1. Mediation and Multi-group Analyses
Lyytinen & Gaskin

2. Mediation
In an intervening variable model, variable X, is postulated to exert an effect on an outcome variable, Y, through one or more intervening variables called mediators (M)
“mediational models advance an X → M → Y causal sequence, and seek to illustrate the mechanisms through which X and Y are related.” (Mathieu & Taylor)
Mediation refers to causal relationships somewhat more complex than the simple “x predicts y” relationship. When hypothesizing and testing for mediation, a mediating variable, “m,” (click) plays some role in the relationship between x and y. Thus we are hypothesizing that the relationship between x and y is somewhat more complex than a simple direct effect from IV to DV. More complex mediation can also take place, for example, with two mediating variables, but testing more complex mediation is not only beyond the scope of a “simple guide to mediation”, there is also not currently an accepted or agreed upon method for testing these types of mediation. So, we’re going to keep it simple. X M Y

3. Why Mediation?
Seeking a more accurate explanation of the causal effect the antecedent (predictor) has on the DV (criterion , outcome) – focus on mechanisms that make causal chain possible
Missing variables in the causal chain
Intelligence  Performance
Intelligence  Work Effectiveness  Performance
So why even consider mediation? Why not just predict and test direct effects? Often the reason is because we believe that some x really does have an effect on some y, but we want to ensure that there is not a more accurate explanation for this cause and effect relationship. For example, let’s say we want to predict what causes people to either (Click) use an umbrella or not. We may say that (click) rain causes this effect. However, this does not explain why many people do not use an umbrella when it rains. Thus we can explain more of the observed behavior in our sample if we include a mediator called, (click) “Desire to stay dry”. Rain causes those who wish to remain dry to use an umbrella when it rains, and those who do not have this desire do not use an umbrella. So in this example “desire to stay dry” mediates the effect rain has on using an umbrella. Without the mediator, we can only explain the behavior of those who use umbrellas in the rain; whereas, with the mediator, we can also explain the behavior of those who do not use umbrellas.

4. Conditions for mediation
(1) justify the causal order of variables including temporal precedence; (2) reasonably exclude the influence of outside factors; (3) demonstrate acceptable construct validity of their measures; (4) articulate, a priori, the nature of the intervening effects that they anticipate; and (5) obtain a pattern of effects that are consistent with their anticipated relationships while also disconfirming alternative hypotheses through statistical tests.

5. Conditions for mediation
Inferences of mediation are founded first and foremost in terms of theory, research design, and the construct validity of measures employed, and second in terms of statistical evidence of relationships.
Mediation analysis requires:
1) inferences concerning mediational X MY relationships hinge on the validity of the assertion that the relationships depicted unfold in that sequence (Stone-Romero & Rosopa, 2004). As with SEM, multiple qualitatively different models can be fit equally well to the same covariance matrix. Using the exact same data, one could as easily ‘confirm’ a YMX mediational chain as one can an XMY sequence (MacCallum, Wegener, Uchino, & Fabrigar, 1993).

6. Conditions for mediation
2) experimental designs is to isolate and test, as best as possible, XY relationships from competing sources of influence. In mediational designs, however, this focus is extended to a three phase XMY causal sequence requiring random assignments to both X and M and related treatments
“Because researchers may not be able to randomly assign participants to conditions, the causal sequence of XMY is vulnerable to any selection related threats to internal validity (Cook & Campbell, 1979; Shadish et al., 2002). To the extent that individuals’ status on a mediator or criterion variable may alter their likelihood of experiencing a treatment, the implied causal sequence may also be compromised. For example, consider a typical: trainingself-efficacyperformance, mediational chain. If participation in training is voluntary, and more efficacious people are more likely to seek training, then the true sequence of events may well be self-efficacytrainingperformance. If higher performing employees develop greater self-efficacy (Bandura, 1986), then the sequence could actually be performanceefficacytraining. If efficacy and performance levels remain fairly stable over time, one could easily misconstrue and find substantial support for the trainingefficacyperformance sequence when the very reverse is actually occurring.” (Mathieu and Taylor 2006)

7. Conditions of mediation
It is a hallmark of good theories that they articulate the how and why variables are ordered in a particular way (e.g., Sutton & Staw, 1995; Whetten, 1989). This is perhaps the only basis for advancing a particular causal order in non-experimental studies with simultaneous measurement of the antecedent, mediator, and criterion variables (i.e., classic cross-sectional designs).
Implicitly, mediational designs advance a time-based model of events whereby X occurs before M which in turn occurs before Y. It is the temporal relationships of the underlying phenomena that are at issue, not necessarily the timing of measurements
In other words, in mediation analyses, omitted variables represent a significant threat to validity of the XM relationship if they are related both to the antecedent and to the mediator, and have a unique influence on the mediator. Likewise omitted variables (and related paths) may lead to conclude falsely that no direct effect XY exists, while in fact it holds in the population

8. Importance of theory – Cause and effect
Training
Self-efficacy
Performance
Training
Performance
Self-efficacy
Performance
Self-efficacy
Training
Just as with other hypotheses, mediation suggests a cause and effect relationship between variables. However, remember from previous lessons that causation is not implied by correlation, and statistically, the IVs, DVs, and mediating variables will likely be correlated statistically. Thus it is critical to develop mediation models based on theory first, then to test and support the theoretical model through statistics. To illustrate this point, consider the relationship shown. Self efficacy mediates the effect training has on performance. In other words, training effects performance through Self-efficacy. That makes sense.
Now consider this alternative model (click). Performance mediates the effect Training has on Self-efficacy. Well, that also makes sense, and it is supported by a statistical test. Now consider these other models (click) (click), they also make sense in certain contexts, and they are also supported by statistical tests. So which model do we choose? Most likely, we have a certain theory and model in mind prior to doing the statistical tests. Our advice to you is to let this a priori theory guide your model development, rather than first seeing what the statistics say, then developing a model. See Mathieu and Taylor 2006 for a more in depth discussion of this specific example.
Self-efficacy
Training
Performance

9. Types of Mediation Indirect Effect Partial Mediation Full Mediation
Significant Path
Insignificant Path M
Indirect Effect X Y M
Partial Mediation X Y
Now, there are three main types of mediation. The first is called indirect effect. Indirect effects predict no direct effect from x to y while leaving the mediator out of the model. However, x has a direct effect on the mediator, and the mediator has a direct effect on y. Thus, x is said to have an indirect effect on y. This hypothesis can only be supported if the direct effect of x on y is insignificant, prior to testing for indirect effects. (Click) The next type of mediation is called Partial Mediation. Partial mediation predicts (click) significant direct (click) and indirect effects from x to y. Thus the unmediated relationship is significant, as well as the x to mediator and mediator to y relationships. In order to avoid concluding that a partially mediated effect is significant, when in fact, only the three direct effects are individually significant, a significance test for mediation must be performed. This will be explained more fully on the next slide. (click) The third type of mediation is called Full Mediation. Full mediation predicts that the direct effect of x on y will be significant, only if the mediator is absent. (click) When the mediator is present, this direct effect becomes (click) insignificant (click), while the indirect effect is significant. Lastly, if the x to mediator, and/or the mediator to y relationships are insignificant, no mediation is taking place. M
Full Mediation X Y

10. More complex mediation structures
Chain Model X M1 M2 M3 Y M1 X Y M2 M3
Parallel Model

11. Hypothesizing Mediation
All types of mediation need to be explicitly and with good theoretical reasons and logic hypothesized before testing them
Indirect Effect
You still need to assume and test that X has an indirect effect on Y, though there is no effect in path XY
“X has an indirect, positive effect on Y, through M.”
Partial or Full
“M partially/fully mediates the effect of X on Y.”
“The effect of X on Y is partially/fully mediated by M.”
“The effect of X on Y is partially/fully mediated by M1, M2, & M3.”

12. Statistical evidence of relationships.
Each type of mediation needs to be backed by appropriate statistical analysis
Sometimes the analysis can be based on OLS, but in most cases it needs to be backed by SEM based path analysis
There are four types of analyses to detect presence of mediation relationships
Causal steps approach (Baron-Kenny 1986) (tests for significance of different paths)
Difference in coefficients (evaluates the changes in betas/coefficients and their significance when new paths are added to the model)
Product of effect approach (tests for indirect effects a*b’- this always needs to be tested or evaluated using bootstrapping)
Sometimes evaluating differences in R squares

13. Statistical evidence of relationships
Convergent validity is critical for mediation tests as this forms the basis for reliability – especially poor reliability of mediator as “to the extent that a mediator is measured with less than perfect reliability, the MY relationship would likely be underestimated, whereas the XY would likely be overestimated when the antecedent and mediator are considered simultaneously” (see Baron & Kenny 1986)
Discriminant validity must be gauged in the context of the larger nomological network within which the relationships being considered are believed to reside. Discriminant validity does not imply that measures of different constructs are uncorrelated – the issue is whether measures of different variables are so highly correlated as to raise questions about whether they are assessing different constructs. It is incumbent on researchers to demonstrate that their measures of X, M, and Y evidence acceptable discriminant validity before any mediational tests are justified.

14. Statistical evidence of relationships

15. Statistical evidence of the relationships
In simple partial mediation βmx is the coefficient for X for predicting M, and βym.x and βyx.m are the coefficients predicting Y from both M and X, respectively. Here βyx.m is the direct effect of X, whereas the product βmx*βym quantifies the indirect effect of X on Y through M. If all variables are observed then βyx = βyx.m + βmx*βym or βmx*βym = βyx - βyx.m
Indirect effect is the amount by which two cases who differ by one unit of X are expected to differ on Y through X’s effects on M, which in turn affects Y
Direct effect part of the effect of X on Y that is independent of the pathway through M
Similar logic can be applied to more complex situations

16. What would be the paths here?

17. Statistical analysis
The testing of the existence of the mediational effect depends on the type of indirect effect
The lack of direct effect XY (βyx is either zero or not significant) is not a demonstration of the lack of mediated effect
Therefore three different situations prevail (in this order)
The presence of a indirect effect (βmx*βym is significant)
The presence of full mediation (βyx is significant but βyx.m is not)
The presence of partial mediation (βyx is significant and βyx.m is non zero and significant)

18. Testing for indirect effect

19. Testing for full mediation

20. Testing for partial mediation

21. Observations of statistical analysis
The key is to test for the presence of a significant indirect effect – just demonstrating the significant of paths βyx, βyx.m,βmx.y, and βmx is not enough
One reason is that Type I testing of statistical significance of paths is not based on inferences on indirect effects as products of effects and their quantities
Can be done either using Sobel test (see e.g. or bootstrapping
Sobel tests assumes normality of product terms and relatively large sample sizes (>200)
Lacks power with small sample sizes or if the distribution is not normal

22. Bootstrapping
Bootstrapping (available in most statistical packages, or there is additional code to accomplish it for most software packages)
Samples the distribution of the indirect effect by treating the obtained sample of size n as a representation of the population as a minitiature – and then resampling randomly the sample with replacement so that a sample size n is built by sampling cases from the original sample by allowing any case once drawn to be thrown back to be redrawn as the resample of size n is constructed
βmx and βym and their product is estimated for each sample recorded
The process is repeated for k times where k is large (>1000)
Hence we have k estimates of the indirect effect and the distribution functions as an empirical approximation of the sampling distribution of the indirect effect when taking the sample of size n from the original population
Specific upper and lower bound for confidence intervals are established to find ith lowest and jst largest value in the ordered rank of value estimates to reject the null hypothesis that the indirect effect is zero with e.g. 95 level of confidence

23. Observations of statistical analysis
In full and partial mediation bivariate XY (assessed via correlation rYX or coefficient βyx) must be nonzero in the population if the effects of X on Y are mediated by M
Hence establishing a significant bivariate is conditional on sample size
For example Assume that N=100 and sample correlations rXM=.30 and rMY =.30 and both would be significant at p<.05. However sample correlation rXY =.09 would not!
Hence tests for full mediation can be precluded if this is the true model in the population
This point become even more challenging when complex mediations XM1M2M3Y are present.
Hence many times full mediations are not detected due to underpowered designs; the same holds for interactions or suppression variables; in fact four step Baron Kenny has power of .52 with a sample size of 200 to detect medium effect!
This can be overcome by bootstrapping

24. Observations of statistical analysis
Testing for full mediation requires that βyx.m is zero. When βyx.m does not drop zero the evidence supports partial mediation. This requires researchers to make a priori hypotheses concerning full or partial mediation and transforms confirmatory tests to exploratory data mining
What counts as significant reduction in βyx vs. βyx.m is not clear (c.f. from .15 to .05 vs. .75 to .65)
Typically the baseline model for mediation is partial mediation while theoretical clarity and Ockam’s razor would speak for full mediation

25. Testing for Mediation in AMOS
Direct Effects First
Using the Bencare Constructs dataset, we can test for mediation in AMOS. You may find it helpful to reproduce this model and table on your own. You should already be familiar enough with AMOS to do this without a step by step guide. Just as a reminder, don’t forget to use standardized estimates, and to estimate means and intercepts (because some data is missing from the dataset). Also, as you already know, the regression weights table is found in the estimates section of the output window.
Regression Weights
Estimate
S.E.
C.R. P
loylong
<---
ctrust
.282
.048
5.812
***
atrust
.184
3.850

26. Testing for Mediation in AMOS
Add Mediator
Next include the mediating variable, and run the model again. Notice the change in path coefficients when comparing this mediated model to the direct effects model on the previous slide. You’ll notice that the direct effect from Trust in Company to Loyalty for Longterm has dropped out of significance, whereas, the direct effect from Trust in Agent to Loyalty for Longterm has simply been reduced, but is still significant. The reason for the reduction in the path coefficients is due to the mediating variable explaining some of the variance in the dependent variable that had previously been explained by the independent variables, but is more appropriately explained through the mediator.
Regression Weights
Estimate
S.E.
C.R. P
value
<---
atrust
.210
.048
4.400
***
ctrust
.602
12.452
loylong
.089
.056
1.592
.111
.123
.047
2.638
.008
.312
.052
5.935

27. Testing significance of partially mediated paths – Sobel Test
Use for partially mediated relationships.
Use the Sobel Test online calculator
Assumes normal distribution and sufficiently large sample
Regression Weights
Estimate
S.E.
C.R. P
value
<---
atrust
.210
.048
4.400
***
ctrust
.602
12.452
loylong
.089
.056
1.592
.111
.123
.047
2.638
.008
.312
.052
5.935
Now, just because there appears to be mediation, does not mean that mediation is significant. As explained earlier, the apparent mediation may actually just be three significant direct effects, rather than a mediated effect. In order to test the significance of a mediated relationship, you’ll need to find the regression weights table in the estimates section of the AMOS output window, and use the values indicated in this slide to fill in the appropriate boxes on the Sobel calculator. Then press calculate and use the two tailed p-value to determine significance.
(This test actually has it’s limitations, for example, it is sensitive with sample size less than Next semester we will offer a more detailed explanation, including how to conduct bootstrapping in AMOS.)

28. Testing significance of indirect effects– Bootstrapping
At least 1000
No Missing Values Allowed!

29. Testing significance of indirect effects– Bootstrapping
p- values

30. Given the direct effects were significant prior to adding the mediator
Direct Effects - Two Tailed Significance wu wf
aut
burnm
burnc
0.003
0.033
0.026
...
0.004
0.969
0.435
satc
0.845
0.260
0.016
0.007
satw
0.836
0.020
0.011
0.009
Indirect Effects - Two Tailed Significance
0.005
0.546
0.115
Total Effects - Two Tailed Significance
0.024
0.174
No Mediation
If Indirect is > 0.05
Full Mediation
Given the direct effects were significant prior to adding the mediator
If Indirect < 0.05 and Direct is > 0.05
Partial Mediation
If Direct & Indirect < 0.05, check Total.
If Total < 0.05 then partial mediation is significant.

31. Findings Partial Mediation Full Mediation WORDING
.23***
.37***
.20**
.17*
.08
Full Mediation
So, we found that (click) overall value partially mediates the effect of trust in agent on loyalty for longterm (click), and (click) fully mediates the effect of trust in company on loyalty to longterm. (click) (click) Please follow this style of wording both for your hypotheses and for reporting your findings.
WORDING
Overall value partially mediates the effect of trust in agent on loyalty for longterm (p < 0.000).
Overall value fully mediates the effect of trust in company on loyalty for longterm (p < 0.000).

32. Using AMOS for testing chain models and parallel models

33. Moderation concept
Based on the observation that independent-dependent variable relationship is affected by another independent variable
This situation is called moderator effect which occurs when a moderator variable, a second independent variable changes the form of the relationship between another independent variable and the DV
Can be expanded to a situation where the mediated relationship is moderated

34. Moderation: affecting the effect
Moderating variables must be chosen with strong theoretical support (Hair et al 2010)
The causality of the moderator cannot be tested directly
Becomes potentially confounded as moderator becomes correlated with either of the variables in the relationship
Testing easiest when moderator has no significant relationship with other constructs
This assumption is important in distinguishing moderator from mediators which (by definition) are related to both constructs of the mediated relationship

35. Moderation: Multi-group
Non-Metric moderators: categorical variables are hypothesized as moderators (gender, age, turbulence vs. non-turbulence, non customer vs. customer)
For non-metric variables a multi-group analysis is applied i.e. data is split for separate groups for analysis based on variable values and tested for statistical difference (both for measurement and structural model)

36. Multi-group example Weight Loss Exercise Weight Loss Exercise
Low Protein
High Protein

37. Moderator vs. Mediator Mediator: the means by which IV affects DV
Moderator: a variable that influences the magnitude of the effect an IV has on a DV K E M C A B K E M

38. Mediation vs. Moderation Example
Notice that the mediator and the moderator can be the same!
Can a mediator also be used as a moderator?
Yes - see Baron and Kenny 1986 for a complex example

39. Some Theory-based Criteria (i. e
Some Theory-based Criteria (i.e., arguments for mediation and moderation are based on theory first, rather than statistical correlations)
Mediator
Logical effect of IV
Logical cause of DV
Moderator
Not logically correlated to IV or DV (if categorical)
Holistic/multiplicative effect (interaction)
Varying effect for different categorical values (multi-group)

40. Driving home the point: Moderator or Mediator?
Either, Neither, One or the Other?
Driving home the point: Moderator or Mediator?
Caloric intake
Exercise partner
Positive reinforcement
Exercise IQ
Gender
Activity level
Age
Protein intake
Heredity
Attitude
Exercise M M
Weight Loss
Exercise
Weight Loss

41. Koufteros & Marcoulides 2006