| Scenario | Inputs | Expected output (typical) |
|---|---|---|
| Nested models | ℓ0 = -120.5, ℓ1 = -110.2, k0 = 3, k1 = 5, n = 200 | LR ≈ 20.6, df = 2, p-value small, reject null. |
| 2×2 G-test | O11=12, O12=18, O21=20, O22=10 | G statistic computed from observed vs expected; compare to χ²(1). |
- LR = 2(ℓ1 − ℓ0)
- df = k1 − k0 (difference in parameters)
- Under standard conditions, LR ~ χ²(df) approximately.
- p-value = P(χ²(df) ≥ LR) (right-tail probability)
- G = 2 Σ Oij ln(Oij / Eij) (terms with Oij=0 contribute 0)
- df = (r−1)(c−1) = 1 for 2×2
- Approximate reference distribution: G ~ χ²(1)
- Select a calculation mode based on your data type.
- Enter the required inputs, then choose your significance level.
- Press Calculate to show results above the form.
- Review the test statistic, degrees of freedom, and p-value.
- Use CSV/PDF buttons to export results for reports.
- Check notes if you have small expected counts or unusual inputs.
1) What does the likelihood ratio test compare?
It compares how well two models explain the same data. One model is usually a restricted special case of the other, so the test checks if extra parameters improve fit significantly.
2) Why is the statistic multiplied by 2?
Using 2(ℓ1−ℓ0) makes the statistic follow a chi-square distribution under standard large-sample conditions, enabling quick p-value calculations.
3) How do I choose degrees of freedom?
For nested models, use the difference in parameter counts: df = k1 − k0. For a 2×2 G-test, df is 1 because (2−1)(2−1)=1.
4) What if my alternative log-likelihood is smaller?
That can happen from convergence issues, different data, or non-nested models. The calculator truncates negative statistics to zero, but you should verify model fitting and assumptions.
5) When is the chi-square approximation unreliable?
It may be unreliable with small samples, boundary parameters, non-regular models, or very small expected counts in tables. In those cases, consider exact methods or simulation-based approaches.
6) Is the G-test the same as Pearson’s chi-square test?
They are closely related and often give similar conclusions. The G-test uses log ratios, while Pearson’s uses squared differences. Differences shrink as sample size increases.
7) What do ΔAIC and ΔBIC tell me here?
They compare model fit with penalties for complexity. Negative ΔAIC/ΔBIC favors the alternative model, while positive values favor the null. They are not hypothesis tests, but helpful model-selection summaries.
8) What should I report from an LRT?
Report the test statistic, degrees of freedom, p-value, and conclusion. Also mention the models compared and any relevant diagnostics, such as sample size or small expected counts.