Chapter 7 : Mediation Analysis and Hypotheses Testing. chapter 7.pdf · Goodman’s Test * 2 * 2 2...
Transcript of Chapter 7 : Mediation Analysis and Hypotheses Testing. chapter 7.pdf · Goodman’s Test * 2 * 2 2...
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Chapter 7 : Mediation Analysis and Hypotheses Testing
7.1. Introduction
Data analysis of the mediating hypotheses testing will investigate the impact of mediator on the
relationship between independent variables and dependent variable. This study examines
mediating effect on the direct path between the independent variables and the dependent variable
using the Baron and Kenny’s (1986) three-step mediation analysis and chi-square (χ2) difference
test. The results of the mediating effect are further confirmed by Sobel’s (1982) test, the Aroian’s
(1944) test, and the Goodman’s (1960) test.
A variable may be considered a mediator to the extent to which it carries the influence of a given
independent variable to a given dependent variable. Mediation can be said to occur when...
(1) the independent variable significantly affects the mediator,
(2) the independent variable significantly affects the dependent variable in the absence of the
mediator,
(3) the mediator has a significant unique effect on the dependent variable, and
(4) the effect of the independent variable on the dependent variable shrinks upon the addition of
the mediator to the model.
These criteria can be used to informally judge whether or not mediation is occurring, but
MacKinnon & Dwyer (1993) and MacKinnon, Warsi, & Dwyer (1995) have popularized
statistically based methods by which mediation may be formally assessed by using the Sobel’s
(1982) test, the Aroian’s (1944) test, and the Goodman’s (1960) test. These tests consider the
unstandardized regression and standard error for the association between independent variable and
mediator, and also the unstandardized regression and standard error for the association between
mediator and the dependent variable.
We propose the following mediating hypothesis:
Mediating Hypothesis MedH1 : Teachers Job Contribution (Mediator) significantly mediates
the relationship between Teachers Organizational Commitment (Independent Variable) and
Teacher’s Engagement (Dependent Variable).
Mediating Hypothesis MedH2 : Teachers Job Contribution (Mediator) significantly mediates
the relationship between Teachers Perceived Organizational Support (Independent
Variable) and Teacher’s Engagement (Dependent Variable).
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7.2. Baron and Kenny’s (1986) Three-Step Mediating Analysis
A variable may be considered a mediator to the extent to which it carries the influence of a given
independent variable to a given dependent variable. Hence, a mediator accounts for the
relationship between an independent variable and the dependent variable. Mediation can be
said to occur when...
1. the independent variable significantly affects the mediator,
2. the independent variable significantly affects the dependent variable in the absence of the
mediator,
3. the effect of the independent variable on the dependent variable shrinks upon the addition of
the mediator to the model.
Perfect mediation holds if the independent variable has no effect on the dependent variable, when
the mediator is controlled. That is complete mediation or full mediation exists if the independent
variable exerts its total influence through the mediating variable.
Partial mediation is given if the independent variable exerts some of its influence on the
dependent variable through the mediating variable, and it also exerts some of its influence directly
on the dependent variable and not through mediating variable.
7.3. Anderson and Gerbing’s (1988) Chi-square Difference Test
Further Chi-square Difference Test was conducted. Chi-square Difference Test is Anderson and
Gerbing’s (1988) approach to testing nested models to ensure that the mediating models produced a
better fit than non-mediating models. In this process, mediating models and non-mediating models
of two indirect relationship models were tested and evaluated based on χ2 statistics. If the mediating
models are better suited to the data than non-mediating models, the change in χ2 statistic should be
statistically significant (Byrne, 1998).
7.4. Sobel’s (1982) Test, the Aroian’s (1944) Test, and the Goodman’s (1960)
Test
MacKinnon & Dwyer (1993) and MacKinnon, Warsi, & Dwyer (1995) have popularized
statistically based methods by which mediation may be formally assessed by using the Sobel’s
(1982) test, the Aroian’s (1944) test, and the Goodman’s (1960) test. These tests consider the
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unstandardized regression and standard error for the association between independent variable and
mediator, and also the unstandardized regression and standard error for the association between
mediator and the dependent variable.
Sobel’s Test
Researchers argue that it is not enough to report whether the size of the relation between the
predictor and the outcome variable becomes smaller (partial mediation) or insignificant (full
mediation) when the mediator is added to the equation (Frazier, Tix, & Barron, 2004). Thus, the
Sobel (1982) tests were also applied to more thoroughly confirm the significance of the mediated
effect.
The mediated, indirect effect of the predictors on outcome variables is defined as the product of the
predictor-moderator path (a) and the moderator-outcome variable path (b), or ab. The mediated
effect was tested for statistical significance by dividing the estimate of the mediating variable effect
by its standard error and comparing this value to a standard normal distribution (MacKinnon,
Lockwood, Hoffman, West, & Sheets 2002; Sobel, 1982). The standard error of the indirect effect
(Sab) is
222222
babaab SSSaSbS
Where,
a = unstandardized regression coefficient of path a;
b = unstandardized regression coefficient of path b;
Sa = standard error of a;
Sb = standard error of b.
Aroian’s Test
Formulae for the tests provided here were drawn from MacKinnon & Dwyer (1994) and from
MacKinnon, Warsi, & Dwyer (1995):
222222 **** baba SSSaSbbavaluez
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Goodman’s Test
222222 **** baba SSSaSbbavaluez
7.5. Mediating Hypotheses Testing
Mediating Hypothesis MedH1
Null Hypothesis : Teachers Job Contribution (Mediator) does not significantly mediates the
relationship between Teachers Organizational Commitment (Independent Variable) and Teacher’s
Engagement (Dependent Variable).
Alternate Hypothesis : Teachers Job Contribution (Mediator) significantly mediates the
relationship between Teachers Organizational Commitment (Independent Variable) and Teacher’s
Engagement (Dependent Variable).
Figure 7.1 : Independent Variable (Teachers Organizational Commitment), Mediator
(Teachers Job Contribution) and Dependent Variable (Teacher’s Engagement)
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Baron and Kenny’s Mediating Analysis
Condition 1: The independent variable should significantly affect the mediator.
Figure 7.2 : Baron and Kenny’s First Condition : Independent Variable Mediator
(Teachers Organizational Commitment Teachers Job Contribution)
Table 7.1
Standardized Regression Estimate : : Independent Variable Mediator (Teachers
Organizational Commitment Teachers Job Contribution)
Standardized
Regression
Estimate
S.E. C.R. P
Teachers Organizational Commitment
→ Teachers Job Contribution 0.763 0.055 7.949 ***
Teachers Job Contribution regresses significantly on Teachers Organizational Commitment hence
the first condition of Baron and Kenny’s Mediating Analysis is satisfied.
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Condition 2: The independent variable significantly affects the dependent variable in the absence
of the mediator.
Figure 7.3 : Baron and Kenny’s Second Condition : Independent Variable Dependent
Variable (Teachers Organizational Commitment Teacher’s Engagement)
Table 7.2
Standardized Regression Estimate : : Independent Variable Dependent Variable
(Teachers Organizational Commitment Teacher’s Engagement)
Standardized
Regression
Estimate
S.E. C.R. P
Teachers Organizational
Commitment → Teacher’s
Engagement
0.456 0.080 4.873 ***
Teacher’s Engagement regresses significantly on Teachers Organizational Commitment hence the
second condition of Baron and Kenny’s Mediating Analysis is satisfied.
The effect of the independent variable on the dependent variable with the moderator :
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Figure 7.4 : Effect of the Independent Variable on the Dependent Variable with the
Mediator for Mediating Hypothesis MedH1
Table 7.3
Standardized Regression Estimate : : Independent Variable Mediator Dependent
Variable for Mediating Hypothesis MedH1
Standardized
Regression
Estimate
S.E. C.R. P
Teachers Organizational
Commitment → Teachers Job
Contribution → Teacher’s
Engagement
0.187 0.083 1.653 0.098
As the p > 0.050 (p = 0.098), there exists full mediation by the mediating variable, hence the
independent variable exerts its total influence through the mediating variable.
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Chi-square Difference Test : Anderson and Gerbing’s (1988) approach
Chi-square Difference Test is Anderson and Gerbing’s (1988) approach to testing nested models to
ensure that the mediating models produced a better fit than non-mediating models. If the mediating
models are better suited to the data than non-mediating models, the change in χ2 statistic should be
statistically significant (Byrne, 1998). The following table shows the chi-square difference test for
mediating hypothesis MedH1.
Table 7.4
Chi-square Difference Test for Mediating Hypothesis MedH1
Model Chi-square Df p-value Reject / Not
Reject the Null
Hypothesis
Non-Mediating Model 42.451 30
Mediating Model 73.149 47
The chi-square
Difference Test
30.698 17 0.0217289 Reject
The chi-square difference test reveals a significant mediation, thus it can be concluded that the
null hypothesis (MedH1) is rejected, and hence the alternative hypothesis is accepted.
Sobel’s Test, Aroian’s Test and Goodman’s Test
Thus to confirm the above mediating hypothesis we conducted the Sobel’s (1982) test, the
Aroian’s (1944) test, and the Goodman’s (1960) test. These tests were conducted in line with the
z-prime method (MacKinnon, Lockwood, Hoffman, and West & Sheetes, 2002) to check for the
statistical power of our models and discount the possibility of Type I error while exploring the
strength of mediation.
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Table 7.5
Sobel’s Test, Aroian’s Test and Goodman’s Test for Mediating Hypothesis for Mediating
Hypothesis MedH1
Mediating Hypothesis MedH2
Null Hypothesis: Teachers Job Contribution (Mediator) does not significantly mediate the
relationship between Teachers Perceived Organizational Support (Independent Variable) and
Teacher’s Engagement (Dependent Variable).
Alternate Hypothesis: Teachers Job Contribution (Mediator) significantly mediates the
relationship between Teachers Perceived Organizational Support (Independent Variable) and
Teacher’s Engagement (Dependent Variable).
Figure 7.5: Independent Variable (Teachers Perceived Organizational Support), Mediator
(Teachers Job Contribution) and Dependent Variable (Teacher’s Engagement)
Sobel’s
Test
Aroian’s
Test
Goodman’s
Test P Remarks
Teachers Organizational Commitment →
Teachers Job Contribution →
Teacher’s Engagement
4.659 4.634 4.685 *** MedH1
accepted
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Baron and Kenny’s Mediating Analysis
Condition 1: The independent variable should significantly affect the mediator.
Figure 7.6 : Baron and Kenny’s First Condition : Independent Variable Mediator
(Teachers Perceived Organizational Support Teachers Job Contribution)
Table 7.6
Standardized Regression Estimate : : Independent Variable Mediator (Teachers
Perceived Organizational Support Teachers Job Contribution)
Standardized
Regression
Estimate
S.E. C.R. P
Teachers Perceived Organizational
Support → Teachers Job Contribution 0.739 0.088 10.916 ***
Teachers Job Contribution regresses significantly on Teachers Perceived Organizational Support
hence the first condition of Baron and Kenny’s Mediating Analysis is satisfied.
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Condition 2: The independent variable significantly affects the dependent variable in the absence
of the mediator.
Figure 7.7 : Baron and Kenny’s First Condition : Independent Variable Mediator
(Teachers Perceived Organizational Support Teachers Job Contribution)
Table 7.7
Standardized Regression Estimate : : Independent Variable Dependent Variable
(Teachers Perceived Organizational Support Teacher’s Engagement)
Standardized
Regression
Estimate
S.E. C.R. P
Teachers Perceived Organizational
Support → Teacher’s Engagement 0.583 0.063 9.632 ***
Teacher’s Engagement regresses significantly on Teachers Perceived Organizational Support
hence the second condition of Baron and Kenny’s Mediating Analysis is satisfied.
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The effect of the independent variable on the dependent variable with the moderator
Figure 7.8 : Effect of the Independent Variable on the Dependent Variable with the
Mediator for Mediating Hypothesis MedH2
Table 7.8
Standardized Regression Estimate : : Independent Variable Mediator Dependent
Variable for Mediating Hypothesis MedH2
Standardized
Regression
Estimate
S.E. C.R. P
Teachers Perceived Organizational
Support → Teachers Job Contribution →
Teacher’s Engagement
0.546 0.094 5.989 ***
As the p < 0.050 (p < 0.001), there exists partial mediation by the mediating variable, hence the
independent variable exerts some of its influence on the dependent variable through the mediating
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variable, and it also exerts some of its influence directly on the dependent variable and not through
mediating variable.
Chi-square Difference Test : Anderson and Gerbing’s (1988) Approach
Chi-square Difference Test is Anderson and Gerbing’s (1988) approach to testing nested models to
ensure that the mediating models produced a better fit than non-mediating models. If the mediating
models are better suited to the data than non-mediating models, the change in χ2 statistic should be
statistically significant (Byrne, 1998). The following table shows the chi-square difference test for
moderating hypothesis MedH1.
Table 7.9
Chi-square Difference Test for Mediating Hypothesis MedH2
Model Chi-square Df p-value Reject / Not Reject
the Null Hypothesis
Non-Mediating Model 43.754
30
Mediating Model 76.456 47
The chi-square Difference
Test
32.702 17 0.01229 Reject
The chi-square difference test reveals a significant mediation, thus it can be concluded that the
null hypothesis (MedH2) is rejected, and hence the alternative hypothesis of is accepted.
Sobel’s Test, Aroian’s Test and Goodman’s Test
Thus to confirm the above mediating hypothesis we conducted the Sobel’s (1982) test, the
Aroian’s (1944) test, and the Goodman’s (1960) test. These tests were conducted in line with the
z-prime method (MacKinnon, Lockwood, Hoffman, West and Sheetes, 2002) to check for the
statistical power of our models and discount the possibility of Type I error while exploring the
strength of mediation.
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Table 7.10
Sobel’s Test, Aroian’s Test and Goodman’s Test for Mediating Hypothesis for Mediating
Hypothesis MedH2
Thus, Teachers Job Contribution (Mediator) significantly mediates the relationship between
Teachers Perceived Organizational Support (Independent Variable) and Teacher’s Engagement
(Dependent Variable).
Table 7.11 : Summary of Mediating Hypotheses Results
Mediating
Hypotheses
Independent
Variable
Mediating
Variable
Dependent
Variable
Result of
Hypothesis
Explanation
MedH1 Teachers
Organizational
Commitment
Teachers Job
Contribution
Teacher’s
Engagement
MedH1
Accepted
Teachers Job Contribution
significantly mediates the
relationship between Teachers
Organizational Commitment and
Teacher’s Engagement.
MedH2 Teachers
Perceived
Organizational
Support
Teachers Job
Contribution
Teacher’s
Engagement
MedH2
Accepted
Teachers Job Contribution
significantly mediates the
relationship between Teachers
Perceived Organizational Support
and Teacher’s Engagement.
Sobel’s
Test
Aroian’s
Test
Goodman’s
Test P Remarks
Teachers Perceived Organizational Support →
Teachers Job Contribution →
Teacher’s Engagement
4.594 4.568 4.621 *** MedH2
accepted