**Quantitative Methods of Uncertainty Analysis**

The WHO (2008) presented three levels of quantitative uncertainty analysis; briefly summarized here. These levels of uncertainty analysis correspond to the tiered approaches (discussed later in this section) presented with detailed examples.

### Quantifying Variability

When only variability is quantified, the output is a single distribution representing a 'best estimate' of variation in the model output.

This approach can be used to make **estimates for different percentiles of the distribution, but provides no confidence intervals**; which may lead to a false impression of certainty (WHO, 2008).

### 1D Monte Carlo

Inputs (e.g. parameteres or data) to the model have distributions that represent both variability and uncertainty. These input distributions are combined in the output as a single distribution representing a mixture of variability and uncertainty.

This approach can be interpreted as an uncertainty distribution for the exposure of a single member of the population selected at random (i.e.*"the probability of a randomly chosen individual being exposed to any given level"*)

### 2D Monte Carlo

Is similar to the 1D approach, but instead, variability and uncertainty are propagated in the model and shown separately in the output.

For example, the output is typically presented as three cumulative curves: a central one representing the median estimate of the distribution for variation in exposure, and two outer ones representing lower and upper confidence bounds for the distribution.

Interpreted as: *"Exposure estimates for different percentiles of the population, together with confidence bounds showing the combined effect of those uncertainties"*.