The test is based on the work of Michael E. Sobel,[1][2] and is an application of the delta method.
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.
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 α term represents the magnitude of the relationship between the independent variable and the mediator.
The Sobel test uses the magnitude of the indirect effect compared to its estimated standard error of measurement to derive a t statistic[1] 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.
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.
[6] The distribution of the product term αβ is only normal at large sample sizes[5][6] 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.
[9] This occurs when the sample size is different in the models used to estimate the mediated effects.