Effect sizes and confidence intervals in a mediation model
Description
Automate the process of simple mediation analysis (one independent variable and one mediator) and effect size estimation for mediation models, as discussed in Preacher and Kelley (2011).
Usage
1 2 3 4 5 
Arguments
x 
vector of the predictor/independent variable 
mediator 
vector of the mediator variable 
dv 
vector of the dependent/outcome variable 
S 
Covariance matrix 
N 
Sample size, necessary when a covariance matrix ( 
x.location.S 
location of the predictor/independent variable in the covariance matrix ( 
mediator.location.S 
location of the mediator variable in the covariance matrix ( 
dv.location.S 
location of the dependent/outcome variable in the covariance matrix ( 
mean.x 
mean of the 
mean.m 
mean of the 
mean.dv 
mean of the 
conf.level 
desired level of confidence (e.g., .90, .95, .99, etc.) 
bootstrap 

B 
number of bootstrap replications when 
which.boot 
which bootstrap method to use. It can be 
save.bs.replicates 
Logical argument indicating whether to save the each bootstrap sample or not 
complete.set 
identifies if the function should report the estimated kappa.squarred (see below) 
Details
Based on the work of Preacher and Kelley (2010) and works cited therein, this function implements (simple) mediation analysis in a way that automates much of the results that are generally of interest, where "simple" means one independent variable, one mediator, and one dependent variable. More specifically, three regression outputs are automated as is the calculation of effect sizes that are thought to be useful or potentially useful in the context of mediation. Much work on mediation models exists in the literature, which should be consulted for proper interpretation of the effect sizes, models, and meaning of results. The usefulness of effect size κ^2 was called into question by Wen and Fan (2015). Further, another paper by Lachowicz, Preacher, and Kelley (submitted) offers a better was of quantifying the effect size and it is developed for more complex models. Users are encouraged to use, instead of or in addition to this function, the upsilon function.
Value
Y.on.X$Regression.Table 
Regression table of 
Y.on.X$Model.Fit 
Summary of model fit for the regression of 
M.on.X$Regression.Table 
Regression table of 
M.on.X$Model.Fit 
Summary of model fit for the regression of 
Y.on.X.and.M$Regression.Table 
Regression table of 
Y.on.X.and.M$Model.Fit 
Summary of model fit for the regression of 
Indirect.Effect 
the product of \hat{a} \times \hat{b}, where \hat{a} and \hat{b} are the estimated coefficients of the path from the independent variable to the mediator and the path from the mediator to the dependent variable 
Indirect.Effect.Partially.Standardized 
It is the indirect effect (see 
Index.of.Mediation 
Index of mediation (indirect effect multiplied by the ratio of the standard deviation of X to the standard deviation of Y) (Preacher and Hayes, 2008) 
R2_4.5 
An index of explained variance see MacKinnon (2008, Eq. 4.5) for details 
R2_4.6 
An index of explained variance see MacKinnon (2008, Eq. 4.6) for details 
R2_4.7 
An index of explained variance see MacKinnon (2008, Eq. 4.7) for details 
Maximum.Possible.Mediation.Effect 
the maximum attainable value of the mediation effect (i.e., the indirect effect), in the direction of the observed indirect effect, that could have been observed, conditional on the sample variances and on the magnitudes of relationships among some of the variables 
ab.to.Maximum.Possible.Mediation.Effect_kappa.squared 
the proportion of the maximum possible indirect effect; Uses the indirect effect in the numerator with the maximum possible mediation effect in the denominator (Preacher & Kelley, 2010) 
Ratio.of.Indirect.to.Total.Effect 
ratio of the indirect effect to the total effect (Freedman, 2001); also known as mediation ratio (Ditlevsen, Christensen, Lynch, Damsgaard, & Keiding, 2005); in epidemiological research and as the relative indirect effect (Huang, Sivaganesan, Succop, & Goodman, 2004); often loosely interpreted as the relative indirect effect 
Ratio.of.Indirect.to.Direct.Effect 
ratio of the indirect effect to the direct effect (Sobel, 1982) 
Success.of.Surrogate.Endpoint 
Success of a surrogate endpoint (Buyse & Molenberghs, 1998) 
SOS 
shared over simple effects (SOS) index, which is the ratio of the variance in Y explained by both 
Residual.Based_Gamma 
A residual based index (Preacher & Kelley, 2010) 
Residual.Based.Standardized_gamma 
A residual based index that is standardized, where the scales of M and Y are removed by using standardized values of M and Y (Preacher & Kelley, 2010) 
ES.for.two.groups 
When X is 0 and 1 representing a two group structure, Hansen and McNeal's (1996) Effect Size Index for Two Groups 
Author(s)
Ken Kelley (University of Notre Dame; KKelley@nd.edu)
References
Buyse, M., & Molenberghs, G. (1998). Criteria for the validation of surrogate endpoints in randomized experiments. Biometrics, 54, 1014–1029.
Ditlevsen, S., Christensen, U., Lynch, J., Damsgaard, M. T., & Keiding, N. (2005). The mediation proportion: A structural equation approach for estimating the proportion of exposure effect on outcome explained by an intermediate variable. Epidemiology, 16, 114–120.
Freedman, L. S. (2001). Confidence intervals and statistical power of the 'Validation' ratio for surrogate or intermediate endpoints. Journal of Statistical Planning and Inference, 96, 143–153.
Hansen, W. B., & McNeal, R. B. (1996). The law of maximum expected potential effect: Constraints placed on program effectiveness by mediator relationships. Health Education Research, 11, 501–507.
Huang, B., Sivaganesan, S., Succop, P., & Goodman, E. (2004). Statistical assessment of mediational effects for logistic mediational models. Statistics in Medicine, 23, 2713–2728.
Lachowicz, M. J., Preacher, K. J., & Kelley, K. (submitted). A novel measure of effect size for mediation analysis. Submited for publication.
Lindenberger, U., & Potter, U. (1998). The complex nature of unique and shared effects in hierarchical linear regression: Implications for developmental psychology. Psychological Methods, 3, 218–230.
MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Mahwah, NJ: Erlbaum.
Preacher, K. J., & Hayes, A. F. (2008b). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40, 879–891.
Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models: Quantitative and graphical strategies for communicating indirect effects. Psychological Methods, 16, 93–115.
Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. In S. Leinhardt (Ed.), Sociological Methodology 1982 (pp. 290–312). Washington DC: American Sociological Association.
Wen, Z., & Fan, X. (2015). Monotonicity of effect sizes: Questioning kappasquared as mediation effect size measure. Psychological Methods, 20, 193–203.
See Also
mediation.effect.plot
, mediation.effect.bar.plot
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84  ## Not run:
############################################
# EXAMPLE 1
# Using the Jessor data discussed in Preacher and Kelley (2011), to illustrate
# the methods based on summary statistics.
mediation(S=rbind(c(2.26831107, 0.6615415, 0.08691755),
c(0.66154147, 2.2763549, 0.22593820), c(0.08691755, 0.2259382, 0.09218055)),
N=432, x.location.S=1, mediator.location.S=2, dv.location.S=3, mean.x=7.157645,
mean.m=5.892785, mean.dv=1.649316, conf.level=.95)
############################################
# EXAMPLE 2
# Clear the workspace:
rm(list=ls(all=TRUE))
# An (Unrealistic) example data (from Hayes)
Data < rbind(
c(5.00, 25.00, 1.00),
c(4.00, 16.00, 2.00),
c(3.00, 9.00, 3.00),
c(2.00, 4.00, 4.00),
c(1.00, 1.00, 5.00),
c(.00, .00, 6.00),
c(1.00, 1.00, 7.00),
c(2.00, 4.00, 8.00),
c(3.00, 9.00, 9.00),
c(4.00, 16.00, 10.00),
c(5.00, 25.00, 13.00),
c(5.00, 25.00, 1.00),
c(4.00, 16.00, 2.00),
c(3.00, 9.00, 3.00),
c(2.00, 4.00, 4.00),
c(1.00, 1.00, 5.00),
c(.00, .00, 6.00),
c(1.00, 1.00, 7.00),
c(2.00, 4.00, 8.00),
c(3.00, 9.00, 9.00),
c(4.00, 16.00, 10.00),
c(5.00, 25.00, 13.00))
# "Regular" example of the Hayes data.
mediation(x=Data[,1], mediator=Data[,2], dv=Data[,3], conf.level=.95)
# Sufficient statistics example of the Hayes data.
mediation(S=var(Data), N=22, x.location.S=1, mediator.location.S=2, dv.location.S=3,
mean.x=mean(Data[,1]), mean.m=mean(Data[,2]), mean.dv=mean(Data[,3]), conf.level=.95)
# Example had there been two groups.
gp.size < length(Data[,1])/2 # adjust if using an odd number of observations.
grouping.variable < c(rep(0, gp.size), rep(1, gp.size))
mediation(x=grouping.variable, mediator=Data[,2], dv=Data[,3])
############################################
# EXAMPLE 3
# Bootstrap of continuous data.
set.seed(12414) # Seed used for repeatability (there is nothing special about this seed)
bs.Results < mediation(x=Data[,1], mediator=Data[,2], dv=Data[,3],
bootstrap=TRUE, B=5000, save.bs.replicates=TRUE)
ls() # Notice that Bootstrap.Replicates is available in the
workspace (if save.bs.replicates=TRUE in the above call).
#Now, given the Bootstrap.Replicates object, one can do whatever they want with them.
# See the names of the effect sizes (and their ordering)
colnames(Bootstrap.Replicates)
# Define IE as the indirect effect from the Bootstrap.Replicates object.
IE < Bootstrap.Replicates$Indirect.Effect
# Summary statistics
mean(IE)
median(IE)
sqrt(var(IE))
# CIs from percentile perspective
quantile(IE, probs=c(.025, .975))
# pvalue.
mean(IE>0)
## End(Not run)

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