Objectives of this programme
Other web-sites exist that can compute Sobel's z for mediation (Preacher and Leonardelli's on-line applet is an excellent example:
These sites offer certain capabilities that are helpful, however, in my opinion, these sites may not give the user all of the information that he or she may require.
For example, writers have criticized researchers relying entirely upon a significant Sobel's z-value rather than using confidence intervals to determine significance for mediated effects.
Also, the user often wishes to know the size of the mediated effect and some of these sites do not provide this information.
The present site asks for more statistical information from the user initially in order to provide the user with more complete information at the end.
In particular, you are asked to input zero order correlations, sample size, both standardized and unstandardized regression information, and overall variance explained.
This information is then combined, computed, and displayed on the last page of this site to provide information concerning the significance of the mediation (Sobel's z-score and confidence interval),
the effect size (both standardized and R2 estimates), and it displays the mediational triangle in graphical form so that the user can double-check that all values have been inputted correctly.
My aim is to provide a programme that will be maximally informative about the nature of the relationships among these three variables.
It is assumed, if you wish to use this programme, that:
1) You are interested in whether a variable (the mediating variable) mediates the covariance relationship between two other variables (the IV and DV). In other words, you are concerned with the relationships among three variables, and these three variables only.
2) Neither the mediating variable nor the dependent variable is categorical. The independent variable can be a dichotomous categorical variable (e.g., gender), but neither the MedV nor the DV can be categorical. Special treatment of these types of variables are required outside of ordinary multiple regression (see my recent book, listed below).
3) Your data conform to acceptable statistical assumptions (avoid excessive skewness, kurtosis, and multivariate non-normality). Also be careful that asymmetrical missing data do not bias your results (i.e., computing correlations and regressions on different sized samples due to list-wise deletion).
In order to generate the figure, you need to input the names of the three variables under consideration: the IV, the mediating variable (MedV), and the DV. Once you do this, the next box (and subsequent pages) reflects this new information, and you are asked to input the zero-order correlations (Pearson) between the pairs of variables. Finally, you are asked to input the sample size. (Compare the sample sizes for your correlations; if they are markedly different, as in N = 189 for one and N = 162 for another, then listwise deletion will likely bias your analyses. If my correlation output indicates different sample sizes for different pairs, I usually choose the smallest N listed in order to be conservative or I perform a missing values imputation.)
When you hit "submit", the programme will take you to the next page which asks for statistical information taken from two different regressions. The user should perform these regressions, and input the requested values. Be careful to perform the regressions correctly and input the requested information accurately. Once the values are entered, click on "do calculations", and the figure on the last page is generated.
You may leave the boxes empty for the two part correlations and the Total R2 if you do not wish to compute the R2 effect size.
Several computed outcomes and the figure will be displayed on this page. The display will inform the user whether significant or non-significant mediation was obtained based on the significance of the Sobel's z-score (described in Baron & Kenny, 1986). A p-value less than .05 conventionally signifies a statistically significant mediation result. In addition, since conventional wisdom suggests that a confidence interval is a more useful estimate of significance than the Sobel's test, a 95% confidence interval will be generated and the upper and lower values will be printed. If the range includes zero, then you have obtained non-significant mediation, but if the range does NOT include zero, then you have obtained significant mediation.
In addition, the unstandardized indirect effect and its standard error are reported.
The next section outputs the effect sizes of the indirect effect. Two methods are presented (see MacKinnon, 2008, for a fuller description of these approaches) and it should be understood at the outset that they do not yield the same results. The first column of numbers reports outputs based on the standardized regression coefficients. The total effect size is composed of the direct and the indirect effects. The direct effect is based on the beta for the IV to DV relationship with MedV included in the equation (i.e., in the figure it is the beta presented in parentheses under the IV to DV arrow, known as c prime or c`). The indirect effect can be computed two ways. First, it can be obtained by subtracting the direct effect (c`) from the total effect (the zero order correlation between the IV and the DV, known as c). And second, one can multiply the beta between the IV and the MedV (known as a) by the beta between the MedV and the DV with the IV in the equation at the same time (known as the b). Both methods of computation should yield the same numerical value. And then last, the "Indirect to Total ratio" reports the division of the indirect effect by the total effect to yield a value that estimates how large the indirect effect is in relation to the total effect. The last value should fall between .00 and 1.00, and higher values indicate larger relative indirect effects. For example, it is helpful to note in one's report, for example, that "the indirect path from the IV through the MedV to the DV accounted for about 56% of the total effect".
If you click on "R2 variance effect sizes", you will see an additional set of effect size estimates. This column of numbers represents the same estimates of effects, but these are based on amount of variance explained instead of beta weights. The derivation of these values come from the part correlations (otherwise known as 'semipartial correlations') entered earlier. Explication of these computations can be found in my book on mediation and moderation (Jose, 2013).
An obvious question can be posed here: which set of effect size estimates should one report? Effect size measures in mediation are not usually reported, but the ones based on standardised coefficients are probably more often used than others. I might mention in passing that MacKinnon (2008) described five or six different approaches. My personal preference is for the set based on variance estimates because these are conceptually more transparent, although they are more arithmetically challenging to produce. However, it must be said that there is at this point little consensus in the field concerning which is preferable. Whatever method you use, be sure to cite your method.
And the last part of the output is the figure, and it should display variable names in the proper places, beta weights, and asterisks indicating significance levels. The two beta weights in parentheses report the beta weights computed after the mediator has been included in the regression equation. The values not in parentheses are the zero order correlations. It is not commonplace (unfortunately) in the literature to publish this type of output because it is seen as costly in terms of space, but I include it here so that the researcher can quickly check that all values were inputted correctly. In many (if not all) of the other mediation applets and web-sites, it is not possible to double-check whether values have been properly inputted. Since a simple transposition of two numerical values can invalidate the result, I think that it is an important check.
For a general orientation to statistical mediation, one should read the original Baron and Kenny article:
Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research:
Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182.
The single most helpful, comprehensive, and up-to-date description of mediation is found in MacKinnon's book:
MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Mahway, NJ:
Lawrence Erlbaum and Associates.
My book on statistical mediation and moderation is more focused on hands-on procedures of how to compute mediation (and moderation):
Jose, P. E. (2013). Doing statistical mediation and moderation. New York: Guilford Press.
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You can use the programme on this web-site, or you can download an Excel version for your own purposes.
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If you have questions, problems, kudos, or suggestions about this programme, please e-mail me at the address given on my home page. I hope that you will find this programme useful. If you can think of ways to improve it, please let me know.