DPI

🛸 The Directed Prediction Index (DPI).

The Directed Prediction Index (DPI) is a simulation-based method for quantifying the relative endogeneity of outcome versus predictor variables in multiple linear regression models.

Author

Han-Wu-Shuang (Bruce) Bao 包寒吴霜

📬 baohws@foxmail.com

📋 psychbruce.github.io

Citation

• Bao, H.-W.-S. (2025). DPI: The Directed Prediction Index. https://CRAN.R-project.org/package=DPI
  • Note: This is the original citation. Please refer to the information when you library(DPI) for the APA-7 format of the version you installed.

Installation

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

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

Computation Details

\[ \begin{aligned} \text{DPI}_{X \rightarrow Y} & = t^2 \cdot \Delta R^2 \\ & = t_{\beta_{XY|Covs}}^2 \cdot (R_{Y \sim X + Covs}^2 - R_{X \sim Y + Covs}^2) \\ & = t_{partial.r_{XY|Covs}}^2 \cdot (R_{Y \sim X + Covs}^2 - R_{X \sim Y + Covs}^2) \end{aligned} \]

In econometrics and broader social sciences, an exogenous variable is assumed to have a unidirectional (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 more plausible direction of influence (e.g., \(\text{DPI}_{X \rightarrow Y} > 0 \text{: } X \rightarrow Y\)) after controlling for a sufficient number of potential confounding variables.

1. It uses \(\Delta R_{Y vs. X}^2\) to test whether \(Y\) (outcome), compared to \(X\) (predictor), can be more strongly predicted by \(m\) observable control variables (included in a regression model) and \(k\) unobservable random covariates (specified by k.cov; see DPI). A higher \(R^2\) indicates relatively higher dependence (i.e., relatively higher endogeneity) in a given variable set.
2. It also uses \(t_{partial.r}^2\) to penalize insignificant partial correlation (\(r_{partial}\), with equivalent \(t\) test as \(\beta_{partial}\)) between \(Y\) and \(X\), while ignoring the sign (\(\pm\)) of this correlation. A higher \(t^2\) (equivalent to \(F\) test value when \(df = 1\)) indicates a more robust (less spurious) partial relationship when controlling for other variables.
3. Simulation samples with k.cov random covariates are generated to test the statistical significance of DPI.