526:). This occurs when the sample size is different in the models used to estimate the mediated effects. Suppose that the independent variable and the mediator are available from 200 cases, while the dependent variable is only available from 150 cases. This means that the α parameter is based on a regression model with 200 cases and the
335:. This total amount of variance in the dependent variable that is accounted for by the independent variable can then be broken down into areas c and d. Area c is the variance that the independent variable and the dependent variable have in common with the mediator, and this is the indirect effect. Area
502:
are assumed to be normally distributed, and the distribution of the product of two normally distributed variables is skewed, unless the means are much larger than the standard deviations. If the sample is large enough this will not be a problem, however determining when a sample is sufficiently large
322:
Another way of thinking about the product of coefficients is to examine the figure below. Each circle represents the variance of each of the variables. Where the circles overlap represents variance the circles have in common and thus the effect of one variable on the second variable. For example
377:
In order to determine the statistical significance of the indirect effect, a statistic based on the indirect effect must be compared to its null sampling distribution. The Sobel test uses the magnitude of the indirect effect compared to its estimated standard error of measurement to derive a t
584:
One strategy to overcome the non-normality of the product of coefficients distribution is to compare the Sobel test statistic to the distribution of the product instead of to the normal distribution. This approach bases the inference on a mathematical derivation of the product of two normally
39:. In mediation, the relationship between the independent variable and the dependent variable is hypothesized to be an indirect effect that exists due to the influence of a third variable (the mediator). As a result when the mediator is included in a
47:
that provides a method to determine whether the reduction in the effect of the independent variable, after including the mediator in the model, is a significant reduction and therefore whether the mediation effect is statistically significant.
493:
The distribution of the product term αβ is only normal at large sample sizes which means that at smaller sample sizes the p-value that is derived from the formula will not be an accurate estimate of the true p-value. This occurs because both
306:) represents the product of these two terms. In essence this is the amount of variance in the dependent variable that is accounted for by the independent variable through the mechanism of the mediator. This is the indirect effect, and the (
597:. Bootstrapping is a non-parametric resampling procedure that can build an empirical approximation of the sampling distribution of αβ by repeatedly sampling the dataset. Bootstrapping does not rely on the assumption of normality.
479:
to determine its significance. Alternative methods of calculating the Sobel test have been proposed that use either the z or t distributions to determine significance, and each estimates the standard error differently.
279:). This represents the change in the magnitude of the effect that the independent variable has on the dependent variable after controlling for the mediator. From examination of these equations it can be determined that (
367:
58:
259:
represent the relationship between the independent variable and the mediator, and the mediator and the dependent variable after controlling for the independent variable, respectively.
43:
model with the independent variable, the effect of the independent variable is reduced and the effect of the mediator remains significant. The Sobel test is basically a specialized
704:
Baron, Reuben M.; Kenny, David A. (1986). "The
Moderator-Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic, and Statistical Considerations".
302:
term represents the magnitude of the relationship between the mediator and dependent variable after controlling for the effect of the independent variable. Therefore (
331:
represent the effect of the independent variable on the dependent variable, if we ignore the mediator, and corresponds to
990:
Bollen, Kenneth A.; Stine, Robert (1990). "Direct and
Indirect Effects: Classical and Bootstrap Estimates of Variability".
669:
Sobel, Michael E. (1986). "Some New
Results on Indirect Effects and Their Standard Errors in Covariance Structure".
621:
Sobel, Michael E. (1982). "Asymptotic
Confidence Intervals for Indirect Effects in Structural Equation Models".
787:"Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models"
749:
Judd, Charles M.; Kenny, David A. (1981). "Process
Analysis: Estimating Mediation in Treatment Evaluations".
541:
are based on regression models with 150 cases. Different sample sizes and different participants means that (
1036:
298:
term represents the magnitude of the relationship between the independent variable and the mediator. The
831:
MacKinnon, David P.; Lockwood, Chondra M.; Hoffman, Jeanne M.; West, Stephen G.; Sheets, Virgil (2002).
1031:
594:
238:
denotes the relationship between the independent variable and the dependent variable in model 1, while
245:
denotes that same relationship in model 3 after controlling for the effect of the mediator. The terms
585:
distributed variables which acknowledges the skew of the distribution instead of imposing normality.
718:
635:
928:"Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods"
713:
630:
571:) is when exactly the same participants are used in each of the models testing the regression.
362:
is sufficiently large then Sobel's test is significant and significant mediation is occurring.
28:
8:
476:
40:
1007:
952:
927:
865:
832:
686:
648:
957:
870:
852:
808:
731:
976:
999:
947:
939:
901:
860:
844:
798:
758:
723:
678:
640:
32:
833:"A comparison of methods to test mediation and other intervening variable effects"
63:
When evaluating a mediation effect three different regression models are examined:
727:
943:
890:"The Probability Function of the Product of Two Normally Distributed Variables"
762:
906:
889:
1025:
856:
848:
925:
961:
874:
812:
36:
735:
366:
57:
803:
786:
20:
1011:
690:
652:
593:
Another approach that is becoming more popular in the literature is
1003:
926:
MacKinnon, David P.; Lockwood, Chondra M.; Williams, Jason (2004).
682:
644:
530:
parameter is based on a regression model with only 150 cases. Both
44:
830:
506:
579:
268:
From these models, the mediation effect is calculated as (
784:
703:
989:
748:
488:
785:Preacher, Kristopher J.; Hayes, Andrew F (2008).
574:
1023:
921:
919:
917:
826:
824:
822:
780:
778:
776:
774:
772:
483:
414:Where SE is the pooled standard error term and
213:represent the intercepts for each model, while
914:
819:
475:This t statistic can then be compared to the
372:
339:corresponds to the product of coefficients (
27:is a method of testing the significance of a
769:
706:Journal of Personality and Social Psychology
664:
662:
616:
614:
612:
610:
354:). The Sobel test is testing how large area
234:represent the error term for each equation.
263:
951:
905:
864:
802:
717:
659:
634:
607:
507:Problems with the product of coefficients
31:effect. The test is based on the work of
580:Product of the coefficients distribution
511:In some situations it is possible that (
317:
1024:
887:
16:Statistical test of a mediation effect
974:
668:
620:
51:
13:
14:
1048:
894:Annals of Mathematical Statistics
192:is the mediator. The parameters
932:Multivariate Behavioral Research
588:
489:Distribution of the product term
365:
185:is the independent variable and
56:
983:
35:, and is an application of the
968:
881:
742:
697:
575:Alternatives to the Sobel test
1:
600:
484:Problems with the Sobel test
7:
728:10.1037/0022-3514.51.6.1173
310:) term has been termed the
178:is the dependent variable,
10:
1053:
977:"An Answer to Julie Maloy"
944:10.1207/s15327906mbr3901_4
763:10.1177/0193841X8100500502
373:Calculating the Sobel test
791:Behavior Research Methods
992:Sociological Methodology
849:10.1037/1082-989X.7.1.83
671:Sociological Methodology
623:Sociological Methodology
503:is somewhat subjective.
468:is the variance of
907:10.1214/aoms/1177730442
888:Aroian, Leo A. (1947).
312:product of coefficients
264:Product of coefficients
837:Psychological Methods
318:Venn diagram approach
804:10.3758/BRM.40.3.879
1037:Regression analysis
477:normal distribution
455:is the variance of
347: −
41:regression analysis
975:MacKinnon, David.
556:). The only time (
1032:Statistical tests
751:Evaluation Review
52:Theoretical basis
1044:
1016:
1015:
987:
981:
980:
972:
966:
965:
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878:
868:
828:
817:
816:
806:
782:
767:
766:
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740:
739:
721:
712:(6): 1173–1182.
701:
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694:
666:
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656:
638:
618:
566:
551:
540:
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445:
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411:
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171:In these models
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60:
33:Michael E. Sobel
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1021:
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608:
603:
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582:
577:
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491:
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467:
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982:
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913:
900:(2): 265–271.
880:
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768:
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741:
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683:10.2307/270922
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581:
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380:
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598:
596:
595:bootstrapping
589:Bootstrapping
586:
572:
570:
563:
559:
555:
548:
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537:
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449:
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429:
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359:
358:is. If area
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348:
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307:
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998:: 115–140.
677:: 159–186.
629:: 290–312.
1026:Categories
601:References
378:statistic
343:) and to (
25:Sobel test
21:statistics
857:1939-1463
714:CiteSeerX
631:CiteSeerX
323:sections
128:Model 3:
98:Model 2:
68:Model 1:
29:mediation
962:20157642
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633:
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206:, and
45:t test
23:, the
1008:JSTOR
687:JSTOR
649:JSTOR
567:) = (
565:'
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550:'
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416:SE =
352:'
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283:) = (
277:'
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958:PMID
871:PMID
853:ISSN
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732:PMID
534:and
498:and
459:and
446:and
404:(αβ)
401:t =
382:t =
252:and
1000:doi
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679:doi
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19:In
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569:αβ
560:–
554:αβ
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389:SE
341:αβ
314:.
308:αβ
304:αβ
287:–
281:αβ
272:–
254:βX
247:αX
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199:,
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513:τ
500:β
496:α
470:α
465:α
461:σ
457:β
452:β
448:σ
440:α
436:σ
433:β
428:β
424:σ
421:α
360:c
356:c
349:τ
345:τ
337:c
333:τ
329:d
325:c
300:β
296:α
289:τ
285:τ
274:τ
270:τ
257:M
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240:τ
236:τ
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