Mediation & Causal Path Analysis Studio

Single Mediator Model

$$M = a(W) \cdot X$$ $$Y = c'(W) \cdot X + b(W) \cdot M$$

Conditional Path Coefficients (when moderated)

$$a(W) = a_1 + a_3 W$$ $$b(W) = b_1 + b_3 W$$ $$c'(W) = c'_1 + c'_3 W$$

Parallel Mediator Model

$$M_1 = a_1 X, \quad M_2 = a_2 X$$ $$Y = c'X + b_1 M_1 + b_2 M_2$$

Serial Mediator Model (PROCESS 6)

$$M_1 = a_1 X$$ $$M_2 = a_2 X + d_{21} M_1$$ $$Y = c'X + b_1 M_1 + b_2 M_2$$

The chained route is $a_1 d_{21} b_2$. It exists only because you placed $M_1$ before $M_2$; reversing that order produces a different model with the same fit statistics, so the data cannot tell you which ordering is right.

Effect	Formula	Interpretation
Direct Effect	$c'(W)$	Effect of X on Y not through M
Indirect Effect	$a(W) \times b(W)$	Effect of X on Y transmitted through M
Total Effect	$c'(W) + a(W) \times b(W)$	Combined direct + indirect effects

For parallel mediators, total indirect effect = $a_1 b_1 + a_2 b_2$. For serial mediators it is $a_1 b_1 + a_2 b_2 + a_1 d_{21} b_2$.

Converting CI to SE

$$SE = \frac{\text{Upper} - \text{Lower}}{2 \times 1.96}$$

Variance of Conditional Path Coefficients

$$\text{Var}(a(W)) = \text{Var}(a_1) + W^2 \cdot \text{Var}(a_3)$$

Similarly for b(W) and c′(W). Independence of coefficients is assumed.

Indirect Effect Variance (Delta Method)

$$SE_{IE(W)} \approx \sqrt{b(W)^2 \cdot \text{Var}(a(W)) + a(W)^2 \cdot \text{Var}(b(W))}$$

Total Effect Variance

$$SE_{TE(W)} \approx \sqrt{\text{Var}(c'(W)) + \text{Var}(IE(W))}$$

Propagated Uncertainty in Visualizations

The charts show error bars that propagate uncertainty through the mediation path:

X → M Plot (vertical error): Uncertainty in predicted mediator

$$SE(\hat{M}) = |X| \cdot SE(a)$$

M̂ → Y Plot: Uncertainty propagates through both the a-path and b-path

• Horizontal error: $SE(\hat{M}) = |X| \cdot SE(a)$
• Vertical error (delta method): $$SE(\hat{Y}_{via \hat{M}}) = \sqrt{(b \cdot SE(\hat{M}))^2 + (\hat{M} \cdot SE(b))^2}$$

This combines: (1) uncertainty in where M̂ lands, and (2) uncertainty in the b slope.

X → Y Decomposition (vertical errors): Effect uncertainty scaled by X

$$SE(\hat{Y}_{direct}) = |X| \cdot SE(c')$$ $$SE(\hat{Y}_{indirect}) = |X| \cdot SE_{IE}$$ $$SE(\hat{Y}_{total}) = |X| \cdot SE_{TE}$$

When your outcome variable ($Y$) is binary (e.g. 0 vs 1, Churned vs Retained, Purchased vs Not Purchased), standard Ordinary Least Squares (OLS) mediation breaks down mathematically. This tool implements full in-browser Maximum Likelihood Logistic Mediation via Iteratively Reweighted Least Squares (IRLS Newton-Raphson).

1. Why Ordinary Least Squares (OLS) Fails for Binary Outcomes

Fitting a linear probability model ($Y = i_2 + c'X + bM + e$) to a binary outcome creates three fundamental violations:

• Nonsensical Probabilities: Linear slopes project predictions outside the valid probability domain $[0, 1]$ for extreme values of $X$ and $M$.
• Severe Heteroscedasticity: The residual variance of a Bernoulli random variable is $p_i(1 - p_i)$, which varies directly with the predictors, violating Gauss-Markov assumptions and invalidating OLS standard errors.
• Ceiling and Floor Compressions: In real customer behavior, shifting brand awareness from 10% to 20% does not yield the same conversion probability gain as shifting from 50% to 60%. Sigmoidal curves are required.

2. Maximum Likelihood Estimation via IRLS Newton-Raphson

The mediator equation remains linear ($M = i_1 + aX + e_1$), while the outcome equation uses the logistic link function:

$$\text{logit}(P(Y=1)) = \ln\left(\frac{P(Y=1)}{1 - P(Y=1)}\right) = i_2 + c'X + bM$$

The solver uses Newton-Raphson gradient descent with Fisher information weighting ($\mathbf{W} = \text{diag}(p_i(1-p_i))$) until parameter estimates converge:

$$\boldsymbol{\beta}^{(t+1)} = \boldsymbol{\beta}^{(t)} + (\mathbf{X}^T \mathbf{W} \mathbf{X})^{-1} \mathbf{X}^T (\mathbf{y} - \mathbf{p})$$

3. The 4 Reporting Scales (How to Interpret Results)

Because the mediator equation is linear (OLS) and the outcome equation is logistic, effects can be expressed across four complementary metrics:

4. Resample Bootstrapping & Asymmetric Odds Ratio Intervals

Standard Wald confidence intervals assume a symmetric normal distribution. However, Odds Ratios are strictly non-negative and highly right-skewed ($e^{ab}$). The non-parametric bootstrap resamples cases, refits both the OLS and logistic equations on every draw, and cuts percentile and BCa confidence limits directly on empirical distributions.

Capabilities

• ✔ Interactive DAG Path Canvas: Draw arbitrary recursive causal graphs from a blank slate
• ✔ Single, parallel, serial (PROCESS 6), and complex multi-mediator path networks
• ✔ Multi-category dummy coding ($K \ge 2$) with customizable baseline reference categories ($K-1$ contrast indicators)
• ✔ Binary outcome (0/1 or two-group) via Maximum Likelihood Logistic Regression (IRLS Newton-Raphson)
• ✔ 4-scale reporting for logistic mediation: Log-odds, Odds Ratios, MacKinnon Latent Standardization ($Y^*$), and Probability ACME
• ✔ Non-parametric bootstrap resampling (1,000 to 20,000 resamples) directly on your rows with percentile and BCa intervals
• ✔ Path midpoint moderators and statistical controls/covariates
• ✔ Covariance-aware inference (bootstrap mode only)
• ✔ Monte Carlo intervals from typed literature estimates
• ✔ Approximate uncertainty via delta method, shown for contrast
• ✔ Conditional effect plots and simple slopes ($\pm 1$ SD)

Current Boundaries & Scope

• ✘ Non-recursive models (feedback loops / cycles are blocked by DAG validation)
• ✘ Count outcomes (Poisson or negative binomial mediation)
• ✘ Multi-categorical outcomes ($K > 2$ categorical outcomes not supported for OLS path regressions)
• ✘ Latent variables and full SEM with measurement models
• ✘ Multilevel or clustered hierarchical data
• ✘ Missing-data imputation (rows with a missing value in active variables are dropped)