6 Point Source Problems 6 Point Source Problems 6.2 A conditional analysis

6.1 Modelling spatial variation in risk

Many studies in environmental epidemiology investigate disease incidence near one or more pre-specified pollution sources (‘point sources’, ‘line sources’, etc.). In this case, more tightly-constrained modelling of $\rho$ may be justifiable. Four plausible models for $\rho$ are given below.

6.1.1 ‘Near vs far’

An elevated (additive) level of risk within distance $\delta$ of a point source (Elliott et al., 1992).

Figure 6.1: Link, Caption: Illustrating the ‘near vs far’ model for

\rho

. With this model the risk (

y

-axis) at a given distance (

x

-axis) is a single step at some distance

\delta

: the risk at distance greater than

\delta

is equal to

1

and at distances less than

\delta

is equal to

1+\epsilon

. The parameter

\delta

can either be fixed or estimated alongside

\epsilon

, which denotes the elevated risk within distance

\delta

of source: clearly this is a very basic model for risk.

\rho(u)=\left\{\begin{array}[]{ll}1+\epsilon&u\leq\delta\\ 1&u>\delta\end{array}\right.

6.1.2 Isotonic Regression

A more general version of ‘near vs far’, with several levels of risk (Stone, 1988) might set $\rho(x)=\rho(||x-x_{0}||)$ , monotone non-increasing (the form of $\rho$ would have to be specified)

Figure 6.2: Link, Caption: Illustrating an isotonic regression model for

\rho

. With this model the risk (

y

-axis) at a given distance (

x

-axis) is a monotone decreasing step function. Although this model has been used in practice (Stone, 1988), as you might imagine, without strong apriori beliefs about the number and location of the steps, these models can be very challenging inferentially.

The above models are simple and parsimonious, but possibly unrealistic. It is more plausible to think that risk will vary continuously according to distance $u$ and direction $\theta$ from the source. A general formulation of a model that takes into account these factors is

\rho(u,\theta)=1+\beta\exp(-f(u,\theta)^{2}),

where $f$ is a function to be specified, as in the two further examples below

6.1.3 Isotropic Gaussian

Continuous decay in risk, depending on distance (but not direction) from the point source (Diggle and Rowlingson, 1994).

Figure 6.3: Link, Caption: Illustrating the isotropic Gaussian model for

\rho

. With this model the risk (

y

-axis) at a given distance (

x

-axis) takes the form

\rho(u)=1+\beta\exp\{-(u/\delta)^{2}

. The parameter

\beta

is the elevated risk at the source and

\delta

governs how quickly the elevated risk decays.

f(u,\theta)=u/\delta

Thus

\rho(u)=1+\beta\exp\{-(u/\delta)^{2}\}

6.1.4 Directional Plume Model

Continuous decay that depends both on distance and direction from the source (Lawson and Williams, 1994).

f(u,\theta)=u\exp\{-\kappa\cos(\theta-\phi)\}/\delta

Thus

\rho(u,\theta)=1+\beta\exp(-[u\exp\{-\kappa\cos(\theta-\phi)\}/\delta]^{2})

Interpretation of model parameters:

•

$\beta$ : elevation in risk at source
•

$\delta$ : rate of decay of risk with distance from source, as in the isotropic model
•

$\phi$ : direction of plume
•

$\kappa$ : degree of directional concentration of plume

This anisotropic model is most often used for modelling the flow of airborne pollutants.

Figure 6.4: Link, Caption: Illustrating a directional plume model for

\rho

. The plots show the form of the risk function four different values of the parameter

\kappa

, which controls the degree of directional concentration of the plume. Clockwise from the top left plot shows the risk surface for

\kappa=

0, 1, 2, 6; in each of these plots, darker areas correspond to greater risk and the lightest areas correspond to areas of no additional risk (i.e.

\rho(u)\approx 1

). When

\kappa=0

(top left), the model reduces to an isotropic Gaussian hence the concentric form of the plot. As

\kappa

increases, the eccentricity increases and also the directional effect.

It is important to emphasise that initially specifying $\rho$ usually amounts to determining a suitable functional form, which depends to a great extent on the context of the exposure being modelled. Once the form of $\rho$ has been determined, its parameters are then estimated using the case-control data. Testing the significance of estimates these parameter then reveals information about the likely elevation in risk according to distance and/or direction from the source. More complex (multi-parameter) models for $\rho$ inevitably require more case-control data in order for their parameters to be estimated precisely.