Likelihood Ratio Confidence Interval

A likelihood ratio confidence interval is a range of parameter values for which the likelihood function is not too much smaller than its maximum, providing an alternative to Wald or bootstrap intervals. It is constructed by finding all values of the parameter that satisfy a likelihood ratio test criterion, typically using a chi-square threshold.

Formula

The likelihood ratio confidence interval is based on the likelihood ratio statistic:

LR(θ) = 2 [ℓ(θ̂) − ℓ(θ)]

where ℓ is the log-likelihood function, θ̂ is the maximum likelihood estimate (MLE), and θ is a candidate parameter value. The interval includes all θ such that LR(θ) ≤ χ²₁(1−α), the (1−α) quantile of a chi-square distribution with 1 degree of freedom (for a single parameter). For example, with α = 0.05, χ²₁(0.95) ≈ 3.841.

Worked Example

Suppose we observe 10 successes out of 20 trials, modeled as Binomial(n=20, p). The MLE is p̂ = 10/20 = 0.5. The log-likelihood is ℓ(p) = 10 ln(p) + 10 ln(1−p). At p=0.5, ℓ(0.5) = 10 ln(0.5) + 10 ln(0.5) = 20 ln(0.5) ≈ −13.8629. Now find p values where ℓ(p) ≥ ℓ(0.5) − 1.9207 (since 3.841/2 = 1.9207). Solving numerically gives p ∈ [0.271, 0.729]. Thus the 95% likelihood ratio confidence interval for p is approximately (0.271, 0.729).

When to Use a Likelihood Ratio Interval

Use a likelihood ratio interval when you have a parametric model and want an interval that respects the parameter space (e.g., probabilities between 0 and 1) and has better coverage than Wald intervals in small samples. It is especially useful for:

• Generalized linear models (logistic, Poisson regression)
• Parameters with asymmetric likelihoods
• Small to moderate sample sizes where the normal approximation is questionable

How to Read the Result

The interval gives the set of parameter values that are “plausible” given the data, in the sense that they would not be rejected by a likelihood ratio test at the chosen significance level. A wider interval indicates more uncertainty; a narrow interval indicates more precision. The interval is not symmetric around the MLE unless the likelihood is symmetric.

Common Mistakes and Misconceptions

• Using the Wald interval (θ̂ ± z·SE) when the likelihood is skewed; the likelihood ratio interval is often more accurate.
• Forgetting that the chi-square threshold depends on the number of parameters: for a single parameter use df=1, for a vector use df = number of parameters.
• Interpreting the interval as a probability statement about the parameter; it is a frequentist interval, not a Bayesian credible interval.
• Using the profile likelihood for one parameter but ignoring that the threshold remains chi-square with 1 df.

Comparison with Other Intervals

Wald intervals are simple but can produce values outside the parameter space (e.g., negative probabilities). Bootstrap intervals are flexible but require computational resampling. Likelihood ratio intervals are invariant to reparameterization, meaning they give the same result whether you work with p or logit(p), which is not true for Wald intervals. They are generally preferred when the sample size is small or the likelihood is highly skewed.

Limitations and Practical Notes

Likelihood ratio intervals require a correctly specified likelihood and numerical optimization to find the cutoff. They can be computationally intensive for complex models. Also, the chi-square approximation relies on large-sample theory; for very small samples, the coverage may not be exact. In such cases, simulation or exact methods may be needed.

Frequently Asked Questions

How is a likelihood ratio interval different from a confidence interval?

It is a type of confidence interval, but constructed via the likelihood ratio test rather than the normal approximation. It often has better coverage properties, especially for small samples or non-normal likelihoods.

Can I use a likelihood ratio interval for any parameter?

Yes, as long as you can write the likelihood function for the parameter and compute the MLE. It works for single parameters and for functions of parameters (using profile likelihood).

What does “profile likelihood” mean?

When there are nuisance parameters, you maximize the likelihood over them for each fixed value of the parameter of interest. The resulting function is the profile likelihood, and the likelihood ratio interval uses this profile.

Why does my interval sometimes not contain the MLE?

That would be unusual; by construction the MLE maximizes the likelihood, so it always satisfies the condition LR(θ̂) = 0 ≤ threshold. If you see this, check your calculations.

Is the likelihood ratio interval always better than the Wald interval?

Not always, but it is generally more reliable for small samples and skewed likelihoods. The Wald interval is simpler and often adequate for large samples with approximately normal likelihoods.

How do I compute it in practice?

Use statistical software like R (confint function with profile likelihood), Python (statsmodels or scipy), or specialized packages. You can also implement the numerical search yourself by finding the roots of LR(θ) − χ²₁(1−α) = 0.

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