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Subject 1. ANOVA Table and Measures of Goodness of Fit PDF Download

R-squared (R2) measures how well an estimated regression fits the data. It is also known as the coefficient of determination. It is a measure of the goodness of fit of the regression line.

R2 = Explained variation/Total variation = 1 - Unexplained variation/total variation

The higher the R2, the better. R2 values range from 0 to 1.

The value of R2 depends on the number of explanatory variables included in the model. This causes a problem when we try to compare the goodness of fit of two models that have the same dependent variable but different number of explanatory variables.

Multiple regression software packages usually produce an adjusted R2 as an alternative measure of goodness of fit. It does not automatically increase as independent variables are added to the model. Rather, it adjusts for the degrees of freedom by incorporating the number of independent variables.

  • R2 is always ≥ adjusted R2.
  • When a new independent variable is added, adjusted R2 can decrease if adding that variable has only a small effect on R2.
  • In fact, adjusted R2 can actually be negative if the correlation between the dependent variable and the independent variables is sufficient low.

We can use the information in an ANOVA table to determine R2.

R2 = 1 - SSE/SST = 1 - 20.8958/(574.7042 + 20.8958) = 0.9649. Adjusted R2 = 1 - [(10 - 1)/(10 - 3)] (1 - 0.9649) = 0.9549.

In fact, R2 and adjusted R2 are often presented in an ANOVA table. Note that adjusted R2 does not indicate whether a regression coefficient's predictions are true or biased. Residual plots and other statistics are required to determine whether or not the predictions are accurate.

Akaike's information criterion (AIC) and Schwarz's Bayesian information criteria (BIC) are also used to evaluate model fit and select the "best" model among a group with the same dependent variable. AIC is preferred if the purpose is prediction, BIC is preferred if goodness of fit is the goal, and lower values of both measures are better.

User Contributed Comments 4

User Comment
danlan2 Adjusted R^2=1-[(n-1)/(n-k)]*(1-R^2) is right,
but k is the number of all variables (including dependant and independant), or it is the number of independant variables + 1.
JimM I just googled adjusted R2 and most sites gave (n-k-1) in the formula.
arudkov 2 danlan - k - is the number of indep variables. +1 means interciept
Adi8232 Which Textbook?
In L2 Vol1, its n-k-1, not n-k.
Guess they corrected it. or maybe AnalystNotes is putting down the book for those who haven't read it, hehe. [i haven't, had to check, but Cmon guys, don't say bad things about CFA (textbook)]
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