Formula
z = ab ÷ √(b²·SEa² + a²·SEb²) (Aroian adds + SEa²·SEb²; Goodman subtracts it)Example: a = 0.50 (SE 0.10), b = 0.40 (SE 0.12). ab = 0.20 and SE = √(0.16 × 0.01 + 0.25 × 0.0144) = 0.072, so z = 2.77, p = .006: a significant indirect effect.
Where a and b come from
- Regression 1: M = i₁ + a·X → take a and its SE
- Regression 2: Y = i₂ + c′·X + b·M → take b and its SE
- Optional total-effect regression: Y = i₃ + c·X → c, for the proportion mediated
- Use unstandardised (B) coefficients, not standardised betas
Full vs partial mediation
If the indirect effect is significant and the direct effect c′ is not, the result is often called full mediation; if both are significant, partial mediation. Current guidance (Zhao, Lynch & Chen, 2010) focuses on whether the indirect effect is significant rather than on these labels.
Sobel test vs bootstrapping
The Sobel test is easy to compute and widely cited, but it has low power and assumes a normal sampling distribution for ab. Preacher and Hayes (2008) recommend bootstrap confidence intervals; if a supervisor or reviewer asks for one, use PROCESS (SPSS) or lavaan (R) and report Sobel alongside it.
How to report it (APA style)
“The indirect effect of study time on exam score through self-efficacy was significant, ab = 0.20, SE = 0.07, z = 2.77, p = .006, accounting for 33% of the total effect.” Report a, b, their SEs, and the method used.
References
- Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13, 290–312.
- Baron, R. M., & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research. Journal of Personality and Social Psychology, 51(6), 1173–1182.
- Preacher, K. J., & Hayes, A. F. (2008). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40(3), 879–891.
- Zhao, X., Lynch, J. G., & Chen, Q. (2010). Reconsidering Baron and Kenny: Myths and truths about mediation analysis. Journal of Consumer Research, 37(2), 197–206.