Distributions: Overview

Distributions for Modeling Variability

Distributions that are used to model variability can be either discrete or continuous. Examples of discrete distributions include the binomial distribution, commonly used to model the occurrence or not of an event of interest from a total sample size, and the Poisson distribution, commonly used to model counts of events. Examples of continuous distributions that are used to model data variability are the normal distribution and gamma distribution. The modeling of variability is particularly important for discrete-event simulation (DES) models, which are often employed to look at service delivery methods that involve queuing problems. For example, patients might be assumed to arrive at an emergency room and queue up to see the receptionist before waiting to see a physician. Arrival times could be modeled as random while following an underlying exponential distribution, and different methods of organizing the procedures for receiving and attending to patients could be modeled to maximize throughput and minimize waiting time. More generally, individual patient simulation models describe medical decision models that model an individual's pathway through disease and treatment. Monte Carlo simulation is typically used to represent the stochastic nature of this process and is termed “first-order” simulation when the focus is on variability in the patient experience rather than uncertainty in the parameters.

Distributions for Modeling Parameter Uncertainty

The use of probability distributions to represent parameter uncertainty in decision models is known as probabilistic sensitivity analysis. Distributions are chosen on the basis of the type of parameter and the method of estimation. Monte Carlo simulation is then used to select parameter values at random from each distribution, and the model is evaluated at this set of parameter values. By repeating this process a large number of times, the consequences of uncertainty over the input parameters of the model on the estimated output parameters is established. In contrast to modeling variability, only continuous distributions are used to model parameter uncertainty. Monte Carlo simulation used in this way is termed “second order” to reflect the modeling of uncertainty of parameters. Probability parameters are commonly modeled using a beta distribution, since a beta distribution is constrained on the interval 0 to 1. Parameters such as cost of quality-of-life disutility, which are constrained to be 0 or positive, are often modeled using the log-normal or gamma distributions since these distributions are positively skewed and can only take positive values. Relative-risk parameters are often used as treatment effects in decision models and can be modeled using a lognormal distribution, reflecting the standard approach to the statistical estimation of uncertainty and confidence limits for these measures.

Central Limit Theorem

The normal distribution is of particular note for two reasons. First, it turns out that many naturally occurring phenomena (such as height) naturally follow a normal distribution, and therefore, normal distributions have an important role in modeling data variability. Second, the central limit theorem is an important statistical theorem that states that [Page 409]whatever the underlying distribution of the data, the sampling distribution of the arithmetic mean will be normally distributed with sufficient sample size. Therefore, the normal distribution is always a candidate distribution for modeling parameter uncertainty, even if the parameters are constrained (in technical terms, if there is sufficient sample size to estimate a parameter, the uncertainty represented as a normal distribution will result in negligible probability that a parameter will take a value outside its logical range).

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