Statistics for Individual Coefficients

Following are the statistics for the p^th coefficient, including the regression constant:

Coefficient

The coefficient of interest is expressed as b_p, the p^th component in the b vector.

Standard Error of Coefficient

The standard error of this coefficient is expressed as se(b_p), or

Formula for the standard error of an individual coefficient, for regression with a constant

where S is the standard error of estimate (SSE) and c_pp is the diagonal element at (p,p) of the matrix (X^TX)^-1.

t

If the F statistic in ANOVA and the corresponding p indicate a significant relationship between the dependent and the independent variables as a whole, you then want to see the significance of the relationship of the dependent variable to each independent variable. The t statistic tests for the significance of the specified independent variable in the presence of the other independent variables.

The formula for the t statistic:

where bp is the coefficient to check and se(bp) is the standard error of the coefficient.

p

The t statistic (t) follows a t distribution with (n – m – 1) degrees of freedom.