math IRT scores (MATH) which is a continuous variable. Three student level (level 1) predictors were examined for this study; (a) weekly amount of time spending on homework (HWORK) – discrete; (b) family size (FSIZE) – discrete; (c) socio-economic status composite (SES) – continuous. On the other hand, two level-2 predictors were: (a) the school’s region (REGION) – categorical with 4 categories; (b) the percentage of students from minority groups in schools (MINOR) – categorical with 7 categories. However, after some preliminary investigations it was decided to have just two categories for the MINOR variable; whether the school has more than 10% minority rate of students or not, and the REGION variable was decided to have just two categories as an indication of whether the school is in the south part of the U.S.A. or not.

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4.3. Model Specification

In this study, it is focused to use a two-level HLM, as the units of observations were schools and students, and the students were nested into schools. Therefore, level-1 is devoted for student related variables and level-2 is devoted for school related variables. A two-level HLM works through the formulas indicated below for level-1 and level-2 accordingly. Modell for level-level-1 (student-level) is formulated as

where is the score of ith person from the jth school for the dependent variable,

is score of ith person from the jth school for the Qth independent variable,

is the regression coefficient for and s are model errors for level-1. Moreover, level-2 model (school-level) is formulated as

where s are the regression coefficients from level-1, is regression

coefficient for independent variable of the jth school and is the error term for level-2 (Raudenbush & Bryk, 2002).

After defining and specifying the variables in the model, then it was decided to specify the HLM model for this study. The outcome variable at the student level was the students’ math achievement score and the full level one model had five predictors:

The full school-level model included three school variables, predicting each student level (level-1) coefficients:

The combined model can be obtained by replacing students level coefficients ( ,

, , ) their predictors indicated above in the full level one model.

4.4. Procedures

After checking the assumptions for Hierarchical Liner Modelling and making any applicable adjustments, data was analyzed by using the free student version of HLM7 to answer the research questions using appropriate organizational level models. Moreover, in this model, student-level weights were used in estimating results. The outcome variable was specified as MATH for the students’ Math IRT scores.

The model building was started by the null model without any level-1 or level-2 predictors, so that effect of clustering in the data to be estimated. Then level-1 variables were added to model one at a time. After adding all the level-1 predictors to model, then level-2 predictors were added to the model and the ones having

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significant interactions or contributions were kept into the model. The full and preliminary models were then checked whether they met the HLM assumptions. Finally, based on the diagnostics and further investigations the final model was constructed to come up with more explanatory and precise results.

5. Results

5.1. Initial Model

The model construction for this study was started from the null model with no predictors, then level-1 and level-2 predictors were added to the model one at a time. All level-1 predictors were statistically significant (p < .001) except FSIZE. Moreover, both level predictors had at least one significant interaction (p < .05). 5.2. The null model

The intra-class correlation (ICC) coefficient was found 35.76% by having the null model variance as 179.78 and tau as 100.09. This suggests that there is a strong clustering effect and level-2 predictors might have some importance to explain the outcome variable.

5.3. Student-level model

Before adding any level-2 predictor, Student-level (level-1) predictors were added to the model and analyzed. After adding all the student level variables to model, the level-1 coefficient on HWORK was significant, (t(980) = 16.14, p < .001), the coefficient on SES was significant (t(980) = 28.27.63, p < .001) and the coefficient on FSIZE was not significant. The random effects on level-1coefficients SES and HWORK were significantly different from zero, but random effect on FSIZE coefficient was not significantly different from zero suggesting that the predictors HWORK and SES should be kept in the model and FSIZE can be dropped from the model. The variables HWORK and SES were positively associated with outcome variable MATH for students’ math scores. Sigma-square for the null model was 179.78 and sigma-square for the model with the level-1 predictors HWORK and SES kept in the model was 165.15. The proportion of variance explained (PVE) by adding these two predictors was 8.14%. If only SES was added to the model, the PVE was 5.33% and if only HWORK was added to the model, the PVE was 3.43%. Reliabilities for the slopes of SES and HWORK were 0.22 and 0.18, respectively, indicating that there was some amount of variance across these slopes to proceed with entering level-2 predictors.

5.4. School-level model

After analyzing level-1 predictors, level-2 predictors were added to model one at a time, starting with REGION variable. After some preliminary investigations, it was found that the school variable REGION for the schools located in the south part of the U.S. has significant effect in eighth grade students’ mathematics scores. REGION was a significant predictor of the intercept (t( 979) = -3.69, p < 0.001), and HWORK (t( 979) = -1.99, p = 0.047) but not significant for SES. Therefore, it was decided to keep the REGION variable in the model as the school variable to show whether schools are in the south part of the U.S. or not. In this model of level-1 variables (SES, HWORK) and level 2 variable REGION was associated with 2.16 points

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decrease in students’ mathematics scores. The variance of the level-1 intercept decreased from 76.96 to 75.97 representing 1.31% of PVE, which showed that REGION explained some of variation in the mean math scores.

Then level-2 variable MINOR was added to the model, and it had some significant interactions with level-1 variables or the intercept. In the model of level-1 variables (SES, HWORK) and level 2 variables REGION and MINOR, the variable REGION was a significant predictor of intercept (t(978) = -3.11, p = 0.002) and HWORK (t(978) = -2.08, p = 0.04). Additionally, the variable MINOR was a significant predictor of intercept (t(978) = 2.01, p = 0.045) and SES (t(978) = -2.14, p = 0.03). However, the variance of the level-1 increased from 165.19 to 165.20 by adding MINOR to the previous model which is an indication of negative PVE. The negative value of PVE showed that minority rate of schools does not account for any variation in the students’ math achievement scores. Therefore, it was decided to drop MINOR from the initial model and make further investigations later.

Based on the results above, the initial model was considered as

Coefficients with standard errors for the fixed effects are shown in Table 1. The cross-level interaction between REGION and HWORK are negative, implying that being in the REGION returns to HWORK decrease.

Table 1. Fixed Effects Model for the Initial Model

Fixed Effect Coefficient s.e t-ratio Approx. d.f. p-value

For INTRCPT, B0

INTRCPT2, G00 38.87 .39 100.63 979 0 REGION, G01 -2.17 .59 -3.69 979 0

For HWORK slope, B1

INTRCPT2, G10 1.29 .09 13.99 979 0 REGION, G11 -.32 .15 -2.11 979 .04

For SES slope, B2

INTRCPT2, G20 4.48 .16 28.29 979 0

For the sample weights, only the robust standard errors are reported above. Moreover, all the variance components were also significant at the p < 0.001 suggesting that all level-1 coefficients should include the random error terms. 5.5. Model Diagnostics

Before conducting HLM analysis into the data set, HLM 2 model assumptions were evaluated. The evaluated HLM assumptions were based on checking normality of level-1 and level-2 residuals, homoscedasticity at level-1 and level-2 residuals, independence of level-1 residuals from all predictors, independence of level-2 residuals from all predictors, independence of level-1 and level-2 residuals.

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The assumption about normality of level-1 residuals that is independent and normally distributed with a mean of 0 and variance of for each level- 1 unit i within each level- 2 unit j was moderately violated. Moreover, Homoscedasticity for level-1 residuals was violated.

The assumption for multivariate normality of level-2 residuals were tested and met. However, homoscedasticity for level-2 residuals was violated. HLM is robust with respect to violations of homogeneity and normality with sufficient number of cases, therefore the HLM analysis was continued since the data set had large number of cases.

Independence of level-1 residuals was checked and level-1 residuals were found to be independent from all level-1 and level-2 predictors. Independence of level-2 residuals was also checked and level-2 residuals were found to be independent from both level-1 predictors (HWORK, SES) and from both level-2 predictors (REGION and MINOR) were found to be independent from all level-1 and level-2 predictors. All predictors were appeared to have correlation coefficients less than .20 with each other. Finally, by merging level-2 residual files into the level -1 residual files, independence of level-1 and level-2 residuals were checked and the assumption was met.

5.6. Final Model

The final model was built based on some further investigations and model diagnostic findings. During the analysis using HLM7, it was figured out that MINOR seems to have some significance to improve the model specification. Thus, two level-1 predictors (SES, HWORK) and the two level-2 predictors (REGION and MINOR) were simultaneously tested in the HLM7 software then non-significant interactions were drooped from the model and the analysis were rerun again. Then, possible significant combinations of level-1 and level-2 variables were tested to evaluate which level-2 predictors and interactions to include in the model. At the end of the analysis, the final model was specified as:

All the random and fixed effects in this model were significant at the p < 0.05. All the variance components were also significant at the p < 0.001. The level-2 variable MINOR was kept in the model because it appeared to be important for understanding the cross-level interactions: interaction between MINOR and SES and, interaction between MINOR and the intercept. Coefficient estimates with standard errors, degrees of freedom and p-values for the final model are shown in the Table 2.

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Table 2. Fixed Effects Model for the Final Model

Fixed Effect Coefficient s.e t-ratio Approx. d.f. p-value

For INTRCPT, B0

INTRCPT2, G00 39.40 .47 84.69 978 0 REGION, G01 -1.95 .63 -3.11 978 0 MINOR, G02 -1.21 .60 -2.01 978 .04

For HWORK slope, B1

INTRCPT2, G10 1.24 .12 10.71 978 0 REGION, G11 -.32 .16 -2.09 978 .04

For SES slope, B2

INTRCPT2, G20 4.93 .26 18.97 978 0 MINOR, G21 -.70 .33 -2.14 978 .03

After estimating the final model with all the predictors and their interactions, the PVE was calculated. The PVE was about 9% comparing the final model with the null model. There was an important amount of PVE by adding level-2 variables or their interactions the final model. Therefore, significant interactions were kept in the final model as they were contributing to understand effects of HWORK and SES on students’ math achievement scores. The variance component for HWORK of the final model was 1.21 and it was 1.26 without level-2 variable MINOR. The variance component for SES was also decreased from 5.02 to 4.76 by adding level-2 variable MINOR to the final model.

Figure 1 below shows the plot of estimated regression line form the model for students’ math achievement scores on amount of time spending on homework (HWORK) given whether the school is located at the south part of the U.S.A or not. Additionally, Figure 2 shows the plot of estimated regression line form the model for students’ math achievement scores on SES given whether the school have minority rate above 10% or not.

Figure 1. Estimated regression line for Math Achievement scores on HWORK Given if the

School is Located at the South Part of the U.S.A or not (REGION)

-1.17 -0.17 0. 83 1. 83 2. 83 35. 65 37. 34 39. 03 40. 72 42. 41 HOMEWORK M A TH IR T REG I O N3 = 0 REG I O N3 = 1

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Figure 2. Estimated Regression Line form the Model for Students’ Math Achievement

Scores on SES Given if the School have Minority Rate above 10% or not (MINOR).

6. Discussion

The results of this study indicated that contextual factors can explain some variation in math achievement scores. It is likely that eighth grade students’ math performances are affected by their schools’ location in the U.S.A. and minority rates. However, we cannot conclude any causal relationship for these variables from this HLM study.

Moreover, school level predictors seemed have moderator effects on the relationship between output variable and student level predictors. Specifically, the relationship between math achievement and HWORK may be affected by REGION status, and the relationship between math achievement and SES may be affected by MINOR (minority rate).

It was seen that both student level predictors have positive contributions on math achievement score suggesting that students who spend more time on doing homework or the students who have families at higher SES are more likely to score better on Mathematics. On the other hand, school level predictors of this study seemed to negatively affect the students’ math achievements, which suggests students of the schools in the south part of the country or the schools which have high minority rates significantly tend to score lower in mathematics tests.

By analyzing the NELS data, it was investigated that whether there is at least one significant interaction between school variables (region, minority percentage) and the student level variables (weekly amount of time spending on homework, family size, and socio-economic status) to predict students’ math achievement scores. However, some of the suggested interactions in the full model are not significant. Moreover, the directions of the significant interactions were also explored. For example, it was observed that REGION has negative contribution on HWORK, and MINOR also has negative contribution on SES for the eighth-grade students’ math achievement IRT scores. By controlling school level variables, both student-level

-1.10 -0.56 -0.02 0.52 1.06 32.92 35.66 38.40 41.14 43.88 SES M A T H IR T MINOR = 0 MINOR = 1

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variables had positive relationships with the outcome variable (students’ math scores).

7. Limitations and Future Research

One of the important limitations for this study was a possible bias that might come from Math IRT scores. Students from different cultural and personal backgrounds might have different tendencies of approaching mathematical concepts. Therefore, it is very important that the items to measure students’ math achievement are created with the consideration of these acknowledgements, as it is challenging to completely get rid of that bias.

Another limitation for this study was the fact that the school level variables were categorical, and number of these variables were just two. Decreasing the categories into two helped for simplicity of analysis; on the other hand, it prevented the analysis from obtaining more detailed and precise results. Therefore, it might be beneficial for a future study to extend the categories and variables to come up with more extensive results.

It is also important to note that while the variables used in this study explained some variation for eighth grade students’ math achievement, there is still significant amount of variation to be explained. Some future study might be done to explore more about the unexplained factors which have importance on eighth grade students’ math achievement.

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