Concept Architecture
Concept
Theoretically, Maximum Likelihood Estimation (MLE) is a statistical estimation method that determines the values of unknown model parameters that maximise the probability of observing the available data. It is founded on likelihood theory and provides a general framework for parameter estimation across a wide range of statistical models. MLE exists to identify the parameter values that make the observed sample most probable under a specified probability model and forms the basis of modern statistical inference.
Mathematically, Maximum Likelihood Estimation constructs a likelihood function from the joint probability distribution of the observed data and estimates parameters by maximising this function. Because likelihoods often involve products of probabilities, the natural logarithm of the likelihood is usually maximised instead, producing an equivalent optimisation problem that is computationally more stable. Estimates are typically obtained using iterative numerical optimisation algorithms such as Newton-Raphson, Fisher Scoring or Expectation-Maximisation when closed-form solutions are unavailable.
In practice, MLE is used to estimate parameters in regression models, survival analysis, mixed-effects models, latent variable models, discrete choice models and many other statistical techniques. In health economics, MLE is fundamental to logistic regression, Cox proportional hazards models, multinomial logit models, generalised linear models and structural equation models. The resulting parameter estimates support hypothesis testing, prediction, probabilistic sensitivity analysis and economic decision modelling.
Purpose
Used to estimate unknown model parameters by maximising the likelihood of the observed data, providing statistically efficient estimates for a wide range of models used in health economics and health outcomes research.
Mathematical Formulae
Primary Formula
Likelihood function:
L(?) = ????� f(x? | ?)
Maximum Likelihood Estimator:
?? = arg max? L(?)
Supporting Formulae
Log-likelihood:
?(?) = ln L(?) = ????� ln[f(x? | ?)]
Score function:
U(?) = ??(?) / ??
Observed Fisher Information:
I(?) = ??�?(?) / ??�
Asymptotic variance:
Var(??) � I(??)??
Related Mathematical Methods
- Likelihood Ratio Test
- Fisher Information
- Maximum A Posteriori Estimation
- Generalised Linear Models
- Logistic Regression
- Cox Proportional Hazards Model
- Expectation-Maximisation Algorithm
Example
A health economist models the probability of hospital readmission using logistic regression.
The likelihood function is constructed from the observed readmission outcomes of 2,000 patients.
Using Maximum Likelihood Estimation, the estimated treatment coefficient is:
?? = ?0.58
The corresponding odds ratio is:
OR = e??�?? = 0.56
This indicates that the intervention is associated with approximately a 44% reduction in the odds of readmission. The coefficient estimate is obtained by maximising the log-likelihood across all observed patient outcomes.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LN | =LN(B2) | Calculate log-likelihood contributions for individual observations. |
| SUM | =SUM(C2:C2001) | Calculate the total log-likelihood. |
| EXP | =EXP(A2) | Convert regression coefficients to odds ratios or probabilities. |
| Solver | Maximise the total log-likelihood by varying parameter values. | Estimate model parameters using Maximum Likelihood Estimation. |
| IF | =IF(D2>0.5,1,0) | Classify predicted outcomes during likelihood-based modelling. |
VBA (Optional)
A VBA routine can automate iterative likelihood maximisation, monitor convergence criteria and generate parameter estimates with associated standard errors for likelihood-based statistical models.
Sources
- Casella G, Berger RL. Statistical Inference. Cengage Learning.
- Pawitan Y. In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford University Press.
- Cox DR, Hinkley DV. Theoretical Statistics. Chapman & Hall.
- McCullagh P, Nelder JA. Generalized Linear Models. Chapman & Hall.
- Greene WH. Econometric Analysis. Pearson.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is maximum likelihood estimation?
A method estimating a model's parameters by finding the values that make the observed data most probable.
Source: Fisher 1922
What values does maximum likelihood estimation choose for a model?
Maximum likelihood estimation chooses the parameter values that make the observed data most probable under the assumed model. It treats the data as fixed and searches for the settings under which those particular observations would have been most likely to arise, taking those as the best estimates. This principle produces estimators with good large-sample properties and applies across a vast range of models, from simple proportions to complex regressions. Picking the parameters that best explain the data is its logic. Kirkwood and Sterne (2003) describe this method.
Source: Kirkwood & Sterne 2003
How does maximum likelihood estimation work?
Maximum likelihood estimation works by constructing the likelihood function, which expresses the probability of the observed data given the parameters, and finding the parameter values that maximise it, often by maximising the log-likelihood for convenience, using calculus or numerical optimisation. So maximum likelihood estimation works by identifying the parameter values that make the observed data most probable, typically by setting the derivatives of the log-likelihood to zero or optimising numerically when no closed form exists, which yields the maximum likelihood estimates, the parameter values best supported by the data under the assumed model.
Source: Fisher 1922
What are the properties of maximum likelihood estimators?
Maximum likelihood estimators have desirable large-sample properties: they are consistent, converging to the true parameter as the sample grows; asymptotically efficient, achieving the smallest possible variance in large samples; and asymptotically normal, so their distribution is approximately normal, enabling standard errors and confidence intervals. So maximum likelihood estimators are valued for their consistency, efficiency, and asymptotic normality, which make them reliable and allow standard inference in large samples, though these properties are asymptotic, so in small samples the estimators can be biased or their normal approximation poor, which is considered when applying maximum likelihood to limited data.
Source: Fisher 1922
Why is maximum likelihood estimation widely used?
Maximum likelihood estimation is widely used because it is a general principle applicable to a vast range of models, it has strong statistical properties such as efficiency in large samples, and it provides a unified basis for estimation and inference, including standard errors from the likelihood. So maximum likelihood estimation is widely used for its generality and good properties, underpinning the fitting of many models, from generalised linear models to complex ones, which is why it is a central pillar of statistical estimation, providing parameter estimates together with the basis for likelihood-based tests and confidence intervals across diverse applications.
Source: Fisher 1922
What are the limitations of maximum likelihood estimation?
The limitations of maximum likelihood estimation include that its desirable properties are asymptotic, so it can be biased or unreliable in small samples; that it requires specifying the model and distribution correctly, since misspecification can mislead; and that maximising the likelihood can be computationally difficult for complex models. So maximum likelihood estimation is used with awareness that it depends on the model being reasonably correct and on adequate sample sizes for its large-sample properties, and that computation may be challenging, which is why model assumptions are checked, small-sample corrections are sometimes applied, and alternative or robust methods are considered when the assumptions are doubtful.
Source: Fisher 1922
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 18 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-109
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