Tests whether X influences Y through an intermediate variable M — e.g., transformational leadership → psychological empowerment → job performance.
Predict & ExplainMultivariatealso known as: Indirect effect analysis, Process analysis
✓ When to use
Theory proposes a mechanism: X affects M, and M in turn affects Y.
'Why/how does it work?' questions that turn a correlation into a process story.
Ideally with temporal separation: X measured before M, M before Y.
✗ When NOT to use
Cross-sectional data where M and Y are simultaneous self-reports — the indirect 'effect' is then a correlational decomposition, not evidence of mechanism; label conclusions accordingly.
The third variable conditions rather than transmits the effect — that is moderation.
M measured with poor reliability — attenuates the a and b paths multiplicatively.
Reverse causality between M and Y is plausible and undesigned-for.
Data requirements
Dependent / outcome variable
Continuous outcome Y (binary Y possible with logistic extensions).
Independent / grouping variable
Predictor X, mediator M (continuous), optional covariates.
Design
Preferably longitudinal or experimental (manipulate X); cross-sectional possible but interpretively weak.
Sample size guidance
Bootstrap tests need N ≥ 100–200 for typical effects; small indirect effects need far more.
Assumptions
Regression assumptions for both equations (M on X; Y on X and M).
No unmeasured confounding of X–M, X–Y, and crucially M–Y — the assumption most often violated and least often acknowledged.
Correct temporal/causal ordering.
Reliable measurement of M (or use latent-variable SEM).
Hypotheses
H₀ — The indirect effect is zero (a×b = 0).
H₁ — The indirect effect differs from zero (X affects Y through M).
The concept
Two regressions estimate the machinery: path a (X→M) and path b (M→Y controlling X); the indirect effect is a×b, and the direct effect c′ is X→Y controlling M. Total effect c = c′ + a×b. Modern practice tests a×b directly with a bootstrap confidence interval (the product's distribution is skewed, so normal-theory Sobel tests are outdated).
The old Baron–Kenny requirement that the total effect must be significant first is obsolete — suppression can hide real indirect effects. Language matters: with cross-sectional data, report 'indirect association consistent with mediation', reserving causal claims for designs that earn them. 'Full vs partial mediation' labels are increasingly discouraged; report the effect sizes instead.
Worked example
Does psychological empowerment (M) transmit the effect of transformational leadership (X) on performance (Y)? N = 310, three-wave survey.
Result: a = 0.51 (p < .001), b = 0.34 (p < .001); indirect effect a×b = 0.17, 95% bootstrap CI [0.10, 0.26] (5,000 resamples); direct c′ = 0.12 (p = .04). The leadership–performance link operates substantially through empowerment.
How to run it
library(mediation)
med <- lm(empower ~ leadership, data = df)
out <- lm(performance ~ leadership + empower, data = df)
res <- mediate(med, out, treat = "leadership", mediator = "empower",
boot = TRUE, sims = 5000)
summary(res) # ACME = indirect, ADE = direct, Total
# alternative: lavaan SEM with ab := a*b and bootstrapped CI
import pingouin as pg
res = pg.mediation_analysis(data=df, x="leadership", m="empower",
y="performance", n_boot=5000, seed=42)
print(res) # paths a, b, direct, indirect with bootstrap CI
Install Hayes's PROCESS macro (processmacro.org); Analyze → Regression → PROCESS.
Model 4; X = leadership, M = empower, Y = performance; bootstrap samples = 5000.
Read paths a, b, c, c′ and the 'Indirect effect(s) of X on Y' with its BootLLCI/BootULCI.
Report the indirect effect with the bootstrap CI; significant when the CI excludes 0.
Not practical: bootstrapping the a×b product requires thousands of resamples.
Excel can compute the point estimates (two regressions, multiply a×b) but no defensible CI.
Use PROCESS in SPSS, or R/Python.
Interpreting the output
The indirect effect a×b with its bootstrap CI is the focal result; CI excluding 0 = significant.
Report a, b, c′, c so readers see the full decomposition.
Effect size: completely standardized indirect effect, or proportion mediated (unstable in small samples).
Match your causal language to your design — cross-sectional mediation describes associations.
APA-style reporting
The indirect effect of transformational leadership on performance through psychological empowerment was significant, ab = 0.17, 95% bootstrap CI [0.10, 0.26] (5,000 resamples). The direct effect remained significant, c′ = 0.12, p = .041, indicating that empowerment accounted for a substantial share of the total effect (c = 0.29).
Common mistakes
Causal mediation claims from single-wave self-report data.
Using the Sobel test instead of bootstrap CIs.
Requiring a significant total effect before testing the indirect path.