Likelihood Ratio Test

Concept

Theoretically, Likelihood Ratio Test is a hypothesis test used to compare the goodness of fit of two nested statistical models by evaluating whether the addition of one or more parameters significantly improves model fit. It is founded on likelihood theory, whereby competing models are compared according to the probability of observing the available data under each model. In health economics, the likelihood ratio test is widely used during regression modelling, survival analysis, generalised linear modelling and other likelihood-based statistical analyses to determine whether additional explanatory variables or model structures are justified.

Mathematically, the likelihood ratio test compares the maximum likelihood values of a restricted model and a more general unrestricted model. The test statistic is based on twice the difference in their log-likelihoods and, under regularity conditions and the null hypothesis, follows an asymptotic chi-square distribution with degrees of freedom equal to the difference in the number of estimated parameters. Larger test statistics indicate stronger evidence that the unrestricted model provides a significantly better fit.

In practice, analysts estimate both nested models using maximum likelihood estimation, calculate the likelihood ratio statistic and compare it with the appropriate chi-square distribution or corresponding p-value. Within health economics, the test is commonly applied when evaluating alternative survival models, selecting regression covariates, comparing disease progression models and validating statistical models used to estimate costs, utilities or transition probabilities.

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Purpose

Used to determine whether a more complex nested statistical model provides a significantly better fit to observed data than a simpler model, thereby supporting model selection and validation in health economic analyses.

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Mathematical Formulae

Primary Formula

LR = ?2[ln L? ? ln L?]

where:

• L? = maximum likelihood of the restricted model
• L? = maximum likelihood of the unrestricted model

Under the null hypothesis,

LR ? ?�?df?

where

df = p? ? p?

• p? = number of parameters in the unrestricted model
• p? = number of parameters in the restricted model

Supporting Formulae

p-value:

p = P(?�?df? � LR)

Related Mathematical Methods

• Maximum likelihood estimation
• Nested model comparison
• Chi-square hypothesis testing
• Logistic regression
• Generalised linear models
• Survival analysis
• Akaike Information Criterion
• Bayesian Information Criterion

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Example

A health economist compares two nested logistic regression models predicting hospital admission.

The restricted model contains age and sex only, with

ln L? = ?482.3

The unrestricted model additionally includes comorbidity score, with

ln L? = ?474.6

The likelihood ratio statistic is

LR = ?2(?482.3 + 474.6) = 15.4

The unrestricted model estimates one additional parameter, so

df = 1

The corresponding p-value is less than 0.001, indicating that inclusion of comorbidity significantly improves model fit.

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Excel Implementation

Function	Example Formula	Health Economics Application
CHISQ.DIST.RT	=CHISQ.DIST.RT(B2,B3)	Calculates the p-value from the likelihood ratio statistic and degrees of freedom.
LN	=LN(B2)	Calculates log-likelihood values where required.
ABS	=ABS(B2-B3)	Examines differences in log-likelihoods between competing models.
IF	=IF(CHISQ.DIST.RT(B2,B3)<0.05,""Improved model"",""No significant improvement"")	Determines whether the unrestricted model significantly improves fit.

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VBA (Optional)

Automate estimation of nested models, calculation of likelihood ratio statistics and production of model comparison reports.

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Sources

• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• Casella G, Berger RL. Statistical Inference. 2nd ed.
• Dobson AJ, Barnett AG. An Introduction to Generalized Linear Models. 4th ed.
• Harrell FE. Regression Modeling Strategies. 2nd ed.
• Wilks SS. The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses. Annals of Mathematical Statistics. 1938;9:60?62.