Gamma Distribution

Gamma Distribution: Concept Architecture

Model positive, often right-skewed values

The gamma distribution is a continuous probability family for positive values whose shape can range from strongly right-skewed to more nearly symmetric. This page explains its parameters, calculates them from a mean and standard deviation, and shows when a gamma model is useful or unsuitable for health economic data.

Define the shape and scale convention

Let $X$ follow a two-parameter gamma distribution with shape $k>0$ and scale $\theta>0$. Its probability density for $x>0$ is:

$$f(x\mid k,\theta) =\frac{x^{k-1}e^{-x/\theta}} {\Gamma(k)\theta^k}, \qquad x>0.$$

Here $\Gamma(k)$ is the gamma function, which normalizes the density. The mean is $E[X]=k\theta$ and the variance is $\operatorname{Var}(X)=k\theta^2$. Some software instead specifies a rate $r=1/\theta$; substituting a rate where a scale is expected changes the distribution. The NIST gamma reference displays both conventions.

Calculate parameters for an illustrative cost input

Suppose an analyst assigns a positive cost input a mean of £200 and a standard deviation of £100. Matching those two moments gives:

$$k=\frac{\mu^2}{\sigma^2}=4, \qquad \theta=\frac{\sigma^2}{\mu}=£50.$$

The resulting gamma model has mean $4(£50)=£200$ and standard deviation $\sqrt{4(£50)^2}=£100$. The parameters and monetary units are illustrative; shape is dimensionless, while scale has the same units as the outcome. NIST provides the corresponding moment-estimation relationships for positive data in its distribution-fitting guidance.

Choose the role of the distribution

One use is to describe variation in positive individual-level costs or resource quantities. Another is to represent uncertainty about a positive model parameter in probabilistic sensitivity analysis. Those are distinct levels of uncertainty: the standard deviation between patients need not equal the standard error of an estimated mean cost.

In a generalized linear model, a gamma response distribution can also be paired with a suitable link, often a log link, to estimate conditional mean costs. This regression specification is not the same as merely drawing uncertain parameter values from a gamma distribution. Methods research on healthcare cost analysis discusses gamma generalized linear models and alternative response families.

Check support, zeros, and tails

The ordinary two-parameter gamma distribution represents values greater than zero and assigns no point mass to exact zero. Healthcare expenditure data often contain non-users with zero cost; fitting a gamma density to every observation without a separate zero mechanism is inappropriate for that setting. A two-part or another appropriate model can handle zero and positive portions explicitly.

Inspect the observed frequency of zeros, right tail, subgroup patterns, and fit to the decision-relevant mean. Compare plausible alternatives such as a lognormal model when tails materially affect cost-effectiveness. Comparative research on cost distributions finds that model choice can affect economic conclusions; a familiar distribution is not automatically the best fit.

Report assumptions that matter to the decision

State whether the gamma describes people, uncertain parameters, or regression residual variation; give the shape–scale or shape–rate convention, units, source data, estimation method, and any zero-handling rule. If correlated parameters enter a model, independent gamma draws can break meaningful relationships and require a joint approach.

The distribution's positive support and adjustable skew are useful, but they do not validate extrapolation beyond observed costs or fix missing and biased data. Show how uncertainty about the shape, scale, and choice of distribution affects expected cost and the final decision.