Simultaneous Confidence Interval for Regression Coefficient

Definition

Simultaneous confidence interval is used when computing confidence intervals for $g$ parameters ${\beta }_1,{\beta }_2,\dots ,{\beta }_g$ simultaneously.

Joint confidence region provides the accurate elliptical area as the confidence region, but the calculations and interpretations are complex. To obtain a rectangular-shaped confidence interval, the simultaneous confidence interval is used. When multiplying the confidence intervals of each coefficient, the resulting confidence region becomes smaller than the desired area. Therefore, to obtain the “at least $100(1-\alpha )\%$” confidence region, correction method is used. Conservative methods like the Bonferroni’s Method, provide a confidence region much larger than our desired $100(1-\alpha )\%$, which satisfies the condition of being at least $100(1-\alpha )\%$ but reduces the power of the test. To address this issue, other methods have been devised.

Bonferroni’s Method

Assume that $\mathbf{y}\sim N_n(\mathbf{X}\mathbf{\beta },{\mathbf{I}}_n{\sigma }^2)$ The $100(1-\alpha )\%$ Bonferroni confidence interval for ${\beta }_j$ is defined as ${\hat{\beta }}_j\pm t_{\alpha /2g}(n-p)SE({\hat{\beta }}_j)$ where $SE({\hat{\beta }}_j)$ is the Standard Error for Regression Coefficient ${\beta }_j$ and $p-1$ is the number of explanatory variables.

Scheffe Method

Assume that $\mathbf{y}\sim N_n(\mathbf{X}\mathbf{\beta },{\mathbf{I}}_n{\sigma }^2)$ The $100(1-\alpha )\%$ Scheffe confidence interval for ${\beta }_j$ is defined as ${\hat{\beta }}_j\pm \sqrt{g{F}_{\alpha }(g,n-p)}SE({\hat{\beta }}_j)$ where $SE({\hat{\beta }}_j)$ is the Standard Error for Regression Coefficient ${\beta }_j$ and $p-1$ is the number of explanatory variables.

Maximum Modulus Method

Assume that $\mathbf{y}\sim N_n(\mathbf{X}\mathbf{\beta },{\mathbf{I}}_n{\sigma }^2)$ The $100(1-\alpha )\%$ Maximum Modulus confidence interval for ${\beta }_j$ is defined as ${\hat{\beta }}_j\pm u_{\alpha }(g,n-p)SE({\hat{\beta }}_j)$ where $u$ is a Random Variable representing the maximum of the absolute of $g$ independent random variables following $t(n-p)$, $SE({\hat{\beta }}_j)$ is the Standard Error for Regression Coefficient ${\hat{\beta }}_j$, and $p-1$ is the number of explanatory variables.

Facts

Maximum Modulus method < Bonferroni’s method < Scheffe’s method Here, the second inequality holds only when the degree of freedom of comparisons is relatively small compared to the number of groups being compared.