Supplemental Materials
“Self-Efficacy in Classroom Management, Classroom Disturbances, and Emotional
Exhaustion: A Moderated Mediation Analysis of Teacher Candidates”
By Theresa Dicke et al., 2013, Journal of Educational Psychology
http://dx.doi.org/10.1037/a0035504
Alternative Mediation Model
For Study 1, we have analyzed an alternative model in which classroom disturbances
predict emotional exhaustion via self-efficacy in classroom management. Results indicate a
model fit (² = 390.137, df = 87, p < .001, CFI = .92, TLI = .91, RMSEA = .05) which is pretty
similar to the model fit of our suggested model. However, as both alternative models have the
same degrees of freedom, they will fit the data similarly. The indirect effect of the alternative
model is β = .103 (SE = 0.021), p < .001, which is slightly lower than our suggested indirect
effect.
Details on Latent Moderated Mediation Analyses
Latent Interaction Modeling
According to our hypothesized model, the prediction of emotional exhaustion by selfefficacy in classroom management not only includes mediation by the severity of classroom
disturbances, it simultaneously involves a moderating component according to which this
mediating process is also moderated by classroom management self-efficacy. Hence, the model
includes not only a latent indirect effect, but additionally a latent interaction effect. The
importance of interaction terms has become increasingly recognized in social science
(particularly in stress-related research), but the testing of latent variable interactions has been
hampered by difficulties in estimating interactions that control for measurement error (Marsh,
Wen, & Hau, 2004, 2006). This is because interactions between latent variables often require
complex procedures for estimating the nonlinear components that make such approaches difficult
for applied researchers (Marsh et al., 2006). Recently, however, there have been advances in this
respect that have overcome the difficulties of estimating nonlinear components (Marsh et al.,
2006). In particular, researchers have advocated the use of unconstrained approaches to latent
interactions that are easier to implement and have been shown to produce solutions that are
consistent with more complex techniques (Marsh et al., 2004, 2006). Using the unconstrained
approach, the observed indicators of interest (self-efficacy in classroom management, classroom
disturbances, and emotional exhaustion) were mean-centred and interaction indicators were
created by calculating interaction terms between the highest loading items of each scale of
interest (loadings obtained from confirmatory factor analysis). Subsequent interaction indicators
were developed by multiplying the second highest loading scale items (and so on) until all scale
items were accounted for. Due to the different number of indicators in both scales (see Measures
section for details), item parcelling was used to create an equal number of item parcels per scale,
resulting in three parcels: one containing two items, the other three items each. The item parcels
were developed by combining the highest loading items into one parcel, the second highest
loading items into another parcel, and the lowest loading items into the remaining parcel in order
to meet the needs of the unconstrained approach. These item parcels were then used to estimate a
latent interaction factor for use in structural equation modelling. Once the new interaction items
were calculated, the latent interaction variable operated in a similar manner to all other latent
variables, with the exception that the nonlinear components of the interaction factor were
constrained.
Power of (Latent) Moderated Mediation
Recent studies have investigated the power of mediation models (Fritz & MacKinnon,
2007) or even more complex models (e.g., latent mediation models or models integrating
moderation and mediation; Fairchild & MacKinnon, 2009; Morgan-Lopez & MacKinnon, 2006;
Thoemmes et al., 2012). Fairchild and MacKinnon (2009) pointed out the difficulties of designs
including moderating as well as mediating effects, as these multiply the disadvantages of the
already low power of simple interactions and indirect effects due to the small effect sizes
observed in real data. Exploring moderated mediation they recommended Ns of up at least 500
for .8 power depending on how well the parameter effect size explains the dependent variable’s
variance. Morgan-Lopez and MacKinnon, (2006) who have focused on mediated moderation
have found similar results. Fairchild and MacKinnon (2009) also pointed out the lack of research
on effect size measures for models that analyze mediation and moderation simultaneously,
making it impossible to rely on these as a criterion for meaningful conclusions. However, both of
these studies are based on (moderated) mediation of manifest (observed) variables. Thoemmes et
al. (2012) took latent mediation models into account and even provided examples for power
analysis of partly latent moderated mediation models based on Monte Carlo methods. In these
simulation studies, Thoemmes et al. (2012) provided a comprehensible method in which new
variables that define the mediated effect in each group and differences in all paths and the
product term are initialized. For the moderation of the mediation effect, it is particularly
interesting how often the variables tapping the differences reach significance across all iterations
of the simulation. However, as this solution indicates, these authors investigated a power analysis
of moderated mediation based on a dichotomous moderator (group differences). Moderated
mediation based on (possibly more than one) continuous moderator(s) could require a far more
complex approach, as calculating simple differences is not possible. Further, following
assumptions of Ledgerwood and Shrout (2011), employing all variables as latent variables will
most likely influence power as well. Ledgerwood & Shrout (2011) have discussed the possible
advantages and disadvantages of manifest versus latent mediation models, including issues of
power. They claimed that despite the advantages of latent mediation in terms of bias, it is often
countered by a loss in efficiency where the increase in the number of parameters estimated
results in greater uncertainty around point estimates (Ledgerwood & Shrout, 2011). Thus, when
sample sizes are large, the tradeoff between bias and efficiency is reasonable. However, when
sample sizes are smaller, manifest moderated mediation is better due to the greater efficiency.
We have taken these assumptions into account by choosing a design consisting of two
complementary studies. The first study was based on a large sample size and therefore applied a
latent approach, while our second study consisted of a smaller longitudinal subsample of Study 1
and therefore applied a manifest approach. Hence, the combination of these two studies enabled
us to boost the advantages and reduce the disadvantages of both approaches.
References of Supplemental Materials
Fairchild, A. J., & MacKinnon, D. P. (2009). A general model for testing mediation and
moderation effects. Prevention Science, 10, 87–99.
Fritz, M., & MacKinnon, D. P. (2007). Power to detect the mediated effect.
Psychological Science, 18, 233–239.
Ledgerwood, A., & Shrout, P. E. (2011). The tradeoff between accuracy and precision in
latent variable models of mediation processes. Journal of Personality and Social Psychology,
101, 1174–1188.
Marsh, H. W., Wen, Z., & Hau, K.-T. (2004). Structural equation models of latent
interactions: Evaluation of alternative estimation strategies and indicator construction.
Psychological Methods, 9, 275–300.
Marsh, H. W., Wen, Z., & Hau, K.-T. (2006). Structural equation models of latent
interaction and quadratic effects. In G. R. Hancock and R. O. Mueller (Eds.), Structural equation
modeling: A second course (pp. 225–265). Greenwich, CT: Information Age.
Morgan-Lopez A. A, & MacKinnon D. P. (2006). Demonstration and evaluation of a
method for assessing mediated moderation. Behavior Research Methods, 38, 77–87.
Thoemmes, F., MacKinnon, D. P., & Reiser, M. R., (2010). Power analysis for complex
mediational designs using Monte Carlo methods. Structural Equation Modeling: A
Multidisciplinary Journal, 17, 510–534.