fbpixel
101 Concepts for the Level I Exam

Essential Concept 3: Analysis of Variance (ANOVA)


Analysis of variance is a statistical procedure for dividing the variability of a variable into components that can be attributed to different sources. We use ANOVA to determine the usefulness of the independent variable or variables in explaining variation in the dependent variable.
ANOVA table

Source of variation Degrees of freedom Sum of squares Mean sum of squares
Regression
(explained variation)
k RSS  MSR=RSS/k
Error
(unexplained variation)
n – 2 SSE  MSE=SSE/(n-k-1)
Total variation n – 1 SST

n represents the number of observations and k represents the number of independent variables. With one independent variable, k = 1. Hence, MSR = RSS and MSE = SSE / (n – 2).
Information from the ANOVA table can be used to compute:

  • item Standard error of estimate ($\mathrm{SEE) =\ }\sqrt{\mathrm{MSE}}$
  • item Coefficient of determination (R${}^{2}$) = RSS / SST

The F-statistic tests whether all the slope coefficients in a linear regression are equal to 0. In a regression with one independent variable, this is a test of the null hypothesis H${}_{0}$: b${}_{1}$ = 0   against the alternative hypothesis H${}_{a}$: b${}_{1}$$\mathrm{\neq}$0. It measures how well the regression equation explains the variation in the dependent variable.
 \begin{enumerate} \item F-stat = MSR / MSE