Používanie Anova a AIC kritérií na porovnanie dvoch modelov

When to use ANOVA for model comparison

(Use ANOVA when models are nested)

Two linear models are nested if one model is a special case of the other, meaning the smaller model can be obtained by removing terms from the larger one.

Example:

modelA <- lm(y ~ x)
modelB <- lm(y ~ x + z)
anova(modelA, modelB)

ANOVA (F-test) answers:

Is the larger model significantly better? i.e., does adding extra predictors reduce residual variance enough to be statistically meaningful?

We should use ANOVA when:

Models differ only by added or removed predictors

We want a hypothesis test

We assume:

• Normal errors
• Homoskedasticity
• Models have the same dependent variable
• Models are fit using ordinary least squares

Cannot use ANOVA when models are not nested. - Do not use by the BCox-Box transformation of the explained variable

Example of non-nested:

model1 <- lm(y ~ x1 + x2)
model2 <- lm(y ~ log(x1) + I(x2^2))

These cannot be compared with ANOVA.

When to use AIC (Akaike Information Criterion) for model comparison

AIC is based on information theory, not hypothesis testing.

AIC can compare:

• nested models,
• non-nested models,
• models with different functional forms,
• models estimated by maximum likelihood (GLM, Poisson, logistic regression, etc.)

Example:

AIC(model1,model2)

AIC answers:

• Which model has the best trade-off between fit and complexity?
• Lower AIC = better model.

AIC does NOT give:

• p-values
• significance tests

It gives a relative measure, not a hypothesis test.

Is not the Adjusted R-squared enough to compare 2 models?

AIC is the standard when comparing models, if:

• testing transformations (like log(Y), Box–Cox, etc.), or
• the models have different sets of regressors.
• using the Least squares method of the estimation, but also other methods (e.g., Maximum Likelihood)

Adjusted R-squared, if:

• only linear models with same dependent variable Y
• simple OLS