Online Supplement A: Additional Information on Bayesian Analysis
Overview of Bayesian Analysis
Given the relative novelty of Bayesian analyses in applied psychology research, we
provide additional details about the procedure. As with likelihood-based approaches (e.g., ML),
Bayesian analysis begins with a descriptive model. The most noteworthy difference between
likelihood-based and Bayesian approaches is with the treatment of the unknown parameters (van
de Shoot, Kaplan, Denissen, Asendorpf, Neyer, & van Aken, 2014). The likelihood-based
framework treats the data as uncertain and the parameters of the model as fixed, thereby
assuming a single true population parameter (e.g., one true mean or correlation). Accordingly,
likelihood-based statistics represent the probability of the observed data given the parameters
(i.e., the likelihood, which is then maximized with a set of parameter estimates for the specified
model). Conversely, in Bayesian modeling, the data are fixed, and the parameters are treated as
uncertain. Parameter values are thus described by a probability distribution (Kruschke, Aguinis,
& Joo, 2012; van de Shoot et al., 2014), called the posterior distribution.
The posterior distribution is computed by combining prior information regarding the
parameter—called a prior distribution—with information derived from the current data. More
specifically, the posterior distribution is the product of the prior distribution and the likelihood of
the data, given the parameter (Zyphur & Oswald, 2015). Because this formula (Bayes’ theorem)
incorporates prior information into the estimation procedure, researchers who wish to implement
a Bayesian approach must decide what form the prior distribution should take—a decision which
involves subjective judgment and therefore has the potential to influence study results.
The analyses are performed using a Markov chain Monte Carlo (MCMC) simulation (van
de Shoot et al., 2014; Muthén, 2010). During this process, posterior values for each parameter
are estimated over the course of several iterations. These estimates form a “chain” and are
eventually combined to develop the posterior distribution. In order to ensure that the iteration
process converges on a stable set of posteriors, two or more starting points (in our case, two,
which is the Mplus default) are used (Zyphur & Oswald, 2015). From the posterior distributions
of the parameters, estimates such as means, medians, and modes are obtained (Muthén, 2010),
and credibility intervals, the Bayesian version of a confidence interval, around these values are
estimated. One advantage to credibility intervals is that they have a more straightforward
interpretation than the confidence intervals used in likelihood-based approaches (Yuan &
MacKinnon, 2009). A 95% confidence interval, for example, is interpreted to mean that, if a
sample is repeatedly taken form a population, and a confidence interval is calculated from each
sample, 95% of these intervals on average will contain the parameter’s true value. Alternatively,
a 95% credibility interval can be interpreted to mean that, based on the observed data, there is a
95% chance that the true value of the parameter is contained within the interval.
Prior Specification
We specified the following prior distributions for our parameters. For auto-regressive and
positively hypothesized regression coefficients, priors were specified as normal distributions
with small positive means of .1 and variances of .5. For negatively hypothesized regression
coefficients, the priors were also specified as normal distributions, but with small negative means
of -.1 and variances of .5. These values were chosen, because prior research has typically found
small positive effects for the relationship among abusive supervision and aggressive (c.f., Lian,
Brown, Ferris, Liang, Keeping & Morrison, 2014) and avoidant (Mawritz, Dust, & Resick, 2014)
responses, and small negative effects for the relationship among abusive supervision and
prosocial responses (Zellars, Tepper, & Duffy, 2002). Likewise, previous research has also
linked anger, fear, and compassion to aggressive, avoidant, and prosocial responses, respectively
(Bossuyt, Moors, & De Houwer, 2014; Eisenberg & Miller, 1991; Rodell & Judge, 2009).
There has been less empirical work on the effects of abuse on discrete emotions and the
effects of subordinates’ behaviors on abusive supervision. There is, however, some evidence to
suggest a positive effect of subordinates’ deviance on abusive supervision (Lian et al., 2014), and
negative effects between abusive supervision and well-being in general (e.g., Tepper, Moss,
Lockhart & Carr, 2007). Theory also provides rationale for expected directionality of the effect
sizes for the hypothesized relationships that lack empirical evidence (and, of course, those that
do not). For regression coefficients between emotions at time t and emotions at time t + 1 that
were not purely auto-regressive (e.g., the effect of anger at time t on compassion at time t + 1;
the effects of citizenship at time t on counterproductive behavior at time t + 1), priors were
specified as normal distributions, but because there was little theoretical guidance regarding the
directionality of these relationships, means were specified at 0, with variances of .5. Finally, for
random variances and residual variances, we used a diffuse prior common in the multilevel
literature (Browne & Draper, 2006)—an inverse gamma distribution, IG (α, β), where α denotes
the shape of the parameter and β represents the scale of the parameter (Spiegelhalter, Thomas,
Best, & Gilks, 1997). An IG (ϵ, ϵ) prior, where ϵ denotes a small value, was specified with ϵ =
.001 (Muthén, 2010). Simulations suggest that the IG (ϵ, ϵ) prior performs best when used to
estimate median point estimates (Browne & Draper, 2006). However, comparing results using
mean and median point estimates revealed the significance of results to be robust regardless of
the choice of point estimate. Therefore, we report our analyses using a mean point estimate in
order to facilitate interpretation of our results.
Alternative Prior Specifications
In Bayesian analysis, one can choose whether to incorporate informative priors, derived
from previous theoretical or empirical work, or diffuse (non-informative) priors, which are used
when little information regarding the parameters is available or when one wishes for prior
information to bear little influence on the parameters (Browne & Draper, 2006). Although we
used informative priors in estimating our regression coefficients, we also estimated our model
using diffuse priors, to assess whether results differed from those obtained using the more
informative priors. If similar results are obtained for different diffuse priors, it is reasonable to
assume that the sample size is large enough and that the priors chosen did not meaningfully alter
the results; that is, there is a lack of “prior dependence” in that posteriors are not overwhelmed
by priors, and the results are data driven (Asparouhov & Muthén, 2010).
Prior distributions for the regression coefficients in these analyses were specified as
normal distributions with means of zero and variances of infinity (numerically represented as
1010), which is the default, diffuse prior in Mplus (Muthén & Muthén, 2012). For random
variances and residual variances, we again used an IG (ϵ, ϵ) prior, where ϵ denotes a small value,
specified as ϵ = .001 (Muthén, 2010). We also, however, ran the analyses using another common
diffuse prior for random variances and residual variances—a uniform distribution U (0, 1/ ϵ),
which was specified in our analysis as ranging from 0 to 1000 (Brown & Draper, 2006). Results
did not meaningfully differ, in terms of the direction and pattern of significance of the effect
sizes, suggesting that our results were robust and that our posterior distributions were not overly
dependent on the priors.
Model Convergence
Convergence of the model was evaluated by calculating the potential scale reduction
score (PSR; Asparouhov & Muthén, 2010). The PSR is the ratio of the total variance across
chains to the pooled within chain variance, which is the inverse of the intraclass correlation.
When variance between chains is low relative to within-chain variance, (i.e., PSR < 1.05),
estimation of the model can be stopped because the separate chains produce equivalent results
(Zyphur & Oswald, 2015). To obtain further evidence of convergence Muthén (2010),
recommends running longer chains to ensure that parameter values do not change with additional
estimation iterations and the PSR values remain close to 1. Accordingly, we estimated models
with 100,000 iterations and checked the stability of parameters by estimating the models again
with 200,000 iterations. Each model had a PSR value of below 1.05, and parameter values were
stable across models. Trace plots also suggested that the model mixed adequately (see associated
online supplement), and the Kolmogorov-Smirnov test (Muthén & Asparouhov, 2012)—a
convergence criterion which compares draws from each of the two MCMC chains for each
parameter to assess whether they are equal—provided further evidence of model convergence.
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