MLR - Analysis of Variance

The ANOVA table above shows the calculations for a multiple regression model with k independent variables and n observations. The following remarks give some hints on how to interpret the ANOVA table:
 

• For k = 1 the table above is reduced to simple linear regression
• The F-ratio tests the hypothesis that all coefficients a₀ .. a_n of the independents variables are zero (null hypothesis). The F-ratio is distributed according to an F distribution with k and n-k-1 degrees of freedom. Also, the F value is related to the goodness of fit, r², through the following equation:
  
• The residual sum of squares SS_res is an estimate of the variability along the regression line. SS_res can be used to find the estimated standard errors of the individual regression coefficients a_i. The estimated standard error follows a t-distribution with n-k-1 degrees of freedom. The confidence interval for the individual coefficients is given by +/- t(α/2, n-k-1)s(a_i).
• If two variables x_i, and x_j, are highly correlated, the regression coefficients are difficult to estimate, and their actual numeric values probably do not reflect real dependencies.