each construct and also between constructs. Using AMOS, the researcher can specify, estimate, assess, and present the model in a causal path diagram to show ...
A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
CHAPTER 1
INTRODUCTION TO STRUCTURAL EQUATION MODELING The Structural Equation Modeling or popularly known as SEM is a second generation statistical analysis techniques developed for analyzing the inter-relationships among multiple variables in a model. The inter-relationships among variables could be expressed in a series of single and multiple regression equations. The Structural Equation Modeling technique employs the combination of quantitative data and the correlational or causal assumptions into the model. SEM is a more powerful statistical technique to solve the following requirements: 1) 2) 3) 4) 5) 6)
Running the Confirmatory Factor Analysis (CFA) Analyzing multiple regression models simultaneously Analyzing regressions with multi-collinearity problem Analyzing the path analysis with multiple dependents Estimating the correlation and covariance in a model Modeling the inter-relationships among variables in a model
1.1
THE CONCEPT OF SEM AND HOW IT WORKS
SEM begins with a theory where the researcher intends to test the relationship among constructs of interest in the study. The relationships are modeled into a theoretical framework represented by a schematic diagram. The schematic diagram presents the hypotheses of interest to be tested in the study. The constructs of interest involved are measured using a set of items in a questionnaire. The measurement scale for each item should be either interval or ratio. The ideal measurement should be in the interval from 1 to 10 so that the data is more independence and thus meet the requirement for parametric analysis. The researcher should develop at least four items to measure each latent construct. Throughout the chapter, the readers would find the term variable and construct are used interchangeably. A variable is the directly measured score, while the construct is meant for an indirectly measured score. In fact the construct is only a hypothetical concept of something, or the respondents’ perception concerning certain issue. A construct is measured through the respondent’s response towards a set of items in a questionnaire.
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1.2
THE ADVANTAGES OF SEM COMPARED TO OLS
SEM is capable of estimating a series of inter-relationships among latent constructs simultaneously in a model. In fact, SEM is the most efficient method to handle the Confirmatory Factor Analysis (CFA) for measurement models, analyze the causal relationships among latent constructs in a structural model, estimating their variance and covariance, and test the hypotheses for mediators and moderators in a model. As has been said earlier, latent constructs could not be measured directly since it is only a hypothetical concept of something. Thus, the researcher could not model them using the Ordinary Least Squares (OLS) regression. The examples of latent constructs measured through a set of items are in a questionnaire are: 1) Service Quality 2) Customer Satisfaction 3) Job Satisfaction 4) Corporate Image 5) Product Image 6) Customer Loyalty 7) Purchase Intention 8) Consumer Behavior 9) Employee Soft Skills 10) Perceived usefulness 11) Relational bond 12) Financial bond 13) Structural bond 14) Relationship quality 15) Attitudinal loyalty 16) Behavioral loyalty Those constructs cannot be measured directly like counting the number of kids in a family, total income of a household, monthly phone bills, daily production, weekly price of chicken, etc. The variable which could be measured directly is called the observed variable, while the variable which could not be measured directly is called latent construct. These latent constructs could only be measured indirectly using a set of items in a questionnaire. Example of items in a questionnaire to measure student satisfaction as a latent construct: In this example, the construct Students Satisfaction is measured using eight items in a questionnaire. 18
A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin As a student of this university, I….. 1
am satisfied with the lecture schedules
Strongly Disagree 1 2 3
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am satisfied with the learning process
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am satisfied with the academic system
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am satisfied with the continuous evaluation
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am satisfied with academic regulations
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am satisfied with the library references
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am satisfied with classroom facilities
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am satisfied with the campus security
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Source: Research Methodology and Data Analysis 2nd Edition by Zainudin Awang (2012)
Other advantages of Structural Equation Modeling (SEM): 1) Could run the Confirmatory Factor Analysis (CFA) to reduce measurement errors 2) Could deal with the problem of multicollinearity among independent constructs 3) Could assess the fitness of measurement model as well as the structural model 4) Could analyze the model with multiple independents as well as multiple dependents 5) Could include the mediating variable in a model and analyze its effects (mediator) 6) Could analyze the effects of moderating variable in certain path of a model(moderator) 7) Could model the error terms and handle the correlated errors among response items 8) Could analyze both First Order and Second Order Constructs in the structural model 9) Could include both observed variables and latent constructs in the structural model
1.3
Converting Regression Models into AMOS Graphic
Modeling the simple linear regression Y = Bo + B1X1 + e1 in AMOS graphic Usually, the researchers could model the above equation using Ordinary Least Squares (OLS) regression and analyze the model using ANOVA since X1 and X2 are observed variables. However, the researchers could also employ AMOS graphic software to model and analyze the regression equation as shown in Figure 1.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
Observed Variable
Residual
e1 1
β1
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Y
Figure 1: The Simple Regression model in AMOS Graphic Key: X1 = Independent variable (observed), Y = Dependent variable (observed), e1= error in the equation or residuals (unobserved).
Note: In Figure 1, the researcher is interested to estimate the causal effect of X1 on Y and subsequently test the hypothesis to prove of its significance. In Figure 1, both X1 and Y are observed variables. In Amos, the observed variables are represented using rectangles.
Modeling the Multiple Linear Regression Y = Bo + B1X1 + B2X2 + B3X3 + e1 The researchers could model the above equation using Ordinary Least Squares (OLS) regression and analyzed the model using ANOVA. However, the researchers could also employ AMOS to model the equation as shown in Figure 2 below.
X1
e1 1
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Y
X3 Figure 2: The Multiple Regression models in AMOS Graphic Note: X1, X2, X3, and Y are represented by rectangles since they are directly observed variables
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1.4
The Concept of Latent Constructs in Research
The Simple Regression with multiple indicators to analyze latent constructs In science and social science researches, most of the times the researchers are dealing with latent constructs. As has been said earlier, these constructs are measured using a set of items in a questionnaire. Since the OLS procedures could not entertain latent constructs, the researchers need to employ SEM for the analysis. Using SEM, the researcher could model the relationship among these constructs together with their respective items in the model and analyze them simultaneously. In this case, at least two measurement models involved – one for independent construct and the other one is for dependent construct. The theorized link between measurement model for independent construct and measurement model for dependent construct is called a structural model. Thus, instead of modeling the Ordinary Least Squares regression (OLS) and analyzed using ANOVA, the researcher is working with the Structure Equation Modeling (SEM) and analyzed using AMOS as shown in Figure 3.
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Figure 3: The structural equation model for analyzing latent constructs in a model
Note: X1 and Y are latent constructs. In Amos syntax, latent constructs are represented by the ellipses. The latent construct X1 is measured using items X11 to X15, while latent construct Y is measured using items Y1 to Y5. These measured items are represented by rectangles in the model. Key: X1 = Exogenous construct, while X11 to X15 is a set of 5 items to measure latent construct X1 e1 to e5 are errors in measurement for items X11 to X15 Y = Endogenous construct, while Y1 to Y5 is a set of 5 items to measure latent construct Y e6 to e10 are errors in measurement for items Y1 to Y5
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin e11 is an error in the equation or the residual.
1.5 THE MINIMUM SAMPLE SIZE REQUIRED FOR SEM There are endless debates in the literatures as to how many respondents should be obtained in order to employ SEM. However, there is no clear-cut answers to it since every research differs (among other things) in term of the population characteristics, and the number of constructs employed in a model. Hair et al. (2010), offer the following suggestion for minimum sample size depending on the model complexity and basic measurement model characteristics. Model Characteristics (Number of latent constructs and items) 1. Five or less latent constructs. Each latent construct has more than three items 2. Seven or less latent constructs. Each construct has more than three items
Minimum Sample Required 100 sample
3. Seven or less latent constructs. Some constructs have less than three items (just identified model) 4. More than seven latent constructs. Some constructs have less than three items (just identified model)
300 sample
1.6
150 sample
500 sample
INTRODUCTION TO AMOS SOFTWARE
AMOS is an acronym for Analysis of Moments Structure – the software developed for analyzing the Structure Equation Modeling (SEM). Synonym to SEM is Covariance Structure Analysis or Covariance Structure Modeling. Other software available to analyze SEM includes LISREL, SEPATH, PRELIS, SIMPLIS, MPLUS, EQS, and SAS. The advantage of AMOS compared to other software in its class is its graphics representation of the model. So, instead of writing instructions through computer program, researchers only need to draw the AMOS graphic identical to the schematic diagram of a model in the study. AMOS software could be utilized to explore statistical relationships among the items of each construct and also between constructs. Using AMOS, the researcher can specify, estimate, assess, and present the model in a causal path diagram to show the hypothesized relationships among constructs of interest. The empirical model can be tested against the hypothesized model for goodness of fit. If the researchers found any path that does not fit with the original model, they could either modify the path to improve the fitness of the model or remove that particular path completely from the hypothesized model.
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1.7
The Variable Terms in SEM using AMOS Graphic
The explanation below refers to Figure 4. 1. Exogenous construct is the independent variable in the Ordinary Least Squares (OLS) regression. In AMOS, the independent variable is drawn as an upstream variable with the causal arrow pointing out to its corresponding dependent variable. In Figure 4: X1 and X2 are exogenous construct with five response items. The arrows from exogenous constructs X1 and X2 are pointing out to their endogenous construct Y to indicate that X1 and X2 are theorized to have some causal effects on Y. 2. Endogenous contruct is the dependent variable in the Ordinary Least Squares regression. In AMOS, the dependent variable is drawn as a downstream variable with the arrow pointing in from its corresponding independent variable. In Figure 4: Y is an endogenous latent construct with five response items. 3. Mediating variable is the variable which has a double role. This variable acts as a dependent variable in the first equation, and acts as an independent variable in the second equation. In theory, the mediator variable mediates the relationship between an independent variable and a dependent variable. In Figure 4, M is the mediating construct with four response items. 4. Moderating variable is the variable that moderates the effects of independent variable on its dependent variable. In the case of latent constructs, the moderating variable is the variable that moderates the effects of exogenous construct on the endogenous construct. The representation for moderating variable is shown in Figure 4. Unlike the mediating variable, the moderating variable is not in the model.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
Exogenous Construct
Endogenous Construct
Residual
Mediator Variable
Measurement Error
Figure 4: The sequence of constructs assembled in a model in AMOS Graphic Note: X1 and X2 are exogenous constructs while Y is an endogenous construct. All constructs are latent.
5. Error in measurement - an error depicted from each measuring item of a variable. In Figure 4 – we can see that e1 to e5 are the measurement errors for construct X1, e11 to e15 are the measurement errors for construct X2, while e6 to e10 are the measurement errors for construct Y. 6. Error in equation – a residual in the respective regression equation. In Figure 4 – we can see that e20 is the residual for the equation Y = f(X1, X2) or Y = Bo + B1X1 + B2X2 + e1
Remember: The numbering for measurement errors as well as residuals in the model are assigned randomly by Amos Graphic. 24
A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
1.8
MODELING THE OBSERVED AND UNOBSERVED VARIABLES IN AMOS Graphic
Let X1 and X2 be independent variables while Y is a dependent variable in a multiple regression model. Both variables are directly observed. The researcher could model this multiple regression in AMOS Graphic as shown in Figure 5
X1 Y
1
e1
X2 Figure 5: Modeling the observed variables in AMOS Graphic for multiple regression models
The model in Figure 5 is equivalent to the following model in a multiple regression equation:
Y = Bo + B1X1 + B2X2 + e1 This model in Figure 5 is valid and workable only if the independent variables X1 and X2 do not have a multi-collinearity problem between them. Remember, one of the main assumptions in the Ordinary Least Squares (OLS) is no significant multi-collinearity exists between the independent variables. The Structural Equation Modeling (SEM) technique could deal with the multi-collinearity problem. In fact, AMOS requires the researcher to estimate the correlation between independent variables as well as between exogenous constructs.
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Figure 6: Modeling the latent constructs in the multiple regression models
As shown in Figure 6, the latent constructs X1 and X2 are measured using five questionnaire items respectively while the latent variable Y are measured using three questionnaire items. However, in reality each latent construct could be measured using as many as ten to twenty questionnaire items. The modeling in Figure 6 is valid only if the latent variables X1 and X2 do not have significant multi-collinearity problem between them. Remember, the main assumption for Ordinary Least Squares Regression (OLS) is no significant multi-collinearity exists among the independent variables or exogenous constructs in a model.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
Taking multi-colllinearity problem into perspective, AMOS software requires the researcher to estimate the covariance between independent variables or between exogenous constructs in a model. The program would not run until the researchers employ the doubleheaded arrow to link the pair of exogenous constructs in a model to set the pair as “free parameter estimates” concerning the multi-collinearity effects between them. The application of double headed arrow linking two independent variables is shown in Figure 7. However, if the correlation between X1 and X2 is greater than 0.85, then the assumption of discriminant validity has failed. It means, one variable is like the mirror of the other. Thus, the researcher needs to drop one of the two variables from the model and continue analysis using a single variable.
X1 Y
1
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X2 Figure 7: Modeling the multiple regressions and estimating the correlation between independent variables in AMOS Graphic
If the reader could still recall, the double-headed arrow is used to estimate the correlational relationship while the single-headed arrow is used to estimate the causal relationship. In the model shown in Figure 7, the researchers could test the significance of covariance between X1 and X2. At the same time, the researchers could also test the significance of causal effect of X1 on Y, and also the causal effect of X2 on Y. All tests are carried out simultaneously.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
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Figure 8: Modeling the multiple regressions and estimating the correlation between exogenous constructs in AMOS Graphic
The analysis of correlational and causal relationship for the model in Figure 8 is equivalent to the analysis stated in Figure 7. The advantage of analysis as stated in Figure 8 is the researcher could assess the importance of each item in measuring their underlying latent construct. In the easier terms, the researcher could assess which item contributes more information in measuring their respective construct. In SEM, the researchers could even test the significance of each response item on its respective construct.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
1.9
Modeling Multiple Variables in AMOS Graphic: The Multiple Regression Model
The Multiple Regression Analysis Y = Bo + B1X1 + B2X2 + B3X3 + e1 Again, the researchers could model the observed variables using Ordinary Least Squares (OLS) regression and analyzed using ANOVA. For the same problem, the researchers could model the equation in AMOS software as shown in Figure 9. The result of two methods would be identical. However, the output from AMOS is much more informative and friendly.
X1 e1 1
X2
Y
X3 Figure 9: The multiple regression models for the observed variables in AMOS Graphic Key: X1, X2, X3 = Independent variables, Y = Dependent variable, e1= residual
The Multiple Regression Models for Latent Constructs AMOS Graphic can model the relationship among latent constructs with multiple items. In this case, more than one measurement model involved. The researchers need to validate each of these measurement models prior to running structural model. Thus, instead of modeling the OLS, the researchers are modeling SEM as shown in Figure 10.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin e1
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Figure 10: The Structural Equation Modeling for the latent constructs in AMOS Graphic Key:
X1 = Exogenous latent construct, X11 to X15 = a set of 5 items to measure X1 e1 to e5 = error in measurement for items X11 to X15 X2 = Exogenous latent construct, X21 to X25 = a set of 5 items to measure X2 e6 to e10 = error in measurement for items X21 to X25 X3 = Exogenous latent construct, X31 to X35 = a set of 5 items to measure X3 e11 to e15 = error in measurement for items X21 to X25 Y = Endogenous latent construct, Y1 to Y5 = a set of 5 items to measure Y e16 to e20 = error in measurement for items Y1 to Y5 e21= residual 30
A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
1.10
Modeling the Mediator Variable in AMOS Graphic
Once the regression relation exists and the direct effect of X1 on Y is significant, the researchers could determine a variable that mediates the relationship between X1 and Y. This variable is called a mediator. The role of a mediator is providing an indirect effect of X1 on Y. Thus, the researcher needs to test the significance of a mediator in the X1 and Y relationship. The method of path analysis using OLS is quite tedious. However, the testing procedure of path analysis is much easier in SEM. Let X1, Y and M be and independent variable, dependent variable, and a moderator variable respectively. Refer to Figure 11. To begin with, the simple effect of X1 on Y has to be significant
Mediator
Mediator M enters the model.
Figure 11: Modeling the mediator variable M in AMOS Graphic. In Figure 11, X1 is an independent variable, Y is a dependent variable, and M is a mediating variable. All variables in the model are directly observed.
The regression equations involved: Y = Bo + B1X1 + B2M + e2 ...(1) Y = Bo + B1X1 + e2 ...(2) Y = Bo + B2M + e2 ...(3) M = Bo + B3X1 + e3 ...(4)
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
Using OLS, the researcher needs to analyze the four regression equations separately in determining the mediating effect of M. The analysis would be quite tedious. However, in SEM the researcher could include those four regression equations simultaneously in one model. Even, the researcher could convert the schematic diagram into a model in AMOS. Furthermore, the output from AMOS and the subsequent analysis is simple, informative, and presentable. Now let’s discuss in detail the process involved in testing the effect of mediating variable. Our discussion centers on the schematic diagram showing the mediating variable in a model, as shown in Figure 12. In the diagram, the researcher is interested to assess the effects of mediator variable M in linking the relationship between X1 and Y.
0-1
Key: The coefficient B1 would reduce when the mediator M enters in the model. If it reduces and become non- significant, then the full mediation occurs. However, if it reduces but still significant, then the partial mediation occurs. As for B2 and B3, both of them must be significant for a mediation to occur.
Figure 12: The Diagram Showing B1, B2, and B3 in the Analysis for Mediator Variable
The schematic diagram in Figure 12 reveals the following regression equations. Y = Bo + B1X1 + e is the path from X1 to Y (represented by B1) Y = Bo + B2M + e is the path from X2 to Y (represented by B2) M = Bo + B3X1 + e is the path from X1 to M (represented by B3) The path analysis to assess the effect of M in mediating the relationship between X1 and Y could result in one of the three following possibilities: 32
A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
1. M plays a complete mediation role in the relationship between X1 and Y 2. M plays a partial mediation role in the relationship between X1 and Y 3. M plays no mediation role in the relationship between X1 and Y The complete mediation role of M occurs only if these conditions are met: Refer to Figure 12:
1. The hypothesis testing for regression coefficient B1 is not significant 2. The hypothesis testing for regression coefficient B3 is significant 3. The hypothesis testing for regression coefficient B2 is significant
The partial mediation role of M occurs only if these conditions are met: Refer to Figure 12:
1. 2. 3. 4.
The hypothesis testing for regression coefficient B1 is still significant The hypothesis testing for regression coefficient B3 is significant The hypothesis testing for regression coefficient of B2 is significant The absolute value of B3 x B2 is higher than the absolute value of B1
The no mediation role of M occurs if at least one of these three conditions is met: Refer to Figure 12:
1. The hypothesis testing for regression coefficient B3 is not significant 2. The hypothesis testing for regression coefficient B2 is not significant 3. Both regression coefficients namely B2 and B3 are not significant Question: What if both coefficients B3 and B2 are significant but B3*B2is lower than B1? In this case, one needs to compare the value of B1 in the single model (X1 alone) with its value when the mediator M enters the model. If its value is reduced when the mediator is included, then the partial mediation occurred. AMOS could also analyze the mediating effects of latent construct in a model. The theoretical model is illustrated in Figure 13. In the model, construct X1 has five items; the mediator M also has five response items, while Y has three response items respectively. In Figure 13, the researcher is modeling the mediating effect of construct M in linking the relationship between construct X1 and construct Y. So, in this diagram, X1 is an exogenous construct (arrow pointing out), and Y is an endogenous construct (arrow pointing in) while M is a mediating construct (two arrows involved - one is pointing in and another one is pointing out). In this model, the researcher is interested to assess whether construct M is a really a significant mediator in the X1 to Y relationship.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
Figure 13: Modeling the mediating effect of latent construct M in AMOS Graphic.
The hypothesis testing involved in determining whether construct M has full mediation, partial mediation, or no mediation role is similar to the explanation given for Figure 12. In Amos Graphic, one can model more than mediator in a model. The model with more than one mediator is shown in Figure 13a. In Figure 13a, the independent variable (Leverage) and dependent variable (Demand) are observed directly while the two mediators are latent constructs.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
Figure 14: The model contain more than one mediators namely RTP and Attitude
1.11 Modeling the Moderating Variable in AMOS Graphic Sometimes the researcher is also interested to assess the moderating effects of certain variable in the model, normally demographic characteristics of the respondents. As its name implies, the role of a moderator variable is to moderate the relationship between the independent and its corresponding dependent variable. The position of a moderating variable in a schematic diagram of a model is presented in Figure 14:
Figure 14: The moderating variable M in a schematic diagram of a model 35
A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
If you could recall from the earlier explanation, the single headed arrow originating from the independent variable and pointing to its dependent variable indicates the causal effects of X on Y is being estimated. Now, the existence of variable M in the path could play a significant role in altering the effects of independent variable X on its corresponding dependent variable Y. For example, the effectiveness of certain teaching method in improving the academic performance of school children could depend on the background of the respondents under study. Here, teaching method applied is an independent variable, academic performance is a dependent variable, while background or demographic characteristics of the children is a moderating variable. Let see how the above theory is presented in a schematic diagram of a model as shown in Figure 15:
Figure 15: The moderating variable M in a schematic diagram of a model
Analyzing the moderation effects using the traditional Ordinary Least Squares (OLS) is quite tedious and sometimes can be misleading. However, AMOS could handle this job quite easily. The researcher needs to draw the AMOS graphic as shown in Figure 17 and Figure 18, execute the software, analyze the output and interpret the results. Let assume that variable T is teaching method, variable M is family background of children, and variable P is their academic performance. The schematic diagram is given in Figure 16:
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
Figure 1615: The model showing the independent, dependent and moderating variable
The AMOS Graphic model representing the schematic diagram in Figure 16 is presented in Figure 17 and Figure 18.
Modeling the moderator in the model of observed variables: First of all, the study needs to prove the regression effect of variable T on variable P is significant. Refer to Figure 17.
Figure 17: Modeling the effect of variable T on variable P in AMOS Graphic
In analyzing the effect of moderator variable B in the observed model, the researcher needs to compute the interaction effect between independent variable T and moderator variable M. The 37
A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
product of T multiply M is termed as TM. Now the model will estimate the effect of T, the effect of M, and the interaction effect between T and M, termed TM as shown in Figure 18.
Figure 18: Modeling the effects of moderating variable M using AMOS Graphic
In order to prove that the child’s family background (M) is a significant moderator in the relationship between teaching method (T) and child’s academic performance (P), the study needs to prove the hypothesis of causal effects as follows: 1) The effect of T on P is reduced when moderator variable M enters the model. 2) At the same time, the effect of interaction TM on P is statistically significant.
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A Handbook on SEM Zainudin Awang - Universiti Sultan Zainal Abidin
Modeling the moderator in the model of latent constructs: Modeling the moderator in the model consisting latent constructs is not as easy as modeling it in the observed variables. First of all, the researcher needs to determine the path where the moderator effect is to be analyzed. In the first place, the effect of exogenous construct on the endogenous construct in that particular path must be significant. The modeling of moderator for latent constructs is shown in Figure 19.
Figure 19: Modeling the moderator M in a model consisting of latent constructs.
Analyzing the moderator in the latent constructs model is not as straight forward as in the observed model. The method used in the analysis is called the Multi-Group CFA. The procedure for analyzing a moderator for latent constructs is explained in chapter 7.
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