sandwich_variance: Sandwich Variance Estimation in fmrireg

In bbuchsbaum/fmrireg: R package for the anlysis of fmri data

Sandwich Variance Estimation in fmrireg

Description

This documentation describes the sandwich variance estimation capabilities in fmrireg, which provide robust standard errors for regression coefficients when model assumptions are violated.

Background

The sandwich variance estimator (also known as the Huber-White estimator) provides valid standard errors even when the residuals exhibit heteroscedasticity or other violations of the classical linear model assumptions.

Mathematical Details

The sandwich estimator is computed as:

V_{sandwich} = (X'X)^{-1} X' \Omega X (X'X)^{-1}

where \Omega is a diagonal matrix with squared residuals on the diagonal. For robust regression with weights w_i, the weighted version is:

V_{sandwich} = (X'WX)^{-1} X'W \Omega WX (X'WX)^{-1}

Usage in fmrireg

Sandwich variance estimation is automatically used when:

• Robust regression is enabled (using M-estimators)

• AR modeling is combined with robust regression

• Heteroscedasticity is suspected in the residuals

Effective Degrees of Freedom

When using robust regression and/or AR models, the effective degrees of freedom are adjusted to account for:

• Downweighting of outliers in robust regression

• Loss of degrees of freedom due to AR parameter estimation

The adjustment formula is:

df_{effective} = df_{base} \times \frac{\sum w_i}{n} \times \frac{n - p_{AR}}{n}

Implementation Notes

• Small sample corrections are applied (n/(n-p) factor)

• For multi-voxel data, computation is vectorized for efficiency

• Compatible with all contrast types (t, F, custom)

References

Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability.

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817-838.

Examples

## Not run: 
# Fit model with robust regression
cfg <- fmri_lm_control(
  robust = list(
    type = "bisquare",
    c_tukey = 4.685
  )
)

fit <- fmri_lm(model, dataset, config = cfg)

# Standard errors in fit$betas will use sandwich variance
# P-values will use effective degrees of freedom

## End(Not run)

bbuchsbaum/fmrireg documentation built on June 10, 2025, 8:18 p.m.