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
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is maximum likelihood fitting?
The process of applying maximum likelihood estimation to determine the parameter values of a chosen model that best fit observed data.
Source: Fisher 1922
What does maximum likelihood fitting do when applied to a model?
Maximum likelihood fitting is the practical act of applying maximum likelihood estimation to a chosen model, finding the parameter values under which the observed data are most probable. In practice this means writing down how likely the data are as a function of the unknown parameters and then searching, usually by computer, for the values that maximise it. Most standard models, from logistic regression to survival models, are fitted this way. Tuning a model to best explain the data is what it does. Kirkwood and Sterne (2003) describe this process.
Source: Kirkwood & Sterne 2003
How is maximum likelihood fitting carried out?
Maximum likelihood fitting is carried out by specifying the model and its likelihood function, then finding the parameter values that maximise the likelihood, usually by maximising the log-likelihood through calculus for simple models or numerical optimisation algorithms for complex ones, which iterate toward the maximum. So maximum likelihood fitting is carried out by optimising the likelihood over the parameters, often computationally, to obtain the estimates that best fit the data, and the process yields the parameter estimates and, from the curvature of the likelihood, estimates of their standard errors, providing the basis for inference about the fitted model.
Source: Fisher 1922
How does maximum likelihood fitting relate to maximum likelihood estimation?
Maximum likelihood fitting relates to maximum likelihood estimation as its application: maximum likelihood estimation is the principle of choosing parameters to maximise the likelihood, and maximum likelihood fitting is the process of carrying this out for a particular model and dataset to obtain the fitted parameters. So maximum likelihood fitting and maximum likelihood estimation are closely tied, with estimation being the underlying method and fitting the act of applying it to fit a model, and the terms are often used together, since fitting a model by maximum likelihood means estimating its parameters through the maximum likelihood principle.
Source: Fisher 1922
What models are fitted by maximum likelihood?
Many models are fitted by maximum likelihood, including generalised linear models such as logistic and Poisson regression, survival models, mixed models, latent variable models, and numerous others, since maximum likelihood provides a general estimation method applicable across model types. So maximum likelihood fitting is used for a wide range of models, from standard regressions to complex hierarchical and latent variable models, which is why it is a central estimation approach in statistics, providing a consistent way to fit diverse models to data, with the specifics of the likelihood and the optimisation varying by model but the underlying principle remaining the same.
Source: Fisher 1922
What are the challenges of maximum likelihood fitting?
The challenges of maximum likelihood fitting include that for complex models the likelihood may be difficult to maximise, with optimisation algorithms possibly failing to converge or reaching local rather than global maxima; that the model must be specified correctly; and that small samples can give unreliable estimates. So maximum likelihood fitting can pose computational and statistical difficulties, which is why care is taken with optimisation, starting values, and convergence, and why model assumptions and sample sizes are considered, since a poorly fitting or misspecified model, or a failed optimisation, can yield misleading estimates, making the fitting process one that requires attention rather than being purely automatic.
Source: Fisher 1922
Trust Record
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-110
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