Estimate Transportable Mean Squared Error in the Target Population

Estimates the mean squared error (MSE) of a prediction model in a target population using data transported from a source population. Supports both counterfactual (under hypothetical intervention) and factual (observational) prediction model transportability.

Usage

tr_mse(
  predictions,
  outcomes,
  treatment = NULL,
  source,
  covariates,
  treatment_level = NULL,
  analysis = c("transport", "joint"),
  estimator = c("dr", "om", "ipw", "naive"),
  selection_model = NULL,
  propensity_model = NULL,
  outcome_model = NULL,
  outcome_type = c("auto", "binary", "continuous"),
  se_method = c("bootstrap", "influence", "none"),
  n_boot = 500,
  conf_level = 0.95,
  boot_ci_type = c("percentile", "normal", "basic"),
  stratified_boot = TRUE,
  cross_fit = FALSE,
  n_folds = 5,
  parallel = FALSE,
  ncores = NULL,
  ps_trim = NULL,
  ...
)

Arguments

predictions

Numeric vector of model predictions.

outcomes

Numeric vector of observed outcomes.

treatment

Numeric vector of treatment indicators (0/1), or NULL for factual prediction model transportability (no treatment/intervention). When NULL, only the selection model is used for weighting.

source

Numeric vector of population indicators (1=source/RCT, 0=target).

covariates

A matrix or data frame of baseline covariates.

treatment_level

The treatment level of interest (default: NULL). Required when treatment is provided; should be NULL when treatment is NULL (factual mode).

analysis

Character string specifying the type of analysis:

"transport": Use source outcomes for target estimation (default)
"joint": Pool source and target data

estimator

Character string specifying the estimator:

"naive": Naive estimator (biased)
"om": Outcome model estimator
"ipw": Inverse probability weighting estimator
"dr": Doubly robust estimator (default)

selection_model

Optional fitted selection model for P(S=0|X). If NULL, a logistic regression model is fit using the covariates.

propensity_model

Optional fitted propensity score model for P(A=1|X,S=1). If NULL, a logistic regression model is fit using source data.

outcome_model

Optional fitted outcome model for E[L(Y,g)|X,A,S]. If NULL, a regression model is fit using the relevant data. For binary outcomes, this should be a model for E[Y|X,A] (binomial family). For continuous outcomes, this should be a model for E[L|X,A] (gaussian family).

outcome_type

Character string specifying the outcome type:

"auto": Auto-detect from data (default)
"binary": Binary outcome (0/1) - uses efficient transformation
"continuous": Continuous outcome - models loss directly

se_method

Method for standard error estimation:

"bootstrap": Bootstrap standard errors (default)
"influence": Influence function-based standard errors
"none": No standard error estimation

n_boot

Number of bootstrap replications (default: 500).

conf_level

Confidence level for intervals (default: 0.95).

boot_ci_type

Type of bootstrap confidence interval to compute:

"percentile": Percentile method (default)
"normal": Normal approximation using bootstrap SE
"basic": Basic bootstrap interval

stratified_boot

Logical indicating whether to use stratified bootstrap that preserves the source/target ratio (default: TRUE). Recommended for transportability analysis.

cross_fit

Logical indicating whether to use cross-fitting for nuisance model estimation (default: FALSE).

n_folds

Number of folds for cross-fitting (default: 5).

parallel

Logical indicating whether to use parallel processing for bootstrap (default: FALSE).

ncores

Number of cores for parallel processing (default: NULL, which uses all available cores minus one).

ps_trim

Propensity score trimming specification. Controls how extreme propensity scores are handled. Can be:

NULL (default): Uses absolute bounds c(0.01, 0.99)
"none": No trimming applied
"quantile": Quantile-based trimming with default c(0.01, 0.99)
"absolute": Explicit absolute bounds with default c(0.01, 0.99)
A numeric vector of length 2: c(lower, upper) absolute bounds
A single numeric: Symmetric bounds c(x, 1-x)
A list with method ("absolute"/"quantile"/"none") and bounds

...

Additional arguments passed to internal functions.

Value

An object of class c("tr_mse", "tr_performance") containing:

estimate: Point estimate of transportable MSE
se: Standard error (if computed)
ci_lower: Lower confidence interval bound
ci_upper: Upper confidence interval bound
estimator: Estimator used
analysis: Analysis type
naive_estimate: Naive MSE for comparison
n_target: Number of target observations
n_source: Number of source observations
treatment_level: Treatment level (NULL for factual mode)

Details

This function implements estimators for transporting prediction model performance from a source population to a target population. It supports two modes:

Counterfactual Mode (treatment provided)

When treatment is specified, estimates the counterfactual MSE under a hypothetical intervention, EL(Y^a, g(X)) | S=0. This requires:

Selection model: P(S=0|X)
Propensity model in source: P(A=a|X, S=1)
Outcome model trained on source data

Factual Mode (treatment = NULL)

When treatment is NULL, estimates the MSE of observed outcomes, EL(Y, g(X)) | S=0. This is appropriate for factual prediction model transportability (no causal/counterfactual interpretation). Only requires:

Selection model: P(S=0|X)
Outcome model trained on source data

Analysis Types

Transportability Analysis (analysis = "transport"): Uses outcome data from the source population to estimate performance in the target population.

Joint Analysis (analysis = "joint"): Pools source and target data to estimate performance in the target population. More efficient when both populations have outcome data.

For observational analysis (single population), use cf_mse() instead.

References

Steingrimsson, J. A., et al. (2023). "Transporting a Prediction Model for Use in a New Target Population." American Journal of Epidemiology, 192(2), 296-304. doi:10.1093/aje/kwac128

Li, S., et al. (2023). "Efficient estimation of the expected prediction error under covariate shift." Biometrics, 79(1), 295-307. doi:10.1111/biom.13583

Voter, S. R., et al. (2025). "Transportability of machine learning-based counterfactual prediction models with application to CASS." Diagnostic and Prognostic Research, 9(4). doi:10.1186/s41512-025-00201-y

Boyer, C. B., Dahabreh, I. J., & Steingrimsson, J. A. (2025). "Estimating and evaluating counterfactual prediction models." Statistics in Medicine, 44(23-24), e70287. doi:10.1002/sim.70287

Examples

# Generate example data with source (RCT) and target populations
set.seed(123)
n <- 1000
# Covariates
x <- rnorm(n)
# Source indicator (S=1 for RCT, S=0 for target)
s <- rbinom(n, 1, plogis(0.5 - 0.3 * x))
# Treatment (randomized in source, confounded in target)
a <- ifelse(s == 1,
            rbinom(n, 1, 0.5),  # Randomized in RCT
            rbinom(n, 1, plogis(-0.5 + 0.5 * x)))  # Confounded in target
# Outcome
y <- rbinom(n, 1, plogis(-1 + x - 0.5 * a))
# Predictions from some model
pred <- plogis(-1 + 0.8 * x)

# Estimate transportable MSE
result <- tr_mse(
  predictions = pred,
  outcomes = y,
  treatment = a,
  source = s,
  covariates = data.frame(x = x),
  treatment_level = 0,
  analysis = "transport",
  estimator = "dr",
  se_method = "none"  # Skip SE for speed
)
print(result)
#> 
#> Counterfactual Transportable MSE Estimation
#> --------------------------------------------- 
#> Analysis: transport 
#> Estimator: dr 
#> Treatment level: 0 
#> N target: 376  | N source: 624 
#> 
#> Estimate: 0.1779
#> 
#> Naive estimate: 0.1577 
#> 

# Factual prediction model transportability (no treatment)
result_trad <- tr_mse(
  predictions = pred,
  outcomes = y,
  source = s,
  covariates = data.frame(x = x),
  analysis = "transport",
  estimator = "dr",
  se_method = "none"
)