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Simultaneous Confidence Bands for Regression Outcomes or Coefficients

Source: R/regression_outcome_scb.R
SCB_regression_outcome.Rd

This function fits a user-specified regression model (linear or logistic), and constructs simultaneous confidence bands (SCBs) for either the mean outcome or the regression coefficients. SCBs are obtained using a nonparametric bootstrap procedure evaluated on a fixed test design matrix, providing simultaneous inference across the entire range of covariates or parameters of interest.

Usage

SCB_regression_outcome(
  df_fit,
  model,
  grid_df = NULL,
  n_boot = 1000,
  alpha = 0.05,
  grid_df_boot = NULL,
  type = "linear",
  fitted = TRUE,
  w = NULL
)

Arguments

df_fit

A data frame containing the training design matrix used to fit the linear model. Acceptable input format includes numeric and factor.

model

A character string representing the formula for the linear model (e.g., "y ~ x1 + x2").

grid_df

A data frame specifying the covariate settings that define the mean outcome or linear combination for which simultaneous confidence bands (SCB) are constructed. Each row represents one covariate combination at which predictions and SCBs are evaluated. Column names should match variables in the fitted model, but grid_df may include only the subset of covariates of interest for the SCB (it is not required to cover all model variables). Default is NULL, in which case the SCB is constructed over the fitted values based on 'df_fit`.

n_boot

Number of bootstrap samples used in the non-parametric bootstrap procedure to generate the empirical distribution. Default is 1000.

alpha

Significance level for the confidence band (e.g., 0.05 for 95% confidence). Default is 0.05.

grid_df_boot

An optional data frame specifying the input grid at which predictions are evaluated during bootstrap resampling. This allows SCBs to be constructed on a denser set of covariate values if desired. If NULL, uses grid_df. If grid_df is set to NULL, grid_df_boot will also be set to NULL. This argument is only for type = linear.

type

A character string specifying the model type. Either "linear" (default) or "logistic".

fitted

Logical. Whether to estimate the simultaneous confidence bands for regression outcomes or coefficients.

  • TRUE - Estimate the simultaneous confidence bands for regression outcomes.

  • FALSE - estimate the simultaneous confidence bands for regression coefficients.

Default is TRUE.

w

A numeric matrix that specifies the linear combinations of regression coefficients for which simultaneous confidence bands (SCBs) are to be constructed. The number of columns should be equal to the number of coefficients in the regression model fitted. Default is NULL, will return SCBs for all coefficients and the intercept. This argument is only for fitted = FALSE.

Value

A data frame with the following columns:

scb_low

Lower bound of the simultaneous confidence band.

Mean

Predicted mean response from the fitted model.

scb_up

Upper bound of the simultaneous confidence band.

...

All columns from grid_df, representing the prediction grid. Optional, if fitted = TRUE

References

Ren, J., Telschow, F. J. E., & Schwartzman, A. (2024). Inverse set estimation and inversion of simultaneous confidence intervals. Journal of the Royal Statistical Society: Series C (Applied Statistics), 73(4), 1082–1109. doi:10.1093/jrsssc/qlae027

Examples

set.seed(262)
x1 <- rnorm(100)
x2 <- rnorm(100)
epsilon <- rnorm(100,0,sqrt(2))
y <- -1 + x1 + x2 + epsilon
df <- data.frame(x1 = x1, x2 = x2, y = y)
grid <- data.frame(x1 = seq(-1, 1, length.out = 100), x2 = seq(-1, 1, length.out = 100))
model <- "y ~ x1 + x2"
w <- matrix(c(0, 1, 1), ncol = 3)
results <- SCB_regression_outcome(df_fit = df, model = model, grid_df = grid,
                                  w = w, n_boot = 100)