Purpose and scope

This workflow estimates an observed-variable path model with two first-stage mediators, one second-stage mediator, two correlated outcomes, direct effects, specific indirect effects, and total effects. It covers identification, missing-data handling, bootstrap uncertainty, global and local fit, comparison with a defensible alternative, and dynamic reporting.

This is path analysis, not a latent measurement model. Every construct is represented by one observed score. Use the separate psychometrics workflow when reliability, factor structure, or measurement invariance must be modeled. Temporal ordering and causal language still depend on the research design.

User settings

Simulate and import

Data audit

variable n missing_n missing_percent
X 3388 0 0.0
age 3388 0 0.0
M1 3388 24 4.8
M2 3388 26 5.2
M3 3388 19 3.8
Y1 3388 25 5.0
Y2 3388 18 3.6
variable n mean sd min max
X 500 -0.05 1.01 -3.16 2.66
age 500 34.25 7.79 20.00 55.35
M1 476 0.01 1.00 -2.80 3.19
M2 474 -0.04 1.01 -2.29 3.43
M3 481 -0.09 1.16 -3.21 3.61
Y1 475 -0.06 1.16 -3.28 2.80
Y2 482 -0.08 1.03 -3.02 3.97

Full-information maximum likelihood uses each case’s observed outcomes under a missing-at-random assumption conditional on variables in the model. That assumption cannot be proven from observed data. Include defensible auxiliary variables or conduct sensitivity analyses when missingness may depend on omitted information.

Correlations and distribution review

X age M1 M2 M3 Y1 Y2
X 1.00 0.04 0.52 0.48 0.63 0.49 0.39
age 0.04 1.00 0.16 -0.12 0.10 0.20 -0.03
M1 0.52 0.16 1.00 0.31 0.63 0.41 0.32
M2 0.48 -0.12 0.31 1.00 0.56 0.34 0.32
M3 0.63 0.10 0.63 0.56 1.00 0.65 0.54
Y1 0.49 0.20 0.41 0.34 0.65 1.00 0.52
Y2 0.39 -0.03 0.32 0.32 0.54 0.52 1.00

Specify the hypothesized model

The model leaves residual covariances between the parallel first-stage mediators and between the two outcomes. These covariances are specified because the variables share stages and likely omitted causes; they are not added solely to improve fit.

Fit, identification, and convergence

n_used free_parameters degrees_of_freedom converged
500 31 4 TRUE

Global and local fit

measure value
chisq 1.502847493
df 4.000000000
pvalue 0.826136950
cfi 1.000000000
tli 1.010338065
rmsea 0.000000000
rmsea.ci.lower 0.000000000
rmsea.ci.upper 0.040095434
srmr 0.006079334

Global indices are descriptive evidence about covariance reproduction, not pass/fail proof. Cutoffs depend on model complexity, sample size, estimator, and data quality. Review the residual matrix and theory together.

variable_1 variable_2 residual
M1 Y1 -0.022
M2 Y1 -0.018
M1 Y2 -0.018
M2 Y2 -0.012
M1 age -0.001
M1 M2 0.001
M3 Y2 0.000
M2 M3 0.000
M2 age 0.000
M1 M3 0.000

Paths and indirect effects

outcome predictor label estimate std_error p_value conf_low conf_high standardized
M1 X a1 0.502 0.043 0.000 0.417 0.588 0.507
M1 age am1 0.019 0.005 0.000 0.009 0.028 0.146
M2 X a2 0.485 0.041 0.000 0.408 0.566 0.486
M2 age am2 -0.017 0.005 0.000 -0.027 -0.008 -0.134
M3 M1 d1 0.452 0.042 0.000 0.369 0.531 0.389
M3 M2 d2 0.368 0.038 0.000 0.293 0.444 0.319
M3 X a3 0.321 0.046 0.000 0.235 0.411 0.278
M3 age am3 0.008 0.005 0.092 -0.001 0.018 0.055
Y1 M3 b1 0.539 0.046 0.000 0.447 0.634 0.541
Y1 X c1 0.165 0.051 0.001 0.064 0.268 0.144
Y1 age ay1 0.022 0.005 0.000 0.012 0.031 0.147
Y2 M3 b2 0.428 0.046 0.000 0.334 0.516 0.483
Y2 X c2 0.095 0.053 0.070 -0.006 0.198 0.093
Y2 age ay2 -0.012 0.005 0.018 -0.022 -0.003 -0.089
effect estimate std_error p_value conf_low conf_high
ind_Y1_M1_M3 0.122 0.018 0 0.090 0.162
ind_Y1_M2_M3 0.096 0.015 0 0.070 0.129
ind_Y1_M3 0.173 0.028 0 0.121 0.232
total_ind_Y1 0.391 0.038 0 0.320 0.473
total_Y1 0.555 0.045 0 0.465 0.650
ind_Y2_M1_M3 0.097 0.016 0 0.069 0.132
ind_Y2_M2_M3 0.076 0.012 0 0.054 0.101
ind_Y2_M3 0.137 0.025 0 0.093 0.187
total_ind_Y2 0.310 0.037 0 0.240 0.387
total_Y2 0.406 0.042 0 0.327 0.489

Bootstrap intervals quantify sampling uncertainty under the fitted model and observed sample. An indirect-effect interval that excludes zero does not establish causal mediation without defensible temporal ordering, confounding control, measurement, and consistency assumptions.

Theory-led alternative model

The alternative below adds direct paths from both first-stage mediators to both outcomes. It represents a substantive claim that their associations with the outcomes are not fully transmitted through M3. Compare it only if this claim was justified before inspecting modification indices.

model df AIC BIC CFI RMSEA SRMR
Hypothesized 4 10835.41 10966.06 1 0 0.006
Theory-led alternative 0 10841.90 10989.41 1 0 0.000

Avoid iterative modification-index searching on the same sample. If respecification is necessary, label it exploratory and evaluate it in an independent sample when possible.

Path diagram

Dynamic results template

An observed-variable path model was estimated from 500 synthetic cases using full-information maximum likelihood and 1000 bootstrap replications. The model was overidentified with 4 degrees of freedom. Global fit indices were CFI = 1, TLI = 1.01, RMSEA = 0, 90% CI [0, 0.04], and SRMR = 0.006. These values should be considered alongside the largest standardized residuals and the prespecified theory.

The total indirect effect from X to Y1 was 0.391, bootstrap 95% CI [0.32, 0.473], which excluded zero. The total indirect effect from X to Y2 was 0.31, bootstrap 95% CI [0.24, 0.387], which excluded zero. These associations do not by themselves establish causal mediation.

Reporting checklist

  • Call the analysis observed-variable path analysis unless latent variables are explicitly modeled.
  • State the temporal and theoretical basis for every directed path and residual covariance.
  • Report identification, estimator, missing-data handling, bootstrap method, and replications.
  • Present global fit, standardized residuals, unstandardized paths, standardized paths, and interval estimates.
  • Distinguish prespecified alternatives from data-driven respecification.
  • Discuss omitted confounding, measurement error, nonnormality, influential cases, and sample dependence.
  • Validate revised models in an independent sample when feasible.

When this model is not enough

Use a latent-variable SEM when constructs require multiple indicators and measurement error must be modeled. Use longitudinal mediation when temporal ordering requires repeated measures. Use multilevel SEM when observations are nested, and use categorical-data estimators when endogenous variables are ordinal or binary.

Exports

References

  • Kline, R. B. (2023). Principles and Practice of Structural Equation Modeling (5th ed.). Guilford Press.
  • MacKinnon, D. P. (2008). Introduction to Statistical Mediation Analysis. Routledge.
  • Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36.
Paid version: The download includes all editable R code, the synthetic-data generator, documented outputs, and complete software-version details.
Dissertation Stats Helper · Reproducible statistical guidance for doctoral research · Specify paths from theory and design, then evaluate fit and uncertainty.