DPI

🛸 The Directed Prediction Index (DPI).

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

⚠️ Please use version ≥ 2025.11 for correct functionality (see Changelog).

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 causal direction inference from observational data. https://doi.org/10.32614/CRAN.package.DPI
• Bao, H. W. S. (in preparation). The Directed Prediction Index (DPI): Quantifying relative endogeneity for causal direction inference from observational data. (Manuscript in preparation)

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 $\text{DPI} \in (-1, 1)$ as $\text{Direction} \in (-1, 1)$ (relative endogeneity) restricted by $\text{Significance} \in (0, 1)$ (normalized penalty) of the expected $X \rightarrow Y$ relationship:

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

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 and Computation

All steps have been compiled into DPI() and DPI_curve(). See their help pages for usage and illustrative examples. Below are conceptual rationales and mathematical explanations.

Step 1: Relative Endogeneity as Direction Score

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

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

The $\text{Delta}(R^2)$ endogeneity score aims 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 endogeneity in a set of variables.

Notably, as an expected attribute in causal inference, the $\text{Delta}(R^2)$ can also ensure the resulting Directed Acyclic Graph (DAG) structure to be both directed and acyclic, since each direction (edge) has been constrained to go from a lower-R² variable (node) to a higher-R² variable (node) within a specific set of variables. Therefore, it would be impossible to observe any unexpected cyclic structure in the DPI framework.

Step 2: Normalized Penalty as Significance Score

Define $\text{Sigmoid}(\frac{p}{\alpha})$ as normalized penalty for insignificance of the partial relationship between $X$ and $Y$ when controlling for all possible confounders $Covs$:

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

The $\text{Sigmoid}(\frac{p}{\alpha})$ penalty score aims 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. Be careful that it does not suggest the strength or effect size of relationships. It is used mainly for penalizing insignificant partial relationships.

To control for false positive rates, users can set a lower $\alpha$ level (see alpha in DPI() and related functions) and/or use Bonferroni correction for multiple pairwise tests (see bonf in DPI() and related functions).

Notes on transformation among $\tanh(x)$, $\text{sigmoid}(x)$, and $\text{Sigmoid}(\frac{p}{\alpha})$:

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

Wagenmakers (2022) also proposed a simple and useful algorithm to compute approximate (pseudo) Bayes Factors from p values and sample sizes (see transformation rules below).

$$ \text{PseudoBF}_{10}(p, n) = \left{ \begin{aligned} & \frac{1}{3 p \sqrt n} && \text{if} && 0 < p \le 0.10 \\ & \frac{1}{\tfrac{4}{3} p^{2/3} \sqrt n} && \text{if} && 0.10 < p \le 0.50 \\ & \frac{1}{p^{1/4} \sqrt{n}} && \text{if} && 0.50 < p \le 1 \end{aligned} \right. $$

Below we show that normalized penalty scores $\text{Sigmoid}(\frac{p}{\alpha})$ and normalized log pseudo Bayes Factors $\text{sigmoid}(\log(\text{PseudoBF}_{10}))$ have comparable effects in penalizing insignificant p values. However, $\text{Sigmoid}(\frac{p}{\alpha})$ indeed makes stronger penalties for p values when $p &gt; \alpha$ by restricting the penalty scores closer to 0, and it also makes straightforward both the specification of a more conservative α level and the Bonferroni correction of p values for multiple pairwise DPI tests.

Table. Transformation from p values to normalized penalty scores and pseudo Bayes Factors.

+------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | $p\text{ value}$ | $\text{Sigmoid}(p/\alpha)$\ | $\text{Sigmoid}(p/\alpha)$\ | $\text{PseudoBF}{10}$\ | $\text{PseudoBF}{10}$\ | | | ($\alpha = 0.05$) | ($\alpha = 0.01$) | ($n = 100$) [sigmoid(logBF)] | ($n = 1000$) [sigmoid(logBF)] | +==================+=============================+=============================+==============================+===============================+ | (~0) | (~1) | (~1) | ($+\infty$) [~1] | ($+\infty$) [~1] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.0001 | 0.999 | 0.995 | 333.333 [0.997] | 105.409 [0.991] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.001 | 0.990 | 0.950 | 33.333 [0.971] | 10.541 [0.913] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.01 | 0.900 | 0.538 | 3.333 [0.769] | 1.054 [0.513] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.02 | 0.803 | 0.238 | 1.667 [0.625] | 0.527 [0.345] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.03 | 0.709 | 0.095 | 1.111 [0.526] | 0.351 [0.260] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.04 | 0.620 | 0.036 | 0.833 [0.455] | 0.264 [0.209] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.05 | 0.538 | 0.013 | 0.667 [0.400] | 0.211 [0.174] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.10 | 0.238 | 0.00009 | 0.333 [0.250] | 0.105 [0.095] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.20 | 0.036 | 0 | 0.219 [0.180] | 0.069 [0.065] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.50 | 0.00009 | 0 | 0.119 [0.106] | 0.038 [0.036] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 0.80 | 0 | 0 | 0.106 [0.096] | 0.033 [0.032] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+ | 1 | 0 | 0 | 0.100 [0.091] | 0.032 [0.031] | +------------------+-----------------------------+-----------------------------+------------------------------+-------------------------------+

Step 3: Data Simulation

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

(2) Robustness check using DPI_curve(): Run a series of DPI simulation analyses respectively with 1~k.covs (usually 1~10) random covariates, producing a curve of DPIs (estimates and 95% CI; usually getting closer to 0 as k.covs increases) that can indicate its sensitivity in identifying the directed prediction (i.e., How many random covariates can DPIs survive to remain significant?).

(3) Causal discovery using DPI_dag(): Directed acyclic graphs (DAGs) via the DPI exploratory analysis for all significant partial correlations.

Other Functions

This package also includes other functions helpful for exploring variable relationships and performing simulation studies.

• Network analysis functions

  • cor_net(): Correlation and partial correlation networks.

  • BNs_dag(): Directed acyclic graphs (DAGs) via Bayesian networks (BNs).

• Data simulation functions

  • sim_data(): Simulate data from a multivariate normal distribution.

  • sim_data_exp(): Simulate experiment-like data with independent binary Xs.

• Miscellaneous functions

  • cor_matrix(): Produce a symmetric correlation matrix from values.

  • p_to_bf(): Convert p values to pseudo Bayes Factors ($\text{PseudoBF}_{10}$).