Ronald H. Heck
EDEP 606: Multivariate Methods (F2016)
The University of Hawai‘i at Mānoa
1
November 25, 2016
Testing Mediated Effects
With a mediator (B) between A and C, we are trying to determine whether or not the mediator
reduces or increases the initial relationship observed between A and C. Let’s look at how that
might work. In this case, we know there is a moderate negative correlation between female and
current salary (r = -0.450, p < .05). We’ll use beginning salary as the mediator, since we know it
positively affect one’s current salary (r = 0.880, p < .05). Our initial hypothesis is that beginning
salary is a mediator between gender and current salary (e.g., others might be experience and
education) It is not easy to find a mediator that will completely eliminate (or wash out) female’s
negative direct effect on current salary (full mediation) in this example data set.
female
beginning
salary
current
salary
It is more typical that the mediator may reduce the size of the negative effect on Y somewhat
(partial mediation).
female
beginning
salary
current
salary
Let’s say we assume full mediation is the desired model over partial mediation).
We obtain the following results. Full mediation assumes the path between female and current
salary is 0. How do we determine it is the better model of the two?
Ronald H. Heck
EDEP 606 (F2016): Multivariate Methods
The University of Hawai‘i at Mānoa
female
-.457*
beginning
salary
2
November 25, 2016
Testing Mediated Effects
.880*
current
salary
We can propose and alternative (partial mediation) model. This essentially provides a test that
the path from female to current salary is 0. If the path is not statistically significant, we would
have preliminary evidence favoring the full mediation model. We see below we must reject that
hypothesis, however (-0.061, p < .05).
-.457*
female
beginning
salary
.852*
current
salary
-.061*
Note: We would likely eliminate the regression model (with just beginning salary and female as
predictors of current salary), since it does not optimally address the temporal relationships
between female and beginning salary and between beginning salary and current salary.
Given the results of partial mediation model test, is the effect of female on current salary mostly
direct or indirect? Is the effect significant at both points in time? Let’s recapture the total effect
of female on current salary (-0.450). This should be the sum of the direct and indirect effects.
The direct effect is -0.061. The indirect effect can be calculated as the effect of female on
beginning salary multiplied by the effect of beginning salary on current salary (-0.457*0.852 =
-0.389). The total effect is then -0.389 + (-0.061) = -0.450. This matches the original correlation
coefficient.
Part of the analysis of different models, therefore, depends on technical issues (e.g., how well the
model accounts for variance in the outcome, are all the paths necessary?), and part may depend
on which one seems more consistent with one’s proposed theory being tested and is more
Ronald H. Heck
EDEP 606 (F2016): Multivariate Methods
The University of Hawai‘i at Mānoa
3
November 25, 2016
Testing Mediated Effects
complete in terms of having the important variables included in the model and in the correct
positions (e.g., like beginning salary).
How many paths are possible?
If p is the total number of variables in a particular correlation or covariance matrix, then the total
number of variances and covariances (or correlations) is the following:
p(p+1)/2
With three variables in the correlation matrix, we will then have
3(4)/2 = 6
The six elements in the correlation matrix are the three variances (the 1s in the main diagonal)
and the three correlations. When estimating our model below, six will be the total number of
parameters estimated. We can see below that for partial mediation we have estimated three paths,
plus two residual variances. The last variance is the one for female (which is exogenous, so its
variance is determined outside of the model). Altogether, this makes a total of six parameters.
How many parameters are estimated in the full mediation model?
Here, we can see why full mediation is considered a more parsimonious theoretical model. More
specifically, it has one degree of freedom (df), which is the result of fixing the path from female
to current salary to 0. This suggests it is more parsimonious than the partial-mediation model
with 0 df.
Ronald H. Heck
EDEP 606 (F2016): Multivariate Methods
The University of Hawai‘i at Mānoa
4
November 25, 2016
Testing Mediated Effects
Testing the Significance of the Indirect Effect in SPSS
In many cases, the significance of the indirect effect of X on Y through M may be of importance.
In SPSS, we can test this by obtaining the standard error (SE) of the indirect effect. To do so, we
use the relevant t-tests of the direct effects for the effect of X on the mediator and the mediator
on Y. In this case, the indirect effect of female on current salary is -0.457*0.852 = -0.389.
If we examine the regression tables, we can see that the t value for the relationship between
female and beginning salary is -11.162 (path a) and for the relationship between beginning salary
and current salary (path b) it is 34.866.
Coefficientsa
Standardized
Unstandardized Coefficients Coefficients
Model
B
Std. Error
Beta
1
(Constant)
.000
.041
female
-.457
.041
-.457
a. Dependent Variable: begsal
t
.000
-11.162
Sig.
1.000
.000
t
.000
34.866
-2.473
Sig.
1.000
.000
.014
Coefficientsa
Standardized
Unstandardized Coefficients Coefficients
Model
B
Std. Error
Beta
1
(Constant)
.000
.022
begsal
.852
.024
.852
female
-.060
.024
-.061
a. Dependent Variable: cursal
We can then estimate the standard error as follows (-0.389 is the indirect effect):
𝑆𝐸 =
𝑎𝑏√𝑡𝑎2 + 𝑡𝑏2 −.389√124.590 + 1215.638 −14.241
=
=
= 0.037
𝑡𝑎 𝑡𝑏
−11.162(34.866)
−389.174
Ronald H. Heck
EDEP 606 (F2016): Multivariate Methods
The University of Hawai‘i at Mānoa
5
November 25, 2016
Testing Mediated Effects
We can then divide the indirect effect coefficient ab (-0.389) by its standard error (0.037) to
obtain a t-ratio (t = -0.389/0.037 = 10.51). In this case, with a relatively large sample size (N =
474), the required t-ratio is 1.96 or larger @ p = .05. So we can reject the null hypothesis that the
indirect effect = 0.
Examining a Mediated Effect at Level 2
We can also test the significance of an indirect (or mediated) effect at level 2 in much the same
way. We will propose the following model at level 2. We might first pose a multilevel partial
mediation model. This is a saturated model since there are no degrees of freedom.
V3
DEP
V2
------------------------------------------------------------------------------------------------------------------------------
female
DEP
.
The alternative model is a full mediation model. This model has one degree of freedom, since the
path between variable 3 and the dependent variable has been fixed to 0. We can next test the two
models and compare the parameter estimates.
V3
DEP
V2
------------------------------------------------------------------------------------------------------------------------------
female
DEP
.
Ronald H. Heck
EDEP 606 (F2016): Multivariate Methods
The University of Hawai‘i at Mānoa
6
November 25, 2016
Testing Mediated Effects
The partial mediation model provides the following standardized estimates using Mplus. We can
see that the path from V3 to the dependent variable is not significant (0.257, p > .05). We can fix
this estimate to zero and re-estimate the alternative model (i.e., the full mediation model).
.257
V3
DEP
1.042*
(.124)
V2
.757*
(.725)
------------------------------------------------------------------------------------------------------------------------------
.223*
female
DEP
.918
The alternative model is shown below. Further, the chi-square test in Mplus, which can be used
to compare the two models was 2.036 (1df) with a significance level of 0.1537. Since the chisquare test was not significant, it suggests the fully-mediated model is consistent with the data.
We can therefore accept it, as it is more parsimonious than the partially-mediated model.
Ronald H. Heck
EDEP 606 (F2016): Multivariate Methods
The University of Hawai‘i at Mānoa
7
November 25, 2016
Testing Mediated Effects
Full-Mediation Model Mplus Output (120 individuals nested in 20 groups).
________________________________________________________________
Two-Tailed
Estimate
S.E. Est./S.E.
P-Value
________________________________________________________________
Within Level
ZDEP
ON
FEMALE
0.224
0.066
3.413
0.001
Residual Variances
ZDEP
0.140
0.019
7.532
0.000
Between Level
ZDEP
ON
ZV2
0.825
0.092
8.947
0.000
ZV2
ON
V3
1.042
0.379
2.752
0.006
Intercepts
ZV2
-0.522
0.175
-2.981
0.003
ZDEP
-0.109
0.079
-1.383
0.167
Residual Variances
ZV2
0.717
0.188
3.819
0.000
ZDEP
0.100
0.027
3.786
0.000
R-SQUARE
Within Level
ZDEP
0.082
0.045
1.831
0.067
Between Level
ZV2
0.275
0.183
1.501
0.133
ZDEP
0.870
0.038
23.091
0.000
INDIRECT EFFECTS
Effects from V3 to ZDEP
Total
0.859
0.349
2.462
0.014
Total indirect
0.859
0.349
2.462
0.014
________________________________________________________________
Chi-Square= 2.036 (1 df), p = 0.1537
Partial Mediation Model SPSS Output (2 steps)
Step 1: Estimate the regression of the dependent variable on Female and Zv2.
Estimates of Fixed Effectsa
Parameter
Estimate Std. Error
Intercept
-.110
.087
female
.224
.075
Zv2
.824
.080
a. Dependent Variable: Zscore(dep).
df
28.987
104.192
20.652
t
-1.270
2.983
10.329
Sig.
.214
.004
.000
Step 2: Estimate the regression of Zv2 on V3.
To examine the effect for V2 regressed on V3, you examine the unit level only (i.e., level 2)
data. You can create a small syntax file to eliminate the within-groups data (so N = 20 groups).
Ronald H. Heck
EDEP 606 (F2016): Multivariate Methods
The University of Hawai‘i at Mānoa
8
November 25, 2016
Testing Mediated Effects
compute ran1 = uniform(1).
rank variables = ran1 by id
/ran1 into ran2.
select if ran2 eq 1.
execute.
Coefficientsa
Standardized
Unstandardized Coefficients Coefficients
Model
B
Std. Error
Beta
1
(Constant)
-.521
.283
v3
1.042
.400
.523
a. Dependent Variable: Zscore(v2)
T
-1.842
2.605
Sig.
.082
.018
Estimating the Indirect Effect in SPSS
We can estimate the indirect effect between V3 and DEP as 1.042*0.825 = 0.859. We can next
test the significance of the indirect effect using the following formula once again. We can make
use of the relevant t-tests from the SPSS output. The t-test between V3 and V2 is 2.605. The ttest between V2 and Dep is 10.329.
𝑆𝐸 =
𝑎𝑏√𝑡𝑎2 + 𝑡𝑏2 . 859√6.786 + 106.688 9.150
=
=
= 0.340
𝑡𝑎 𝑡𝑏
2.605(10.329)
26.907
The standard error is then 0.340 (i.e., the estimate is 0.349 in the Mplus output). If we divide the
parameter estimate (0.859) by the standard error, we obtain a t-ratio of 2.526. This well exceeds
the required t-value of roughly 2.0.
SPSS Output Variance Components (Step 1)
Estimates of Covariance Parametersa
Parameter
Estimate Std. Error Wald Z
Residual
.139600
.019763
7.064
Intercept [subject = id] Variance
.100126
.039379
2.543
a. Dependent Variable: Zscore(dep).
Sig.
.000
.011
Ronald H. Heck
EDEP 606 (F2016): Multivariate Methods
The University of Hawai‘i at Mānoa
9
November 25, 2016
Testing Mediated Effects
Step 2 Model Summary
Model Summary
Model
R
R Square
a
1
.523
.274
a. Predictors: (Constant), v3
Adjusted R Std. Error of
Square
the Estimate
.233
.89455
We can conclude that the results across both SPSS and Mplus are quite consistent.