Another characteristic of a good model is that it inheres mediation. Mediation can be thought of as the specification of mechanism. Mediation occurs when the effect of one variable on another operates — either partly or completely — through a third variable. We saw this earlier (Page 79) where the influence of DL on PGL was fully mediated through GBL. Without the presence of GBL, the β from DL to PGL was -0.151 with a t-value of 4.136. As seen in Figure 3.3, with the GBL present, the direct path from DL to PGL was not statistically significant.
1Parameter estimates and bootstrap t-values are automatically generated in SmartPLS, for all possible
Path β value t-value Asian → GBL 0.26 4.51 Asian → PBL 0.16 4.24 Asian → PGL 0.20 4.39 EAT → GBL -0.08 2.89 EAT → PBL -0.05 2.69 EAT → PGL -0.06 2.82 LIR → PBL -0.27 6.41 LIR → PGL -0.33 7.14 RQ → GBL -0.07 2.44 RQ → PBL -0.04 2.35 RQ → PGL -0.05 2.42
Table 3.12: Total Effect Parameter Estimates for BIGL16+ - Total effects for direct
paths are merely the direct effects found in Figure 3.9 and are thus excluded from this table. Those t-values that exceed 1.97 indicate p < .05; t-values that exceed 2.59 indicate p < .01; and t-values that exceed 3.33 indicate p < .001.
Previously I used a Sobel test for testing for mediation. For complex models, one must decide on whether to leave the three variables embedded in the full model, or isolate them out from the model. According to Iacobucci, Saldanha & Deng (2007), a traditional Sobel test is not permissible when the three constructs are embedded in a structural model and the independent variable and the moderator variable have three or more indicators—which is the case with the present model. I will leave all the variables embedded because I am interested in the model as a whole. The BIGL16+ model has instances of multiple mediation. For example, Asian → PGL can be mediated by GBL through two different paths, Asian → GBL and Asian → LIR → GBL. Isolating each triad of possible mediation has the effect of removing other explanatory variables, and so it is possible to inflate the effects — either direct or indirect.
One can speculate with some accuracy when there is full mediation by observing whether a direct effect between two variables becomes statistically non-significant after the addition of the indirect path through the mediating variable1.
1Preacher & Leonardelli (2011, n.p.) has this to say regarding mediation testing: “A variable may be
considered a mediator to the extent to which it carries the influence of a given independent variable (IV) to a given dependent variable (DV). Generally speaking, mediation can be said to occur when (1) the IV significantly affects the mediator, (2) the IV significantly affects the DV in the absence of the mediator,
In Table 3.13, I provide a list of all possible mediated relationships in the model, along with the β and t-value for the direct effect path with both the mediator present (see columns 3a and 3b) and with mediator absent (see columns 4a and 4b). Of 11 possible mediated relationships in the model, five have no direct effect so there can be no mediation (although there can still be a total effect, and in fact there is for all paths). Two of the possible mediated relationships are likely to be partially mediated, and four are likely to be fully mediated. This can be inferred by the reduction in the β and t-values for the direct effect, when the mediator(s) is present.
There are several observations that can be made from the information in Table 3.13. Looking first at the top two lines of Table 3.13, when there are no mediators are present, Asian has a direct on PGL and PBL. (As modelled in Figure 3.9, there are two possible mediating paths from Asian to PGL and from Asian to PBL, both of which includes GBL, and only one of which includes LIR.) However, when the mediators are taken into account, the direct effect of Asian on PGL and PBL is diminished to a negligible level. So meditation is highly probable based on this informal, “reduction in β” test.
What happens when LIR is removed from this grouping, leaving only GBL as a me- diator? The t-values for the direct path remain unchanged from those for the two paths in column 3b. The reverse is not true when GBL is removed. The β and t-values for direct paths from Asian to PGL and PBL, when LIR is present, do not differ significantly from those values in column 4a and 4b. So, GBL is the mediator in operation for the top two paths in Table 3.13.
What does this mediation mean though? The effect of cultural background (as measured by Asian language spoken at home) on the belief in personal good and bad luck is explainable by the effect of cultural background on a general belief in luck. To put this another way, being from an Asian cultural background has an influence on the beliefs in personal good and bad luck, but only insofar as it has an influence on a general belief in luck. This provides strong support for the model as proposed, and further indication that the dimensionality of luck beliefs has a structural arrangement. Recall that the Fornell-Larcker Table (Table 3.11) indicated the GBL items had low discriminant validity in relation to the PGL and PBL construct, despite a factor analysis
(3) the mediator has a significant unique effect on the DV, and (4) the effect of the IV on the DV shrinks upon the addition of the mediator to the model. These criteria can be used to informally judge whether or not mediation is occurring, but MacKinnon & Dwyer (1993) and MacKinnon & Dwyer (1995) have popularized statistically based methods by which mediation may be formally assessed.”
1 2 3a 3b 4a 4b 5 Direct effect Direct effect Probable
Path Mediator(s) w/ mediator(s) w/o mediator(s) Mediation
β t-value β t-value
Asian → PGL GBL, LIR+GBL 0.03 0.61 0.17 3.24 Full
Asian → PBL GBL, LIR+GBL 0.02 0.35 0.14 2.28 Full
Asian → GBL LIR 0.20 4.79 0.26 6.38 Partial
EAT → GBL LIR -0.04 1.10 -0.13 3.10 Full
LIR → PGL GBL -0.15 2.89 -0.45 8.75 Partial
LIR → PBL GBL -0.07 1.14 -0.32 4.80 Full
EAT → PGL LIR+GBL 0.03 0.63 0.00 0.04 NDE
EAT → PBL LIR+GBL 0.00 0.02 -0.02 0.29 NDE
RQ → GBL LIR 0.04 0.92 -0.03 0.62 NDE
RQ → PGL LIR+GBL 0.02 0.45 0.04 0.74 NDE
RQ → PBL LIR+GBL 0.01 0.12 0.03 0.39 NDE
Table 3.13: Possible Mediated Relationships in the BIGL16+ - Column 1 lists the begin-
ning and ending construct in possible paths that contain at least one potential mediator. Column 2 lists the potential mediators. Note that some paths have more than one me- diator. Refer back to the model in Figure 3.9 to see the configuration. Columns 3a and 3b list the β and t-values (respectively) for the path with all possible mediators present. Columns 4a and 4b list the β and t-values (respectively) for only the direct path with all possible mediators have been removed. Column 5 proposes full or partial mediating based on whether the direct effect with mediators present remains statistically significant. See the footnote on page 110 for explanation, and note in particular the fourth condition listed there. Those t-values that exceed 1.97 indicate p < .05; t-values that exceed 2.59 indicate p < .01. Paths in the lower block are disregarded, as there is no direct effect.
assigning the items to a separate factor. Perhaps mediation of the Asian → PGL and Asian → PBL paths reflects discrimination problems inherent to the GBL items? Note however, that EAT has no direct effect on either PGL or PBL (see top two rows on the bottom section of Table 3.13), whereas EAT does have a direct effect on GBL (see row four of the top section). So, concerns about the discriminant validity of the GBL items can be further quelled.
Of the six paths listed in the top section of Table 3.13, only one can be subjected to a traditional Sobel test without any modification of the original model. A Sobel test requires that all three constructs have paths connected them, as was the case in the model in Figure 3.2. Note that the path Asian → GBL has both the direct path and the path via the mediator, LIR. The other three paths in Table 3.13 with only a single mediator can be subjected to a traditional Sobel test for mediation without any modification to the model other than inserting a direct path from the IV to the DV. For columns 5 and 6 of Table 3.14, I have removed the path from Asian to LIR and inserted direct paths from Asian to PGL and PBL. (Recall that LIR was not an active mediator based on the “β reduction test”.)
Making the necessary alterations to the model, I conducted Sobel tests for each of the six possible paths: the two for which LIR is a potential mediator (i.e., Asian → GBL; EAT → GBL); the two for which GBL is a mediator for LIR and the personal luck beliefs (i.e., LIR → PGL; LIR → PBL); and the two for which GBL is a mediator for Asian and the personal luck beliefs (i.e., Asian → PGL; Asian → PBL). The results are presented in Table 3.14. The result for each path in Table 3.14 concurs with the result for that same path in Table 3.13. Note that any alterations to the model were made only for the test of the path in question, and then the model was returned to the original form. I have also calculated the variance accounted for (VAF). The VAF is the ratio of the mediated effect to the total effect, and is calculated as (a∗b)+ca∗b 0, where a and b are
equivalent to the Sobel test notation, and c’ is the direct effect when the mediated path is present. The VAF explains the effect of an independent variable through a given mediator, as a percentage of the total effect on a dependent variable in question.
The VAF for the path presented in the leftmost column indicates only a minor por- tion (24%) of the effect of Asian on GBL is mediated through LIR (Column 1). This is an interesting finding, suggesting that most of the impact of Asian cultural background
1 2 3 4 5 6
IV: Asian EAT LIR LIR Asian Asian
Mediator: LIR LIR GBL GBL GBL GBL
DV: GBL GBL PGL PBL PGL PBL Sobel Test Inputs and Result a -0.139 0.187 -0.438 -0.438 0.202 0.202 b -0.438 -0.431 0.681 0.572 0.741 0.600 sa 0.064 0.059 0.057 0.057 0.055 0.055 sb 0.058 0.058 0.042 0.051 0.030 0.046 p < 0.037 0.004 0.001 0.001 0.001 0.001 VAF a -0.139 0.187 -0.438 -0.438 0.202 0.202 b -0.438 -0.431 0.681 0.572 0.741 0.600 c0 0.202 -0.041 -0.146 0.071 0.032 0.023 VAF 23% 66% 67% 78% 82% 84%
Table 3.14: Sobel Tests for Mediation in the BIGL16+ - In the first block are the Sobel
test inputs and result for each path. The second block provides the inputs for a Variance Accounted For (VAF) calculation, which estimates the ratio of the mediated effect to the total effect, providing an indication of the strength of mediation. Columns 1 - 4 are mod- elled in Figure 3.9 with only a single mediator. For columns 5 - 6 the paths contain two possible mediators and the path from Asian to LIR has been removed (see discussion in Section 3.11) because a Sobel test cannot be conducted when there is multiple mediation paths.
is not due to LIR, but due to something else which I have not measured. The VAF’s for the other five paths are much higher (ranging from 66% to 84%).
Column 2 examines the path from EAT to GBL, with LIR as a mediator, concluding that mediation is present, and that about 66% of the effect from EAT to GBL is mediated via LIR. In other words, the effect of intelligence (as measured by verbal ability) on a general belief in luck is mostly mediated through the belief that relates luck and chance. Above, I indicated that the effect of Asian on PGL and PBL was mediated only through GBL and not LIR. In light of the mediating role LIR evidenced in columns 1 and 2 of Table 3.14, that previous result should not be taken as evidence that LIR is not an important construct. On the contrary, for EAT and RQ, which are intelligence and knowledge based, LIR is an important construct. This is further support that the validating constructs have demonstrated a nomological validity of the luck belief di- mensions.
Looking now to columns 3 and 4, GBL plays an important mediating role in the rela- tionship between LIR and both PGL and PBL. This was already evidenced in a proto-LIR construct seen in Part One of this chapter, disbelief in luck (DL). That earlier mediation is replicated here, although the item content has changed considerably. A disbelief in luck (DL) and a belief that luck is nothing more than random chance (LIR) are very conceptually similar. This is part replication, and part extension of the finding from Part One. Not only is the mediating role of GBL preserved for an alternate luck scepti- cism construct (LIR), the meditating role is extended to the companion personal belief to PGL, the belief in personal bad luck.
In columns 5 and 6, the mediating role of GBL for Asian and both PGL and PBL is clearly demonstrated to be in agreement with the results of the earlier “reduction in β” test. The VAF’s for these two columns are the highest of the six, with one four- fifths of the effect of Asian operating on PGL and PBL via GBL. So, GBL is an important mediator is the model not only for the LIR belief construct, but also the trait-based Asian construct. This is an important insight into the luck beliefs dimensionality and argues strongly for any model of luck beliefs to situate GBL as antecedent to PGL and PBL.
I also note there is a symmetry between PGL and PBL. Even though the PGL and PBL are correlated at 0.42, they appear to have the same antecedents in the model, with very similar parameter estimates. For instance, the VAF’s in columns 3 and 4 of Table
3.14 differ by very little in relative terms, and this is the case for the VAF’s in columns 5 and 6. When considering only direct effects, the direct effect of Asian on PGL and PBL (without mediators present) is 0.17 and 0.14 respectively and the direct effect of LIR on PGL and PBL (without mediators present) is -0.45 and -0.32 respectively. It remains to be seen whether PGL and PBL will have differential prediction in models to come. That is, will they predict risky choice in the same or opposite directions?