DPI

🛸 The Directed Prediction Index (DPI).

The Directed Prediction Index (DPI) is a quasi-causal inference method for cross-sectional data designed to quantify the relative endogeneity (relative dependence) of outcome (Y) versus predictor (X) variables in regression models.

Author

Bruce H. W. S. Bao 包寒吴霜

📬 baohws@foxmail.com

📋 psychbruce.github.io

Citation

• Bao, H. W. S. (2025). DPI: The Directed Prediction Index for quasi-causal inference with cross-sectional data. https://doi.org/10.32614/CRAN.package.DPI
• Bao, H. W. S. (Manuscript in preparation). The Directed Prediction Index (DPI): Quasi-causal inference for observational data from relative endogeneity of variables.

Installation

## Method 1: Install from CRAN
install.packages("DPI")

## Method 2: Install from GitHub
install.packages("devtools")
devtools::install_github("psychbruce/DPI", force=TRUE)

Algorithm Details

Define \(\text{DPI}\) as the product of \(\text{Direction}\) (relative direction) and \(\text{Strength}\) (absolute strength) of the expected \(X \rightarrow Y\) relationship:

\[ \begin{aligned} \text{DPI}_{X \rightarrow Y} & = \text{Direction}_{X \rightarrow Y} \cdot \text{Strength}_{XY} \\ & = \text{Delta}(R^2) \cdot \text{Sigmoid}(\frac{p}{\alpha}) \\ & = \left( R_{Y \sim X + Covs}^2 - R_{X \sim Y + Covs}^2 \right) \cdot \left( 1 - \tanh \frac{p_{XY|Covs}}{2\alpha} \right) \\ & \in (-1, 1) \end{aligned} \]

In econometrics and broader social sciences, an exogenous variable is assumed to have a directed (causal or quasi-causal) influence on an endogenous variable (\(ExoVar \rightarrow EndoVar\)). By quantifying the relative endogeneity of outcome versus predictor variables in multiple linear regression models, the DPI can suggest a plausible (admissible) direction of influence (i.e., \(\text{DPI}_{X \rightarrow Y} > 0 \text{: } X \rightarrow Y\)) after controlling for a sufficient number of possible confounders and simulated random covariates.

Key Steps of Conceptualization

1. Define \(\text{Direction}_{X \rightarrow Y}\) as relative endogeneity (relative dependence) of \(Y\) vs. \(X\) in a given variable set involving all possible confounders \(Covs\):

\[ \begin{aligned} \text{Direction}_{X \rightarrow Y} & = \text{Endogeneity}(Y) - \text{Endogeneity}(X) \\ & = R_{Y \sim X + Covs}^2 - R_{X \sim Y + Covs}^2 \\ & = \text{Delta}(R^2) \\ & \in (-1, 1) \end{aligned} \]

• It uses \(\text{Delta}(R^2)\) to test whether \(Y\) (outcome), compared to \(X\) (predictor), can be more strongly predicted by all \(m\) observable control variables (included in a given sample) and \(k\) unobservable random covariates (randomly generated in simulation samples, as specified by k.cov in the DPI() function). A higher \(R^2\) indicates higher dependence (i.e., higher endogeneity) in a given variable set.

2. Define \(\text{Sigmoid}(\frac{p}{\alpha})\) as absolute strength of the partial relationship between \(X\) and \(Y\) when controlling for all possible confounders \(Covs\):

\[ \begin{aligned} \text{Sigmoid}(\frac{p}{\alpha}) & = 2 \left[ 1 - \text{sigmoid}(\frac{p_{XY|Covs}}{\alpha}) \right] \\ & = 1 - \tanh \frac{p_{XY|Covs}}{2\alpha} \\ & \in (0, 1) \end{aligned} \]

• It uses \(\text{Sigmoid}(\frac{p}{\alpha})\) to penalize insignificant (\(p > \alpha\)) partial relationship between \(X\) and \(Y\). Partial correlation \(r_{partial}\) always has the equivalent \(t\) test and the same \(p\) value as partial regression coefficient \(\beta_{partial}\) between \(Y\) and \(X\). A higher \(\text{Sigmoid}(\frac{p}{\alpha})\) indicates a more likely (less spurious) partial relationship when controlling for all possible confounders.
• Notes on transformation among \(\tanh(x)\), \(\text{sigmoid}(x)\), and \(\text{Sigmoid}(\frac{p}{\alpha})\):

\[ \begin{aligned} \text{sigmoid}(x) & = \frac{1}{1 + e^{-x}} \\ & = \frac{\tanh(\frac{x}{2}) + 1}{2}, & \in (0, 1) \\ \tanh(x) & = \frac{e^x - e^{-x}}{e^x + e^{-x}} \\ & = 1 - \frac{2}{1 + e^{2x}} \\ & = \frac{2}{1 + e^{-2x}} - 1 \\ & = 2 \cdot \text{sigmoid}(2x) - 1, & \in (-1, 1) \\ \text{Sigmoid}(\frac{p}{\alpha}) & = 2 \left[ 1 - \text{sigmoid}(\frac{p}{\alpha}) \right] \\ & = 1 - \tanh \frac{p}{2\alpha}. & \in (0, 1) \end{aligned} \]

3. Simulate n.sim random samples, with k.cov (unobservable) random covariate(s) in each simulated sample, to test the statistical significance of DPI().