# 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

 1 2 3 4 5 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

 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 (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 B 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)

### 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$Regression.Table Regression table of Y conditional on X Y.on.X$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

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

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)) # p-value. mean(IE>0) ## End(Not run) 

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