🛸 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.
Bruce H. W. S. Bao 包寒吴霜
## Method 1: Install from CRAN
install.packages("DPI")
## Method 2: Install from GitHub
install.packages("devtools")
devtools::install_github("psychbruce/DPI", force=TRUE)Define \(\text{DPI} \in (-1, 1)\) as the product of \(\text{Direction} \in (-1, 1)\) (relative endogeneity as direction score) and \(\text{Significance} \in (0, 1)\) (normalized penalty as significance score) of the expected \(X \rightarrow Y\) influence (quasi-causal) 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.
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.
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.
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 \(p > \alpha\) 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.
| \(p\) | \(\text{Sigmoid}(\frac{p}{\alpha})\) (α = 0.05) |
\(\text{Sigmoid}(\frac{p}{\alpha})\) (α = 0.01) |
\(\text{PseudoBF}_{10}\) (n = 100) [sigmoid(logBF)] |
\(\text{PseudoBF}_{10}\) (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] |
(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.
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 (\(\text{PseudoBF}_{10}\)).