Conﬁrming the Mediation Eﬀect of A Structural Model By Using Bootstrap Approach: A Case Study of Malaysian 8th Grade Students’ Mathematics Achievement

The performance of Malaysian students in Trends In Mathematics and Science Study (TIMSS) was not up to par. The rank of Malaysian students in the assessment is the bottom one third, not the top. However, there are still a very low number of studies have been conducted with the TIMSS finding. This study intends to a) confirm items reliability included in the study( for Malaysian students), b) confirm the significance of latent exogenous constructs toward latent endogenous construct and c) confirm the type of mediation exist in the structural model. Secondary data obtained from TIMSS 2011 database has been used in this study. Items have been grouped into latent exogenous constructs; school, teacher, motivation, self-confidence; latent mediator construct; attitude and latent endogenous construct; achievement. From the analysis, it is found that only a few items included are reliable to represent respective constructs. Furthermore, it is explored that all latent exogenous constructs have significant direct effects toward latent endogenous construct. And lastly, it has also been confirmed that, all latent exogenous constructs except school latent construct have indirect effects toward endogenous latent construct; students’ achievement through the mediator latent construct; attitude. Contribution/ Originality This study is one of very few studies which have investigated the usefulness of bootstrap approach in confirming the type of mediation exist in a model. Commonly, bootstrap approach is only used when the data is small and/or the multivariate normal distribution is not fulfilled.


INTRODUCTION
Structural Equation Modeling or also known as SEM has gained popularity among researchers, academicians and students nowadays. It is due to its flexibility and generality besides the ability to generate an accurate and precise estimation in making prediction. SEM analysis goes through the steps of model specification, data collection, model estimation, model evaluation and also model modification (Zainudin, 2012). SEM enables researcher to modify the structural and measurement models. Structural model may consist of mediator and/or moderator. Mediation analysis is an analysis that is commonly performed in order to identify the type of mediation exists in the model. However, in ISSN(e): 2312-0916/ISSN(p): 2312-5772 URL: www.pakinsight.com this study, an extension analysis known as bootstrap approach has been used to confirm the type of mediation exists in the model.

SAMPLE OF THE STUDY
The sample of this study is obtained from Trends in International Mathematics and Science Study (TIMSS) international database where Malaysia respondents have been selected as the case study. Respondents in this study are normally selected through a two stage stratified cluster sampling technique; cluster sampling for the first stage, school sampling for the second stage and class sampling for the third stage.

Confirmatory Factor Analysis (CFA)
CFA is a special form of factor analysis, most commonly used in social research. It is different compared to Exploratory Factor Analysis (EFA) since it is used to test whether measures of a construct consistent with a researchers' understanding of the nature of that construct (or factor). As such, the objective of CFA is to test whether the data fit a hypothesized measurement model.
CFA is a crucial part for the measurement model in SEM that is used to obtain the acceptable model fit before modeling the structural model. There are two methods in running CFA; Individual CFA and Pooled CFA (Chong et al., 2014). To achieve unidimensionality of construct, the factor loading of every all items must be higher or equal to 0.6 (Hair et al., 2006). Model fit measures could be obtained to assess how well the proposed model captured the covariance between all the items or measures in the model. All redundant items exist in latent constructs should be either removed or constrained. Factor Loading, Absolute, Incremental and Parsimonious fitness indexes must achieve the acceptance level. According to Zainudin (2012) the fitness indices estimations are as follows:

Discriminant Validity
Discriminant validity is a degree to which scores on a test do not correlate with scores from other tests that are not designed to assess the same variable. Correlation coefficients between measures of a construct and measures of conceptually different variables are usually given as evidence of discriminant validity (Lewis-Beck et al., 2004). It is a procedure of linking exogenous constructs in a model to examine whether they are highly correlated to each other or otherwise. Exogenous constructs must be independent to each other, in which, the correlation value between them should not exceed 0.85 in order to achieve discriminant validity of construct (Zainudin, 2012). If the correlation(s) is/are higher than 0.85, one of the highly correlated constructs must be removed or multi-collinearity problem is exist.
However, proper analysis and investigation need to be adopted first before removing any constructs.

Path Analysis
Path Analysis may consist of simple or multiple statistical regression models. Besides the ability to determine the causal path, it is able to identify the type of mediation exist in a model. The difference between path analysis and conventional regression is the causal path in the model can be modified according to the researcher desire. According to Zainudin (2012) there are three types of mediation: i. Complete mediation: Occur when the independent variable links towards the dependent variable only through mediator variable and there is no direct effect of independent variable towards dependent variable.
ii. Partial mediation: Occur when independent variable links towards the dependent variable through mediator variable and there is also a direct effect of independent variable towards dependent variable. iii.
No mediation: Occur when independent variable does not link to the dependent variable through mediator variable but has a direct effect towards dependent variable.

Bootstrap
Bootstrap is one of the crucial parts in modeling structural model when it comes to confirm the type of mediation. In addition, it also allows researcher to assess the stability of parameter estimates that can be applied when the assumptions of large sample size and multivariate normality may not hold. In order to perform this approach, two models need to be built; one with the existence of mediator construct and the other, otherwise. The type of mediation should then be confirmed based on the direct and indirect effects reported.   Figure 1 and Table 2 show that the fitness indexes of the model achieved the required level after the Pooled CFA has been conducted. The factor loading of all items toward its' respective constructs are greater than 0.6, therefore it can be concluded that the models' unidimensionality is achieved. Besides that, all latent exogenous constructs are correlated with the correlation strength of less than 0.85. Therefore, no multi-collinearity problem exists and the discriminant validity is achieved.  Table 3 shows the reliable items for Malaysian students that have been obtained for measurement model through the Pooled CFA. Initially, school, teacher, motivation, self-confidence and attitude constructs consist of nine(9), eight(8), six(6), seven(7) and five (5) items respectively. The other items were dropped from the model for having factor loadings of low than 0.6. The low reliability of the items may jeopardize the model fit indexes measurement, hence were dropped from the model.

Path Analysis
In bootstrap approach, two structural models have been constructed; one with the absence of mediator construct and the other one with the existence of mediator construct.    Table 4 and Table 5 show the direct effect and the significance of exogenous latent constructs toward endogenous latent construct in the absence of mediator latent construct. All exogenous latent constructs are identified to have significant influence toward the endogenous latent construct. This finding indicates that school, teacher, selfconfidence and motivation contribute in affecting the students' achievement. It has been also found that two exogenous latent constructs; school and teacher have negative estimates value.  .018 17.520 *** Attitude <---Motivation .241 .028 8.650 *** Attitude <--Confidence .574 .021 28.000 *** Attitude <---School -.032 .023 -1.426 .154 Achievement <---School -.178 .019 -9.501 *** Achievement <---Teacher -.181 .018 -10.356 *** Achievement <---Confidence .179 .020 9.095 *** Achievement <---Motivation .069 .023 2.984 .003 Achievement <---Attitude .177 .019 9.528 *** Source: AMOS Graphic Output Table 6 and Table 7 show the direct effect and the significance of exogenous latent constructs toward endogenous latent construct in the existence of mediator latent construct. From the table, it is found that all exogenous latent constructs except school latent construct have significant influence toward mediator latent construct.
It is also found that all exogenous latent constructs have significant influence toward endogenous latent construct in the presence of latent mediator construct.

CONCLUSIONS AND DISCUSSION
From the Pooled Confirmatory Factor Analysis finding, it has been found that most of the items included in the study have factor loadings of low than 0.6. Hence, these items were dropped from the model because they are not reliable and may jeopardize the model estimates.
In the discriminant validity test, it has been explored that the correlation of all latent constructs are below 0.85.
The low correlations indicate that all constructs are independent to each other. In other word, there is no constructs are measuring the same thing. In achieving discriminant validity, redundant items in achievement latent construct; ach01 and ach04 have been also constrained.
The path analysis has been conducted in the absence and existence of mediator latent construct. In the absence of mediator latent construct, all exogenous latent constructs show significant direct effects toward endogenous latent construct. This indicates that school, teacher, self-confidence and motivation have direct contribution toward students' achievement. In the existence of mediator latent construct, all exogenous latent constructs show significant indirect effects toward endogenous latent construct through mediator latent construct except for school. This indicates that teachers, self-confidence and motivation have indirect contributions or partial mediation toward students' achievement through attitude while there is no mediation occurs for school construct. In addition, this finding also indicates that teacher, self-confidence and motivation have significant contributions toward students' attitude.
However, there are two types of effect exist in the model; positive and negative effects. School and teacher have negative effects toward students' achievement, the same goes to the effect of school towards attitude. The negative effects indicate that increase in unit of school and teacher will decrease in the students' attitude and/or achievement.
For example, the more a student likes his/her mathematics teacher, the lower the achievement of the student in mathematics will be.