Distribution Fitting

Distribution Fitting: Concept Architecture

Describe variation with a probability model

Distribution fitting chooses a probability family and estimates its parameters from data so that it represents the observed variation. This page follows the choice of family, estimation, checks of fit, and consequences when the fitted distribution supplies inputs to a health economic model.

Match the distribution to the variable

Specify what is measured, its units, observation period, and possible values before fitting. Positive costs, proportions bounded between zero and one, counts, and event times have different supports and data-generating processes. A normal distribution may assign impossible negative costs; a gamma model for strictly positive costs cannot by itself represent a substantial mass of zero costs.

Check missingness, censoring, truncation, repeated measurements, and whether values are patient-level outcomes or estimates of uncertain parameters. A distribution of individual patient costs represents variation between people, whereas a distribution assigned to an estimated mean in probabilistic sensitivity analysis represents uncertainty about that mean. The NIST handbook describes fitting a probability distribution to univariate data.

Estimate parameters from observed values

After selecting a defensible family $f(x\mid\boldsymbol{\theta})$, estimate its parameters using a criterion suited to the data and observation process. Under independent fully observed values, maximum likelihood chooses parameters that maximize their joint likelihood:

$$\widehat{\boldsymbol{\theta}} =\underset{\boldsymbol{\theta}}{\operatorname{argmax}} \sum_{i=1}^{n}\log f(x_i\mid\boldsymbol{\theta}).$$

This expression must be modified for censored event times, grouped data, dependence, or a mixture with zeros. Other approaches include moments and Bayesian estimation, with different assumptions and uncertainty descriptions. The NIST maximum-likelihood reference distinguishes estimating parameters from subsequently checking goodness of fit.

Work through a small cost example

Suppose five illustrative annual costs are £0, £0, £100, £200, and £700. Two people have no cost, while the positive values are right-skewed; the sample mean is £200. A single gamma distribution for all five observations is unsuitable because it gives no point mass at zero.

A simple two-part description would estimate a 2/5 probability of zero and fit a suitable positive-cost distribution to £100, £200, and £700. The mean positive cost in this tiny sample is £1,000/3, so multiplying by the 3/5 probability of positive cost recovers the sample mean of £200. This arithmetic illustrates the components, not a reliable fitted model from only five observations. Research on semicontinuous healthcare costs discusses the mixture of zeros and positive skew.

Check fit where decisions depend on it

Compare empirical and fitted densities or cumulative probabilities, use quantile or probability plots, and examine tails and subgroup patterns. A single summary score or test cannot establish that extrapolation beyond observed data is credible. The NIST probability-plot guide shows a graphical check for candidate families.

For survival or other long-term projections, compare predicted values with clinical knowledge and external evidence, not only in-sample fit. NICE guidance calls for clinical and biological plausibility and coherence with external data when evaluating extrapolation. Explore plausible alternative families when their tails materially change costs, quality-adjusted life years, or the preferred strategy.

Use the fitted result carefully

Report the dataset and population, family and parameters, treatment of zeros and censoring, estimation method, diagnostics, and uncertainty in estimated parameters. In simulation, sample patient-level values from the individual distribution only for the process it represents; carry uncertainty in its fitted parameters separately when relevant.

Distribution fitting differs from general curve fitting because it specifies probabilities over possible values, not merely a function that follows a plotted trend. A smooth visual fit does not establish causal validity, transportability, or a trustworthy unobserved tail. Test how the final decision changes under defensible alternatives.