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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().

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.