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The Predictive Information Index (PII) quantifies how much outcome-relevant information is retained when reducing a set of predictors (e.g., items) to a composite score.
PII is defined as:
\text{PII} = 1 - \frac{\text{Var}(\hat{Y}_{\text{Full}} - \hat{Y}_{\text{Score}})}{\text{Var}(\hat{Y}_{\text{Full}})}Where: - \(\hat{Y}_{\text{Full}}\): predictions from a full model (e.g., all items or predictors) - \(\hat{Y}_{\text{Score}}\): predictions from a reduced score (e.g., mean or sum)
A PII of 1 means no predictive information was lost. A PII near 0 means the score loses most predictive information.
pii()library(piiR)
# Simulate two prediction vectors
set.seed(123)
full_model_preds <- rnorm(100)
score_based_preds <- full_model_preds + rnorm(100, sd = 0.5)
# Compute PII
pii(full_model_preds, score_based_preds)## [1] 0.7721124These binaries (installable software) and packages are in development.
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