Likelihood-based tests
Objective. Build likelihood-ratio tests and intervals.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Likelihood-based tests compare how well competing parameter values explain the data using the likelihood-ratio statistic Λ = L(θ0)/L(θ̂), where θ̂ is the maximum likelihood estimate and θ0 the null value. Under standard conditions, −2 ln Λ follows approximately a chi-squared distribution with degrees of freedom equal to the number of constrained parameters, giving a test of H0: θ = θ0. Confidence intervals can be built from the same principle, taking all θ0 values for which −2 ln Λ falls below the relevant chi-squared cutoff.
Worked example
In a likelihood-ratio test, the statistic −2 ln Λ equals 6.5 with 1 degree of freedom, and the chi-squared critical value at 5% is 3.84. State the conclusion of this test.
- 1Compare the test statistic 6.5 to the critical value 3.84
- 2Since 6.5 > 3.84, the statistic exceeds the critical threshold
- 3This means the result falls in the rejection region
- 4State the conclusion for H0
Answer: Reject H0 at the 5% level
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Start practisingCommon mistake. A common mistake is using the chi-squared approximation for −2 ln Λ with very small sample sizes, where the asymptotic result is unreliable. The approximation requires sufficiently large n to be trustworthy.
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