Likelihood Ratio Test

Computes the likelihood ratio test (LRT) statistic and p-value for comparing nested models.

lrt(null_loglik, alt_loglik, dof)

Arguments

null_loglik

Numeric. The maximized log-likelihood under the null (simpler) model.

alt_loglik

Numeric. The maximized log-likelihood under the alternative (more complex) model.

dof

Positive integer. Degrees of freedom, typically the difference in the number of free parameters between models.

Value

A hypothesis_test object of subclass likelihood_ratio_test containing:

stat

The LRT statistic \(\Lambda = -2(\ell_0 - \ell_1)\)

p.value

P-value from chi-squared distribution with dof degrees of freedom

dof

The degrees of freedom

null_loglik

The input null model log-likelihood

alt_loglik

The input alternative model log-likelihood

Details

The likelihood ratio test is a fundamental method for comparing nested statistical models. Given a null model \(M_0\) (simpler, fewer parameters) nested within an alternative model \(M_1\) (more complex), the LRT tests whether the additional complexity of \(M_1\) is justified by the data.

The test statistic is:

$$\Lambda = -2 \left( \ell_0 - \ell_1 \right) = -2 \log \frac{L_0}{L_1}$$

where \(\ell_0\) and \(\ell_1\) are the maximized log-likelihoods under the null and alternative models, respectively.

Under \(H_0\) and regularity conditions, \(\Lambda\) is asymptotically chi-squared distributed with degrees of freedom equal to the difference in the number of free parameters between models.

Assumptions

1. The null model must be nested within the alternative model (i.e., obtainable by constraining parameters of the alternative).

2. Both likelihoods must be computed from the same dataset.

3. Standard regularity conditions for asymptotic chi-squared distribution must hold (true parameter not on boundary, etc.).

Relationship to Other Tests

The LRT is one of the "holy trinity" of likelihood-based tests, alongside the Wald test (wald_test()) and the score (Lagrange multiplier) test. All three are asymptotically equivalent under \(H_0\), but the LRT is often preferred because it is invariant to reparameterization.

See also

wald_test() for testing individual parameters

Examples

# Comparing nested regression models
# Null model: y ~ x1 (log-likelihood = -150)
# Alt model:  y ~ x1 + x2 + x3 (log-likelihood = -140)
# Difference: 3 additional parameters

test <- lrt(null_loglik = -150, alt_loglik = -140, dof = 3)
test
#> Hypothesis test ( likelihood_ratio_test )
#> -----------------------------
#> Test statistic:  20 
#> P-value:  0.0001697424 
#> Degrees of freedom:  3 
#> Significant at 5% level:  TRUE 

# Is the more complex model significantly better?
is_significant_at(test, 0.05)
#> [1] TRUE

# Extract the test statistic (should be 20)
test_stat(test)
#> [1] 20

# Access stored inputs for inspection
test$null_loglik
#> [1] -150
test$alt_loglik
#> [1] -140