Standard Errors of Standardized Estimates of Regression Coefficients (Yuan and Chan (2011))

.sehatslopeshatprimedelta(
  slopeshat,
  sigma2hatepsilonhat,
  SigmaXhat,
  sigmayXhat,
  sigma2yhat,
  adjust = FALSE,
  n,
  X,
  y
)

Arguments

slopeshat	Numeric vector of length `p` or `p` by `1` matrix. $p \times 1$ column vector of estimated regression slopes $\left( \boldsymbol{\hat{\beta}}_{2, 3, \cdots, k} = \left\{ \hat{\beta}_2, \hat{\beta}_3, \cdots, \hat{\beta}_k \right\}^{T} \right)$ .
sigma2hatepsilonhat	Numeric. Estimate of error variance.
SigmaXhat	`p` by `p` numeric matrix. $p \times p$ matrix of estimated variances and covariances between regressor variables $X_2, X_3, \cdots, X_k$ $\left( \boldsymbol{\hat{\Sigma}}_{\mathbf{X}} \right)$.
sigmayXhat	Numeric vector of length `p` or `p` by `1` matrix. $p \times 1$ vector of estimated covariances between the regressand $y$ variable and regressor variables $X_2, X_3, \cdots, X_k$ $\left( \boldsymbol{\hat{\sigma}}_{\mathbf{y}, \mathbf{X}} = \left\{ \hat{\sigma}_{y, X_2}, \hat{\sigma}_{y, X_3}, \cdots, \hat{\sigma}_{y, X_k} \right\}^{T} \right)$.
sigma2yhat	Numeric. Estimated variance of the regressand $\left( \hat{\sigma}_{y}^{2} \right)$
adjust	Logical. Use $n - 3$ adjustment for small samples.
n	Integer. Sample size.
X	`n` by `k` numeric matrix. The data matrix $\mathbf{X}$ (also known as design matrix, model matrix or regressor matrix) is an $n \times k$ matrix of $n$ observations of $k$ regressors, which includes a regressor whose value is 1 for each observation on the first column.
y	Numeric vector of length `n` or `n` by `1` matrix. The vector $\mathbf{y}$ is an $n \times 1$ vector of observations on the regressand variable.

Details

The $p$th estimated standard error is calculated using $$ \mathbf{\widehat{se}}_{\boldsymbol{\hat{\beta}}_{\text{p}}^{\prime}} = \sqrt{ \frac{\hat{\sigma}_{X_{p}}^{2} \hat{c}_{p} \hat{\sigma}_{\hat{\varepsilon}}^{2}}{n \hat{\sigma}_{y}^{2}} + \frac{\hat{\beta}_{p}^{2} \left[ \hat{\sigma}_{X_{p}}^{2} \left( \boldsymbol{\hat{\beta}}^{T} \boldsymbol{\hat{\Sigma}}_{X} \boldsymbol{\hat{\beta}} \right) - \hat{\sigma}_{X_{p}}^{2} \hat{\sigma}_{\hat{\varepsilon}}^{2} - \hat{\sigma}_{y, X_{p}}^{2} \right]}{n \hat{\sigma}_{y}^{4}} } $$ where

$p = \left\{2, 3, \cdots, k \right\}$
$\hat{\sigma}_{\hat{\varepsilon}}^{2}$ is the estimated residual variance
$\boldsymbol{\hat{\beta}}_{2, 3, \cdots, k} = \left\{ \hat{\beta}_{2}, \hat{\beta}_{3}, \cdots, \hat{\beta}_{k}\right\}^{T}$ is the $p \times 1$ column vector of estimated regression slopes
$\hat{\sigma}_{y}^{2}$ is the variance of the regressand variable $y$
$\boldsymbol{\hat{\Sigma}}_{\mathbf{X}}$ is the $p \times p$ estimated covariance matrix of the regressor variables $X_2, X_3, \cdots, X_k$
$\hat{\sigma}_{X_p}^{2}$ is the variance of the corresponding $p$th regressor variable.
$\hat{\sigma}_{y, X_{p}}^{2}$ is the covariance of the regressand variable $y$ and the regressor variables $X_2, X_3, \cdots, X_k$
$c_p$ is the diagonal element that corresponds to the regressor variable in $\boldsymbol{\Sigma}_{\mathbf{X}}^{-1}$
$n$ is the sample size

References

Yuan, K., Chan, W. (2011). Biases and Standard Errors of Standardized Regression Coefficients. Psychometrika 76, 670-690. doi:10.1007/s11336-011-9224-6.

Author

Ivan Jacob Agaloos Pesigan

Examples