How to Test for Mediation with SEM Model

The research by Baron and Kenny (1986) was one of the fundamental frameworks for how to test mediation. Over the years, research has refined their initial work on testing media- tion. I think it is a worthy pursuit to discuss where mediation testing started and where it has progressed today. Baron and Kenny (1986) stated there were four steps to testing mediation:

Step 1—make sure that X has a significant influence onY (C path; absent of M at this point).

Step 2—test that X has a significant influence on M (A path; no Y included); this needs to be significant for mediation to be present.

Step 3—test that X has an influence on M and that M has an influence onY (A and B paths); both paths need to be significant.

Step 4—test the direct and indirect relationships simultaneously and determine if and what type of indirect effect is present (A, B, and C paths are all being evaluated).

The Baron and Kenny method was based on finding the unstandardized coefficients for each relationship and then determining significance using a Sobel test. As research has progressed, this method of testing mediation has changed, and Sobel testing has been rejected as a valid means of testing mediation. For a good discussion on why Sobel testing should no longer be used in assessing mediation, see Zhao et al. (2010). Even the initial steps outlined by Baron and Kenny have changed as well. The first step that the C path needs to be significant is not a requirement anymore. Indirect effects can be present even if a non-significant C path is initially found. The justification is based on the idea that there are suppressor effects that prevent the C path from being significant, but the indirect effect is still present. The idea that the A path and the B path have to individually be significant has been rejected as well. Hayes (2018) notes that an indirect effect is the product of the A and B paths, and statistical significance of either the A path or the B path is not a requirement for mediation.

The revised method is now concerned with assessing the indirect effect by examining the product of the A path and the B path while controlling for the direct effect of the C path. Since the Sobel test is flawed for this type of test, the more accepted approach in mediation testing is to use a bootstrap technique to determine significance. A bootstrap technique treats your data sample like a pseudo-population and then takes a random sample with replacement to determine if your indirect effect falls within a confidence interval.You can request the number of bootstrap samples to increase the accuracy of predictions (the higher, the better). I find that a bootstrap sample of 5,000 is sufficiently large, and any greater number of samples will produce very little difference. Note that with a bootstrap sample, the computer program will generate a completely different sample every time you run the analysis. With 5,000 samples, the differences will be small, but the exact numbers in the results will not be the same if you run the analysis twice. To control for this, you can ask AMOS to always use the same “seed” number, which will produce the exact same results if you run the bootstrap analysis again. Having a set “seed number” is a good idea because it keeps you from getting slightly different results every time you run the analysis for the exact same model. I will discuss how to set the seed number in AMOS later in the chapter on page 194. Lastly, you need to be aware that when you sample with replacement, the same case can appear in more than one generated data set. Overall, the bootstrap technique has become the accepted method because of its ease and accuracy of results.

Let’s look at an example in AMOS of a mediation test. Using the full structural model example from earlier, we want to examine if the construct of Adaptive Behavior has an indirect effect through Customer Delight to the construct of Positive Word of Mouth. Notice that I am including a direct path from Adaptive Behavior to Positive Word of Mouth. This will allow us to see what type of mediation is present in the analysis.

Figure 6.1 Mediation Test of Adaptive Behavior Through Customer Delight to Positive Word of Mouth

To determine if the indirect effect of Adaptive Behavior to Positive Word of Mouth is significant, we need to request from AMOS the indirect, direct, and total effects in the output.This will give all pos- sible indirect effects in the model. To do this, select the Analysis Properties button , and when the Analysis Properties pop-up window appears, go to the Out- put tab at the top. On that tab, you will see at the top right an option for “Indi- rect, direct, and total effects”. Select this option.

Next, we need to request a boot- strap analysis in AMOS. To do this, go to the bootstrap tab at the top of the Analysis Properties window. On that tab will be a checkbox called “Perform bootstrap”; click that box. AMOS will initially give you a default number of 200 samples. This is way too small. Change the number of samples to 5,000. You will also need to select the “Bias-corrected confidence intervals” checkbox. AMOS will default a 90% confidence interval, but significance in most research is at the .05 level, so you need to change this to a 95% confidence level. I typically leave all the other options on this page blank. See Figure 6.3. After selecting the option of indirect effects in the output and asking AMOS to perform a bootstrap, you can cancel out of the Analysis Properties window and then run the analysis.

Figure 6.2 Request Indirect Effect in Analysis Properties Window

Figure 6.3 Request a Bootstrap and Confidence Intervals

Let’s look at the output to determine if mediation is present. In the Estimates link, you want to select the “Matrices” link. This will let you see the total effects, direct effects, and indirect effects for each relationship in your model. We want to select the indirect effects. AMOS will give you the option to examine the unstandardized or standardized indirect effect. With most mediation analyses, you will see the unstandardized indirect effect reported. If you were look- ing to compare indirect effects within a model, you could easily do so with the standardized indirect effects, but normally, the unstandardized indirect effects are reported. In the “Indirect Effects” tab, you will see all the possible indirect effects in your model. We are concerned only with the relationship of Adaptive Behavior to Positive Word of Mouth through Customer Delight. In our model, we have only one possible mediator from Adaptive Behavior to Posi- tive Word of Mouth, so the indirect effect listed must be through the mediator of Customer Delight. (If you have more than one mediator, I will discuss this on page 182.) If we look at the intersection of Adaptive Behavior and Positive Word of Mouth, the unstandardized indirect effect is .333. Again, to calculate an indirect effect, it is a very simple process. The indirect effect is the product of the A path and the B path.The unstandardized regression coefficient for the relationship from Adaptive Behavior to Customer Delight (A path) was .450.The unstand- ardized regression coefficient for the relationship from Customer Delight to Positive Word of Mouth (B path) was .740. Multiplying these two values together gives us the indirect effect (.450*.740 =.333).

Figure 6.4 Indirect Effects Results in Matrices Link

We now know the indirect effect, but we still need to know if the indirect effect is sig- nificant and if it falls within the 95% confidence interval generated by our bootstrap. The indirect effects tab will give us the indirect effect but nothing else. We need to go to the bootstrap analysis section to find the other information. On the left-hand side, in a box below the output links is a section called “Estimates/Bootstrap”. Under that link will be another option called “Bias-corrected percentile method”. This is where we will find all the informa- tion we are looking for in regards to confidence intervals and significance levels. With the confidence intervals, you are going to get an upper bound and lower bound estimate of the indirect effect based on your bootstrap of 5,000 samples. If the range for the upper and lower bound estimates do not cross over zero, then the indirect effect is considered significant. AMOS will also give you a p-value if you need to show the specific level of significance in the bootstrap test.

Note, you have to be in the “Indirect effects” link at the top of the window to even access the bootstrap analysis in the bottom window. If you are not on the indirect effects tab at the top, the bootstrap analysis will be grayed out and inaccessible.

Figure 6.5 Accessing Confidence Intervals in Results

Figure 6.6 Upper and Lower Confidence Intervals Along With Significance

Based on these results, we can conclude that Adaptive Behavior has a significant indirect effect on PositiveWord of Mouth through the construct of Customer Delight.We know that the indirect effect is significant, but now we need to assess what type of mediation is present. Is it full media- tion or partial mediation through the Customer Delight construct?To accomplish this, we need to examine the C path, or the direct path, from Adaptive Behavior to Positive Word of Mouth in the “Estimates” link in the output.We can see that Adaptive Behavior has a non-significant relationship to PositiveWord of Mouth (p = .455).This means that the influence of Adaptive Behavior on Posi- tive Word of Mouth is fully mediated through the construct of Customer Delight.

Figure 6.7 Examining the Direct Effect in Mediation Test

Since we have only one mediator, we can easily assess if a significant indirect effect is from Adaptive Behavior to the other dependent variable of Tolerance to Future Failures.We need to change the C path, or the direct effect, from Adaptive Behavior to Tolerance to Future Failures, and then we can run the exact same indirect, direct, and total effects analysis as we did before.

Figure 6.8 Mediation Test From Adaptive Behavior Through Customer Delight to Tolerance to Future Failures

The indirect effect of Adaptive Behavior to Tolerance to Future Failures is .313. Next, let’s look at the same output for the confidence intervals generated by our bootstrap. The lower bound confidence interval is .231 and the upper bound is .426. Since this confidence interval did not cross zero, we know that the indirect effect is significant. Examining the two-tail significance test in the output, the indirect effect is significant at the p < .001 level.

The last thing we need to do is assess the C path or direct effect from Adaptive Behavior to Tolerance to Future Failures to determine the type of mediation that is present. In the output of the Estimates link (Fig- ure 6.11), you will see that the direct relationship from Adaptive Behavior to Tolerance to Future Failures has a non- significant relationship (p. = .119).

Figure 6.9 Indirect Effects Results for Adaptive Behavior to Tolerance to Future Failures

Thus, we have a significant indirect effect and non-significant direct effect, meaning that Cus- tomer Delight fully mediates the relationship of Adaptive Behavior to Tolerance to Future Fail- ures. If you have only one potential mediator in your model, AMOS will give you all possible indirect effects, which is a nice function and can save you time especially if you have a large number of independent and dependent variables.

Figure 6.10 Upper and Lower Bound Estimates of Confidence Interval

Figure 6.11 Direct Effect From Adaptive Behavior to Tolerance to Future Failures

Source: Thakkar, J.J. (2020). “Procedural Steps in Structural Equation Modelling”. In: Structural Equation Modelling. Studies in Systems, Decision and Control, vol 285. Springer, Singapore.