Latent Variable Measurement Error Correction with lpmec

Data Simulation

Simulate data with a latent predictor and observed binary indicators:

set.seed(123)
n <- 1000  # Number of observations
d <- 10    # Number of observable indicators

# Generate latent variable and observed outcomes
x_true <- rnorm(n)
Yobs <- 0.4 * x_true + rnorm(n, sd = 0.35)

# Generate binary indicators of latent variable
ObservablesMat <- sapply(1:d, function(j) {
  p <- pnorm(0.5 * x_true + rnorm(n, sd = 0.5))
  rbinom(n, 1, p)
})

Running the Analysis

Use lpmec to estimate corrected coefficients:

library(lpmec)

# Run analysis with bootstrap uncertainty estimates
results <- lpmec(
  Y = Yobs,
  observables = as.data.frame(ObservablesMat),
  n_boot = 10,      # Use 0 to disable bootstrap resampling
  n_partition = 5,  # Reduced for demonstration
  estimation_method = "em"
)

## Warning: n_boot < 199 gives coarse bootstrap confidence intervals; increase
## n_boot for interval estimation.

## {booti_ 1 of 11} -- {parti_ 1 of 5}

## {booti_ 1 of 11} -- {parti_ 2 of 5}

## {booti_ 1 of 11} -- {parti_ 3 of 5}

## {booti_ 1 of 11} -- {parti_ 4 of 5}

## {booti_ 1 of 11} -- {parti_ 5 of 5}

## {booti_ 2 of 11} -- {parti_ 1 of 5}

## {booti_ 2 of 11} -- {parti_ 2 of 5}

## {booti_ 2 of 11} -- {parti_ 3 of 5}

## {booti_ 2 of 11} -- {parti_ 4 of 5}

## {booti_ 2 of 11} -- {parti_ 5 of 5}

## {booti_ 3 of 11} -- {parti_ 1 of 5}

## {booti_ 3 of 11} -- {parti_ 2 of 5}

## {booti_ 3 of 11} -- {parti_ 3 of 5}

## {booti_ 3 of 11} -- {parti_ 4 of 5}

## {booti_ 3 of 11} -- {parti_ 5 of 5}

## {booti_ 4 of 11} -- {parti_ 1 of 5}

## {booti_ 4 of 11} -- {parti_ 2 of 5}

## {booti_ 4 of 11} -- {parti_ 3 of 5}

## {booti_ 4 of 11} -- {parti_ 4 of 5}

## {booti_ 4 of 11} -- {parti_ 5 of 5}

## {booti_ 5 of 11} -- {parti_ 1 of 5}

## {booti_ 5 of 11} -- {parti_ 2 of 5}

## {booti_ 5 of 11} -- {parti_ 3 of 5}

## {booti_ 5 of 11} -- {parti_ 4 of 5}

## {booti_ 5 of 11} -- {parti_ 5 of 5}

## {booti_ 6 of 11} -- {parti_ 1 of 5}

## {booti_ 6 of 11} -- {parti_ 2 of 5}

## {booti_ 6 of 11} -- {parti_ 3 of 5}

## {booti_ 6 of 11} -- {parti_ 4 of 5}

## {booti_ 6 of 11} -- {parti_ 5 of 5}

## {booti_ 7 of 11} -- {parti_ 1 of 5}

## {booti_ 7 of 11} -- {parti_ 2 of 5}

## {booti_ 7 of 11} -- {parti_ 3 of 5}

## {booti_ 7 of 11} -- {parti_ 4 of 5}

## {booti_ 7 of 11} -- {parti_ 5 of 5}

## {booti_ 8 of 11} -- {parti_ 1 of 5}

## {booti_ 8 of 11} -- {parti_ 2 of 5}

## {booti_ 8 of 11} -- {parti_ 3 of 5}

## {booti_ 8 of 11} -- {parti_ 4 of 5}

## {booti_ 8 of 11} -- {parti_ 5 of 5}

## {booti_ 9 of 11} -- {parti_ 1 of 5}

## {booti_ 9 of 11} -- {parti_ 2 of 5}

## {booti_ 9 of 11} -- {parti_ 3 of 5}

## {booti_ 9 of 11} -- {parti_ 4 of 5}

## {booti_ 9 of 11} -- {parti_ 5 of 5}

## {booti_ 10 of 11} -- {parti_ 1 of 5}

## {booti_ 10 of 11} -- {parti_ 2 of 5}

## {booti_ 10 of 11} -- {parti_ 3 of 5}

## {booti_ 10 of 11} -- {parti_ 4 of 5}

## {booti_ 10 of 11} -- {parti_ 5 of 5}

## {booti_ 11 of 11} -- {parti_ 1 of 5}

## {booti_ 11 of 11} -- {parti_ 2 of 5}

## {booti_ 11 of 11} -- {parti_ 3 of 5}

## {booti_ 11 of 11} -- {parti_ 4 of 5}

## {booti_ 11 of 11} -- {parti_ 5 of 5}

Compare naive and corrected estimates:

print(results)

## Latent Predictor Measurement Error Correction (LPMEC) Model Results
## -------------------------------------------------------------------
## Resampling: n_out_of_n, m = 1000, CI = percentile
## Uncorrected Coefficient (OLS): 0.301 (SE: 0.022)
## Corrected Coefficient: 0.425 (SE: 0.027)
## Bayesian OLS (Outer): NA (SE: NA)
## Use summary() for detailed results.

summary(results)

## Latent Predictor Measurement Error Correction (LPMEC) Model Summary
## ====================================================================
## Resampling: n_out_of_n, m = 1000 (m/n = 1.000), CI = percentile, replace = TRUE
## Bootstrap success rate: 1.000
##                       Estimate         SE  CI_Lower  CI_Upper
## OLS                  0.3008490 0.02179329 0.2608019 0.3227255
## IV                   0.7411981 0.05831891 0.6598075 0.8314390
## Corrected IV         0.4250383 0.02723953 0.3916324 0.4690671
## Corrected OLS        0.4250383 0.02723953 0.3916324 0.4690671
## Bayesian OLS (Outer)        NA         NA        NA        NA
## Bayesian OLS (Inner)        NA         NA        NA        NA

You can visualize the relationship between split-half estimates:

plot(results)

Inference for Median Partition Aggregation

The default partition aggregation is the finite-partition median. Because the median is nonsmooth, the formal route for uncertainty is subsampling or m-out-of-n bootstrap with root-scaled intervals:

results_med <- lpmec(
  Y = Yobs,
  observables = as.data.frame(ObservablesMat),
  n_boot = 499L,
  n_partition = 10L,
  estimation_method = "em",
  partition_aggregation = "median",
  bootstrap_method = "subsampling",
  boot_m_rule = "power",
  boot_m_exponent = 0.70,
  boot_ci_type = "root",
  seed = 123
)

This path holds the same realized partition set fixed across the original sample and all subsamples, reruns the full latent-score and correction pipeline inside each subsample, and rescales the empirical root statistic from m to the original n rate. Inspect boot_m, boot_m_ratio, valid_partitions_by_boot, bootstrap_success_rate, and sensitivity across a grid of boot_m values in applied work.

Using Different Estimation Methods

The package supports multiple estimation approaches:

# Bayesian MCMC estimation with the default pscl backend
if(FALSE){
mcmc_results <- lpmec(
  Y = Yobs,
  observables = as.data.frame(ObservablesMat),
  estimation_method = "mcmc"
)

# To use the optional NumPyro backend, first run build_backend()
# or provide your own Python environment with JAX and NumPyro.
numpyro_results <- lpmec(
  Y = Yobs,
  observables = as.data.frame(ObservablesMat),
  estimation_method = "mcmc",
  mcmc_control = list(backend = "numpyro"),
  conda_env = "lpmec"
)
}