CFA Level II Exam · Basics of Multiple Regression and Underlying Assumptions

ANOVA, F-Test, R-squared and Adjusted R-squared

Updated 7 October 2026 · Fact-checked

The ANOVA table splits total variation in Y into explained (regression) and unexplained (residual) parts. F = MSR ÷ MSE tests whether all slope coefficients are jointly zero. R-squared = SSR ÷ SST measures fit. Adjusted R-squared penalises extra variables, so it can fall when a useless variable is added.

Understand ANOVA, F-Test, R-squared and Adjusted R-squared

Multiple regression tries to explain variation in a dependent variable Y using k independent variables. The total variation in Y is the sum of squared deviations from its mean, called SST. The regression splits it into two pieces: SSR (the part the model explains) and SSE (the part left in the residuals). So SST = SSR + SSE.

The ANOVA table lays this out. For each source (regression, error, total) it gives degrees of freedom (df), sum of squares (SS) and mean square (MS). Regression df = k. Error df = n − k − 1. Total df = n − 1. A mean square is a sum of squares divided by its df. So MSR = SSR ÷ k and MSE = SSE ÷ (n − k − 1).

The F-test asks one question: do the slope coefficients together explain anything? H0: all slopes equal zero. Ha: at least one slope is not zero. F = MSR ÷ MSE, with k and n − k − 1 degrees of freedom. It is a one-tailed, right-tail test. Reject H0 if F is above the critical value. Use it for joint significance, not for single coefficients (that is the t-test's job).

R-squared = SSR ÷ SST is the share of variation in Y explained by the model. It never falls when you add a variable, even a useless one. So it rewards complexity. Adjusted R-squared corrects this by scaling with degrees of freedom. It rises only if the new variable improves fit enough to offset the lost df. It is never above R-squared in a model with k ≥ 1.

The standard error of estimate (SEE) = √MSE. It is the typical size of a residual, in the units of Y. Lower is better. For comparing models, AIC and BIC are also used. Both start from SSE and add a penalty for the number of parameters. Lower values are better. AIC suits prediction. BIC penalises extra parameters more heavily, so it suits finding the best-fitting parsimonious model. Both compare models with the same dependent variable and sample.

Key formulas to remember

Variation decomposition: SST = SSR + SSE; SST is total, SSR is explained (regression), SSE is unexplained (residual).
Degrees of freedom: Regression = k; Error = n − k − 1; Total = n − 1; k is the number of independent variables, n the number of observations.
Mean squares: MSR = SSR ÷ k; MSE = SSE ÷ (n − k − 1); Mean square = SS ÷ df.
F-statistic: F = MSR ÷ MSE, df = k and n − k − 1; One-tailed right-tail test of H0: all slopes = 0.
R-squared: R² = SSR ÷ SST = 1 − SSE ÷ SST; Never decreases when a variable is added.
Adjusted R-squared: Adj R² = 1 − [(n − 1) ÷ (n − k − 1)] × (1 − R²); Can fall when a variable is added. Not above R² when k ≥ 1.
Standard error of estimate: SEE = √MSE = √[SSE ÷ (n − k − 1)]; In the units of the dependent variable.
AIC: AIC = n × ln(SSE ÷ n) + 2(k + 1); Lower is better. Compare models on the same data and dependent variable.
BIC: BIC = n × ln(SSE ÷ n) + ln(n) × (k + 1); Lower is better. Heavier penalty than AIC when n ≥ 8.

How to solve ANOVA, F-Test, R-squared and Adjusted R-squared questions

Most questions give an ANOVA table, partial numbers, or R-squared with n and k. Pull out what you have and fill in the rest.

1Read n and k from the vignette. If only df is given, use total df = n − 1 and regression df = k.
2Write the table skeleton: SSR, SSE, SST, with df k, n − k − 1, n − 1. Fill known cells and use SST = SSR + SSE to find missing ones.
3Compute mean squares: MSR = SSR ÷ k, MSE = SSE ÷ (n − k − 1).
4For joint significance, compute F = MSR ÷ MSE and compare with the critical F at k and n − k − 1 df. Reject H0 if F is larger.
5For fit, compute R² = SSR ÷ SST, then adjusted R² with the n − 1 and n − k − 1 terms.
6For SEE take √MSE. Do not use SSE or MSR by mistake.
7When asked to choose between models, pick the one with the lower AIC or BIC, or higher adjusted R², and check the criterion matches the purpose.
8Answer in the form asked: reject or fail to reject, a number, or a comparison. State the conclusion in context.

Quickest way: Table-fill shortcut

When to use it: Use when the vignette gives an ANOVA table with one or two blank cells, or gives R² with n and k.

Write k and n − k − 1 beside the table first.
Use SST = SSR + SSE to get the missing sum of squares.
Divide each SS by its df, then divide MSR by MSE for F.
For adjusted R², use 1 − (1 − R²) × (n − 1) ÷ (n − k − 1).
Eliminate options that put adjusted R² above R² or give a negative F.

Common mistakes in ANOVA, F-Test, R-squared and Adjusted R-squared

Using n − k instead of n − k − 1 as the error degrees of freedom.
Simple regression uses n − 2, and students drift to a memorised pattern.
Fix: Always count the intercept. Error df = n − (k + 1). Check that regression df + error df = n − 1.
Treating F as a test of each coefficient.
Students link significance with any single test statistic.
Fix: F tests all slopes jointly. Use t-statistics for individual slopes. A significant F does not mean every slope is significant.
Taking SEE as MSE or SSE.
The names sound alike and the table shows MSE directly.
Fix: SEE = √MSE. Take the square root before answering.
Assuming a higher R-squared means a better model after adding variables.
R² always rises or stays the same when variables are added.
Fix: Compare adjusted R², AIC or BIC when models differ in the number of variables.
Choosing the higher AIC or BIC as better.
Students are used to higher being better, as with R².
Fix: For AIC and BIC, lower is better. Remember BIC penalises extra parameters more heavily.
Using a two-tailed critical value for F.
The t-test is usually two-tailed, so students carry over the habit.
Fix: The F-test is one-tailed. Only a large F rejects H0.

Worked examples

Example 1

An analyst regresses monthly returns of a global equity fund on 3 factors using 40 observations. The ANOVA table shows SSR = 90 and SSE = 72. (1) Compute R². (2) Compute F. (3) The critical F at 5% with 3 and 36 df is about 2.87. What is the conclusion?

Show the solution

k = 3, n = 40. Error df = 40 − 3 − 1 = 36. SST = 90 + 72 = 162.
R² = 90 ÷ 162 = 0.5556.
MSR = 90 ÷ 3 = 30. MSE = 72 ÷ 36 = 2.
F = 30 ÷ 2 = 15.
15 is greater than 2.87, so reject H0.

Answer: R² ≈ 55.6%, F = 15. Reject H0: at least one slope differs from zero.

Example 2

A researcher fits Model A with 2 variables and Model B with 3 variables on the same 31 observations. Model A has R² = 0.60. Model B has R² = 0.61. (1) Compute adjusted R² for each. (2) Which model does adjusted R² favour? (3) Compute SEE for Model B if SSE = 54.

Show the solution

Model A: k = 2, n − 1 = 30, n − k − 1 = 28. Adj R² = 1 − (30 ÷ 28) × 0.40 = 1 − 0.4286 = 0.5714.
Model B: k = 3, n − k − 1 = 27. Adj R² = 1 − (30 ÷ 27) × 0.39 = 1 − 0.4333 = 0.5667.
Model A has the higher adjusted R² (0.5714 versus 0.5667), so it is favoured.
Model B: MSE = 54 ÷ 27 = 2. SEE = √2 = 1.414.

Answer: Adjusted R² is 57.1% for A and 56.7% for B, so A is favoured. SEE for B ≈ 1.414.

Exam tips

Expect a partly filled ANOVA table. Fill it using SST = SSR + SSE and the df rules before anything else.
When R² rises but adjusted R² falls, the added variable is not worth its cost in degrees of freedom.
For AIC and BIC questions, the answer is the lowest value. Check which criterion the question names.
Read the hypothesis carefully. Joint tests use F. A single coefficient uses t. Do not mix them up.
Scan the vignette for n and k before you start. Many errors come from miscounting the intercept.

ANOVA, F-Test, R-squared and Adjusted R-squared in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

ANOVA, F-Test, R-squared and Adjusted R-squared: frequently asked questions

How do I calculate the F statistic from an ANOVA table?

Divide SSR by k to get MSR. Divide SSE by n − k − 1 to get MSE. Then F = MSR ÷ MSE. Compare it with the critical F value at k and n − k − 1 degrees of freedom.

What is the difference between R-squared and adjusted R-squared?

R-squared is the share of variation explained and never falls when you add variables. Adjusted R-squared adjusts for degrees of freedom and falls if a new variable adds too little. Use adjusted R-squared to compare models with different numbers of variables.

Can adjusted R-squared be negative?

Yes. If R² is low relative to the number of variables, the adjustment can push adjusted R² below zero. It is never above R² when there is at least one independent variable.

AIC or BIC: which one should I use?

Both are lower-is-better. AIC is aimed at prediction and BIC at picking the best-fitting model with fewer parameters, since its penalty is larger. Use the one the question names or fits its stated goal.

If F is significant, are all coefficients significant?

No. A significant F only says at least one slope is not zero. Individual t-tests show which ones matter.