mediation: 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

mediation(x, mediator, dv, S = NULL, N = NULL, x.location.S = NULL, 
mediator.location.S = NULL, dv.location.S = NULL, mean.x = NULL, 
mean.m = NULL, mean.dv = NULL, conf.level = 0.95, 
bootstrap = FALSE, B = 10000, which.boot="both", save.bs.replicates=FALSE,
complete.set=FALSE)

Arguments

vector of the predictor/independent variable

mediator

vector of the mediator variable

vector of the dependent/outcome variable

Covariance matrix

Sample size, necessary when a covariance matrix (S) is used

x.location.S

location of the predictor/independent variable in the covariance matrix (S)

mediator.location.S

location of the mediator variable in the covariance matrix (S)

dv.location.S

location of the dependent/outcome variable in the covariance matrix (S)

mean.x

mean of the x (independent/predictor) variable when a covariance matrix (S) is used

mean.m

mean of the m (mediator) variable when a covariance matrix (S) is used

mean.dv

mean of the y/dv (dependent/outcome) variable when a covariance matrix (S) is used

conf.level

desired level of confidence (e.g., .90, .95, .99, etc.)

bootstrap

TRUE or FALSE, based on whether or not a bootstrap procedure is performed to obtain confidence intervals for the various effect sizes

number of bootstrap replications when bootstrap=TRUE (e.g., 10000)

which.boot

which bootstrap method to use. It can be Percentile or BCa, or both

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)

Value

Y.on.X$Regression.Table

Regression table of Y conditional on X

Y.on.X$Model.Fit

Summary of model fit for the regression of Y conditional on X

M.on.X$Regression.Table

Regression table of X conditional on M

M.on.X$Model.Fit

Summary of model fit for the regression of X conditional on M

Y.on.X.and.M$Regression.Table

Regression table of Y conditional on X and M

Y.on.X.and.M$Model.Fit

Summary of model fit for the regression of Y conditional on X and M

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 Indirect.Effect above) divided by the estimated standard deviation of Y (MacKinnon, 2008)

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 X and M divided by the variance in Y explained by X (Lindenberger & Potter, 1998)

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

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 $\kappa^2$ 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.

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 kappa-squared as mediation effect size measure. Psychological Methods, 20, 193--203.

Examples

Run this code


############################################
# 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))


# Raw data 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))

# Two-sided p-value. 
## First, calculate obseved value of the indirect effect and extract it here. 
IE.Observed <- mediation(x=Data[,1], mediator=Data[,2], dv=Data[,3], 
conf.level=.95)$Effect.Sizes[1,]

## Now, find those values of the bootstrap indirect effects that are more extreme (in an absolute 
## sense) than the indirect effect observed. Note that the p-value is 1 here because the observed
## indirect effect is exactly 0. 
mean(abs(IE) >= abs(IE.Observed))

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