Proportional Marginal Effects (PME)

• Proportional Marginal Effects (PME) are defined by their proportional relationship between inputs and resulting output changes, capturing nuanced impacts in nonlinear models.
• Methodologies incorporate approaches from generalized linear models, causal inference, and global sensitivity analysis to deliver robust and interpretable estimates.
• PME frameworks are widely applied in econometrics, policy evaluation, and machine learning to accurately attribute effect magnitudes amid complex interactions and dependencies.

Proportional Marginal Effects (PME) are a class of effect measures designed to capture how small perturbations—whether in inputs, treatments, or policy interventions—yield marginal changes in outputs, often emphasizing the proportional relationship among these changes. PME arises in diverse contexts, including economic production models, causal inference, global sensitivity analysis, nonlinear prediction functions, and policy evaluation. Methodologically, PME is motivated by the need to interpret and quantify effect magnitudes in models with complex interactions, endogeneity, dependencies, and nonlinearities, frequently leading to more robust and interpretable estimates compared to conventional effect metrics.

1. Key Mathematical Definitions and Structural Foundations

PME are mathematically characterized by models in which the marginal effect of an input is directly proportional to either the input’s level or to model parameters, often enforced through symmetry or invariance properties. In quasi-sum production models (Vîlcu et al., 2014), proportionality is established via the proportional marginal rate of substitution (PMRS), which dictates that the marginal rate of technical substitution between two inputs $x_i$ and $x_j$ satisfies $MRS_{ij} = f_{x_j}/f_{x_i} \propto x_j/x_i$. This directly leads to the class of homothetic generalized Cobb–Douglas functions,

$$f(x_1, ..., x_n) = F\left(\prod_{i=1}^n x_i^k\right),$$

preserving constant elasticity and proportional responsiveness.

In semiparametric generalized linear models (Lee et al., 2022), PME are encapsulated as average derivatives proportional to the regression slope:

$\xi = \mathbf{\beta} \cdot E\{\mathrm{Var}(Y|x)\},$

where $\xi$ denotes the marginal effect.

In causal inference, PME frequently arises under marginal interventional effects (MIE) (Zhou et al., 2022), which for an infinitesimal policy change are identified as weighted averages of conditional average treatment effects (CATE),

$$\mathrm{MIE} = E\left[\frac{\dot{\pi}_0(X)}{E[\dot{\pi}_0(X)]} \cdot \mathrm{CATE}(X)\right],$$

where $\dot{\pi}_0(X)$ is the derivative of the propensity score.

In global sensitivity analysis, PME are derived from proportional values in cooperative games applied to variance decomposition (Herin et al., 2022):

$\mathrm{PME}_i = PV\left((D, S^T)\right),$

where $PV$ denotes the proportional value, $D$ is the set of variables, and $S^T$ is the dual Sobol index.

2. PME in Econometric, Causal, and Policy Models

PME underpin the structural interpretation of marginal effects in causal and policy-related frameworks. Marginality-weighted estimands (Deng, 29 Aug 2025) clarify the average effect among units whose treatment status is altered by a policy:

$$\tau^{\Delta p} = \frac{\int \tau(\theta) \Delta p(\theta) dF(\theta)}{\int \Delta p(\theta) dF(\theta)},$$

where $\Delta p(\theta)$ is the change in participation probability at productivity level $\theta$. This framework generalizes local average treatment effect (LATE) and addresses endogenous selection problems.

Structural characterization in nonseparable models (Wang et al., 13 Jun 2025) relates MPE (marginal policy effect) to functional derivatives:

$$\theta_\Gamma = \Gamma'_{F_Y}\left(E[\omega^f(\cdot, D) \cdot \partial_d m(D, X, \varepsilon)|Y = \cdot]\right),$$

where $\Gamma$ is a Hadamard-differentiable functional, $m$ is the outcome-generating function, and $\omega^f$ encodes policy variation. This establishes PME as a canonical measure of policy impact on arbitrary distributional functionals.

3. PME in Global Sensitivity Analysis

PME indices have become important in variance-based sensitivity analysis for high-dimensional models or correlated inputs (Herin et al., 2022, Foucault et al., 2023). Traditional Sobol indices and Shapley effects can misattribute influence to exogenous (statistically correlated but structurally inert) inputs. PME, constructed via proportional values, overcome this by allocating variance to inputs strictly in proportion to their individual contributions, satisfying the property:

$$\mathrm{PME}_i = 0 \quad \text{whenever input } i \text{ is exogenous},$$

even when $i$ is correlated with endogenous variables. This distinction enhances variable selection and interpretation in models with dependent parameters.

A comparative table between PME and Shapley effects:

Metric	Allocation Principle	Response to Exogenous Correlation	Distribution of Interaction Effects
PME	Proportional to marginal gain	Zero for exogenous inputs	Weighted by marginal contributions
Shapley	Egalitarian (average)	Nonzero if correlated	Equally split among coalition members

In applications—such as CT dosimetry and epidemic modeling (Foucault et al., 2023)—PME provides clearer variable hierarchies, avoiding the "Shapley's joke" of spurious attribution and offering sharper insight for factor fixing and uncertainty quantification.

4. PME in Nonlinear and Complex Prediction Functions

Interpreting feature effects in nonlinear models is often nontrivial, especially in machine learning contexts. Forward marginal effects (fME) (Scholbeck et al., 2022) are defined as finite differences in predicted outcome due to feature perturbation:

$$\mathrm{fME}_{x, h} = \hat{f}(x_1, ..., x_j + h_j, ..., x_p) - \hat{f}(x)$$

for step $h_j$ in feature $x_j$. Multivariate extensions and conditional average marginal effects (cAME) are proposed to address geographical heterogeneity in feature effects.

Nonlinearity is quantified by the non-linearity measure (NLM), which compares the actual change along the path to its linear approximation, enabling rigorous assessment of whether PME interpretation is appropriate locally. Partitioning the feature space yields cAMEs for interpretable effect estimates in subgroups.

5. Practical Estimation and Theoretical Properties

Methodologically, PME estimation leverages modern regression strategies, including regression-with-residuals for marginal effects under time-varying treatments (Wodtke et al., 2018), semiparametric B-spline maximum likelihood (Lee et al., 2022), doubly robust influence-function methods in MIE estimation (Zhou et al., 2022), and plug-in or bootstrapped estimators in nonparametric panel models (Liu et al., 2021).

PME estimators have been shown to attain desirable properties in simulation studies and real data applications:

• Consistency and asymptotic normality (Lee et al., 2022)
• Semiparametric efficiency—estimator variance achieves the minimal bound
• Robustness to treatment-induced confounders and interaction effects
• Ability to deliver unbiased and interpretable effect attribution even in complex correlated input settings or presence of measurement error (Evdokimov et al., 2023)

In nonlinear models, PME estimation must address the potential for bias, particularly in longitudinal clinical trial settings where proportional effect assumptions can inflate Type I error or produce group-label sensitivity (Donohue et al., 31 Jan 2025).

6. Limitations and Implications

While PME offers robust theoretical and practical advantages, certain limitations are observed:

• In longitudinal models with low control group means, proportional parameterizations can yield unstable or unidentifiable PME estimates (Donohue et al., 31 Jan 2025).
• Correct model specification—including handling confounder interactions and measurement error—is critical for accurate PME inference (Wodtke et al., 2018, Evdokimov et al., 2023).
• In global sensitivity analysis, when inputs have complicated dependency structure, PME and Shapley indices may behave divergently, necessitating careful choice of index according to application needs (Herin et al., 2022, Foucault et al., 2023).

Despite these limitations, PME frameworks substantially advance interpretability, causal attribution, and policy relevance in settings with complex selection, endogenous treatment assignment, and input dependencies.

7. Connections to Related Methodologies and Applications

PME bridges multiple methodological paradigms. In economic modeling, PME links to homothetic Cobb–Douglas production functions (Vîlcu et al., 2014), in causal inference it generalizes local and marginal treatment effects (Sasaki et al., 2020, Zhou et al., 2022, Deng, 29 Aug 2025), and in policy analysis it is central to marginal policy effect characterizations (Wang et al., 13 Jun 2025).

Recent empirical applications span:

• Welfare analysis via marginal treatment effects and regret minimization (Sasaki et al., 2020)
• Detection of sensitive input parameters in medical models (Foucault et al., 2023)
• Labor supply, health, and education interventions in economics (Liu et al., 2021, Deng, 29 Aug 2025)
• Machine learning interpretability, in forward difference-based nonlinear effect estimation (Scholbeck et al., 2022)

PME represents a rigorous framework for quantifying, interpreting, and leveraging marginal responses in models with intricate structural and statistical dependencies, underpinning advances in statistical decision theory, global sensitivity analysis, econometric modeling, and policy evaluation.