- CFA Exams
- 2025 Level II
- Topic 1. Quantitative Methods
- Learning Module 2. Evaluating Regression Model Fit and Interpreting Model Results
- Subject 1. ANOVA Table and Measures of Goodness of Fit

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

R-squared (R

^{2}) 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.

^{2}= Explained variation/Total variation = 1 - Unexplained variation/total variation

The higher the R

^{2}, the better. R

^{2}values range from 0 to 1.

The value of R

^{2}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 R**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.

^{2}

- R
^{2}is always ≥ adjusted R^{2}. - When a new independent variable is added, adjusted R
^{2}can decrease if adding that variable has only a small effect on R^{2}. - In fact, adjusted R
^{2}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 R

^{2}.

R

^{2}= 1 - SSE/SST = 1 - 20.8958/(574.7042 + 20.8958) = 0.9649. Adjusted R

^{2}= 1 - [(10 - 1)/(10 - 3)] (1 - 0.9649) = 0.9549.

In fact, R

^{2}and adjusted R

^{2}are often presented in an ANOVA table. Note that adjusted R

^{2}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.

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**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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