Answer First
The Likelihood Ratio Chi‑Square Test compares how well two models fit the data by examining their log‑likelihoods. It tests whether adding parameters significantly improves model fit. Under general conditions, the test statistic follows a chi‑square distribution, making it a powerful and flexible tool for categorical data, logistic regression, and Poisson regression.
Problem Setup
Suppose we compare two nested models:
- Reduced model: fewer parameters
- Full model: more parameters
The likelihood ratio statistic is:
\[ G^2 = -2 \left[ \log L_{\text{reduced}} – \log L_{\text{full}} \right] \]
Under the null hypothesis (reduced model is sufficient):
\[ G^2 \sim \chi^2_{df} \]
where df = difference in number of parameters.
Step-by-Step Solution
1. Fit the reduced model
Compute its log‑likelihood: \(\log L_{\text{reduced}}\).
2. Fit the full model
Compute its log‑likelihood: \(\log L_{\text{full}}\).
3. Compute the likelihood ratio statistic
\[ G^2 = -2(\log L_{\text{reduced}} – \log L_{\text{full}}) \]
4. Compare G² to a chi‑square distribution
If G² is large, the full model fits significantly better.
5. Reject or fail to reject the null
Reject \(H_0\) if G² exceeds the chi‑square critical value.
Intuition
The likelihood ratio test asks: “Does the full model make the observed data much more likely than the reduced model?” If the improvement in likelihood is large relative to the number of added parameters, the full model is justified.
Common Exam Mistakes
- Confusing G² with Pearson’s chi‑square statistic.
- Forgetting that the models must be nested.
- Using G² when sample sizes are extremely small.
- Misinterpreting log‑likelihoods as probabilities.
Why This Matters
The Likelihood Ratio Chi‑Square Test is the backbone of modern statistical modeling. It is used in logistic regression, Poisson regression, survival models, and generalized linear models. Software packages like R, Stata, and Python rely heavily on likelihood ratio tests for model comparison.
Final Summary
The Likelihood Ratio Chi‑Square Test compares nested models by examining their log‑likelihoods. It tests whether adding parameters significantly improves fit, and under broad conditions, the test statistic follows a chi‑square distribution. This makes it a powerful and widely used tool in business analytics and econometrics.
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