ETC3400 Chap.7 Likelihood Ratio, Wald and Score Tests
Likelihood Ratio, Wald and Score Tests
Likelihood Ratio, Wald and Score Tests compares three ways to measure departure from a null hypothesis. Likelihood Ratio uses the objective drop between unrestricted and constrained fits. Wald scales the unrestricted estimate's distance from the null using an estimated covariance. Score, also called Lagrange Multiplier, scales the slope evaluated under the null.
Under regularity conditions the three statistics share a chi squared limiting distribution with degrees of freedom equal to the number of restrictions. In a known variance normal mean model constant curvature makes them exactly equal; estimating variance changes profile curvature and produces a finite sample ordering.
What this chapter covers
- 01
Null and alternative hypotheses as parameter restrictions
- 02
Unrestricted and constrained maximum likelihood estimates
- 03
Likelihood Ratio statistic from the objective difference
- 04
Wald statistic from scaled estimator distance
- 05
Score or LM statistic from null slope and information
- 06
Chi squared reference distribution and restriction count
- 07
Exact equality in the known variance normal model
- 08
Effective curvature and LM less than or equal to LR less than or equal to Wald
Worked example · free
Test a Bernoulli probability with two likelihood methods
- +1The unrestricted maximum likelihood estimate is 16/25=0.64.
- +2Wald equals 25 times (0.64-0.50) squared divided by 0.64 times 0.36, which is about 2.13.
- +2The Bernoulli log likelihood is 16 log p plus 9 log(1-p). Evaluate it at 0.64 and 0.50.
- +1Twice the unrestricted minus constrained log likelihood is about 1.98.
- +2Both statistics are below 3.841, so do not reject the null at the 5% level and explain that finite sample values need not coincide.
Key terms
- Constrained estimator
- The parameter value that maximises the likelihood while satisfying all restrictions imposed by the null hypothesis.
- Likelihood Ratio test
- A test based on twice the difference between unrestricted and constrained maximised log likelihood values.
- Wald test
- A test based on the estimated distance from an unrestricted estimate to the null, scaled by estimated uncertainty.
- Score test
- A test using the log likelihood gradient and information evaluated at the constrained or null estimate.
- Degrees of freedom
- For the regular chi squared test limit, the number of independent restrictions imposed by the null hypothesis.
- Effective curvature
- Local profile likelihood curvature used to interpret why Score, Likelihood Ratio and Wald can differ when nuisance parameters are estimated.
Likelihood Ratio, Wald and Score Tests FAQ
Why is the Likelihood Ratio statistic nonnegative?
The unrestricted parameter space contains the constrained one, so its maximum log likelihood cannot be smaller. Twice the unrestricted minus constrained value is therefore at least zero.
Why can Wald be unstable near a boundary?
It uses an unrestricted estimate and plug in covariance to approximate distance from the null. Near a boundary or in a small sample, that local quadratic approximation can be inaccurate.
How do I choose the chi squared degrees of freedom?
Count the independent restrictions in the null hypothesis, not the total number of free parameters in the unrestricted model. A single scalar equality gives one degree of freedom.
When are the three statistics exactly equal?
For a normal mean with known variance, the log likelihood has constant quadratic curvature, so vertical drop, scaled distance and null slope all reduce to the same expression.
Exam move
Draw one likelihood curve and label the null, unrestricted estimate, constrained objective drop, estimator distance and null slope. Attach each test to the quantity it uses. In calculations, keep the constrained and unrestricted estimates distinct and evaluate both objectives with the same formula and precision. Count restrictions before selecting the reference distribution.
End every test with hypotheses, significance level, statistic, degrees of freedom, threshold or p-value and a restrained decision. Re-derive the known variance normal equality, then study how estimating variance bends the profile and explains the ordering Score less than or equal to Likelihood Ratio less than or equal to Wald.
Build a comparison sheet with one row per statistic and columns for required fit, evaluation point, variance or information estimate, and failure diagnostic. When nuisance parameters are present, write which remain free under the null before computing the constrained estimate. Verify that the unrestricted objective is never below the constrained objective and stop if a negative LR appears.
Practise translating one verbal restriction into a matrix form and calculate its independent rank rather than counting symbols. For each numerical exercise, recompute the decision from both a critical value and a p-value where possible. Conclude with evidence against the null, never with proof that the null is true.
Finally, label which equality or ordering is model-specific so a normal-profile result is not promoted to a universal theorem.
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