Sobel test

In statistics, the Sobel test is a method of testing the significance of a mediation effect. The test is based on the work of Michael E. Sobel, a statistics professor at Columbia University in New York, NY, and is an application of the delta method. 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 regression analysis 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 t test 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.Model 1: YO = γ1 + τXI + ε1Model 2: XM = γ2 + αXI + ε2Model 3: YO = γ3 + τ’XI + βXM + ε3In these models YO is the dependent variable, XI is the independent variable and XM is the mediator. γ1, γ2, and γ3 represent the intercepts for each model, while ε1, ε2, and ε3 represent the error term for each equation. τ denotes the relationship between the independent variable and the dependent variable in model 1, while τ’ denotes that same relationship in model 3 after controlling for the effect of the mediator. The terms αXI and βXM represent the relationship between the independent variable and the mediator, and the mediator and the dependent variable after controlling for the independent variable, respectively. In statistics, the Sobel test is a method of testing the significance of a mediation effect. The test is based on the work of Michael E. Sobel, a statistics professor at Columbia University in New York, NY, and is an application of the delta method. 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 regression analysis 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 t test 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. When evaluating a mediation effect three different regression models are examined: From these models, the mediation effect is calculated as (τ – τ’). 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 (αβ) = (τ – τ’). The α term represents the magnitude of the relationship between the independent variable and the mediatior. The β term represents the magnitude of the relationship between the mediator and dependent variable after controlling for the effect of the independent variable. Therefore (αβ) 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 (αβ) term has been termed the product of coefficients. 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 sections c + d represent the effect of the independent variable on the dependent variable, if we ignore the mediator, and corresponds to τ. 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 c corresponds to the product of coefficients (αβ) and to (τ − τ’). The Sobel test is testing how large area c is. If area c is sufficiently large then Sobel’s test is significant and significant mediation is occurring. 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 statistic Where SE is the pooled standard error term and SE = √α2 σ2β + β2σ2α and σ2β is the variance of β and σ2α is the variance of α. This t statistic can then be compared to the normal distribution 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.