Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

MLR - Analysis of Variance

n  ... number of observations
k  ... number of independent variables

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 .. an 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 SSres is an estimate of the variability along the regression line. SSres can be used to find the estimated standard errors of the individual regression coefficients ai. 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(ai).
  • If two variables xi, and xj, are highly correlated, the regression coefficients are difficult to estimate, and their actual numeric values probably do not reflect real dependencies.