6 Point Source Problems 6.1 Modelling spatial variation in risk 7 Continuous Spatial Variation

6.2 A conditional analysis

Suppose $\rho$ is parametrised by $\theta$ , and that $n$ cases and $m$ controls have been observed.

From Equation (6.1), and conditional on the locations of cases and controls, the probability of event-location $x$ being labelled as a case is $p(x)$ , while the probability of it being labelled as a control is $1-p(x)$ . The (conditional) density function of the case-control labels has Bernoulli form, and

L(\alpha,\theta)=\prod_{i=1}^{n}p(x_{i})\prod_{i=n+1}^{n+m}(1-p(x_{i})).

It is easy to show that when $\rho$ has the form given in Equation (6.1), the log-likelihood has the form

l(\alpha,\theta)=n\log\alpha+\sum_{i=1}^{n}\log\rho(x_{i}-x_{0};\theta)-\sum_{% i=1}^{n+m}\log(1+\alpha\rho(x_{i}-x_{0};\theta)).

(6.2)

We then maximise $l(\alpha,\theta)$ numerically.

6.2.1 Adjustment for further risk factors

In practice we would want to adjust for other known risk factors $z$ , possibly measured at the individual level. One way to do this would be to replace $\rho(x_{i}-x_{0};\theta)$ by

\rho(x_{i}-x_{0};\theta)\exp\left\{\sum_{j}\phi_{j}z_{j}(x)\right\}.

6.2.2 Adjustment for multiple point sources

The exposure model is also easily adjusted to account for multiple point sources, for example by replacing $\rho(x_{i}-x_{0};\theta)$ by

\prod_{k}\rho(x_{i}-x_{0k};\theta)

\prod_{k}\rho(x_{i}-x_{0k};\theta_{k}),

where the sources are located at $x_{0k}$ .

Example 6.1.

Asthma in north Derbyshire, England.

Diggle and Rowlingson (1994) fit the isotropic Gaussian model to data from the following case-control study of asthmatic symptoms in elementary schools in north Derbyshire.

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the study population consisted of all children attending one of 10 schools in the area;
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schools were stratified according to whether the headteacher had previously reported concern about the apparently high level of asthmatic symptoms in the school;
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four potential sources were considered – here, we look only at two:
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  a coking works (point source)
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  the main road network (line source)
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additional binary covariates included:
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  household includes at least one cigarette smoker?
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  child suffers from hay fever?
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overall risk was modelled multiplicatively, with separate terms for each of the two sources, and for log-linear covariate adjustment

Likelihood ratio comparisons:

Risk factors in model	$-2\times$ log-likelihood	Parameters
None	1165.9	2
Coking works	1160.7	4
Coking works, main roads	1160.6	6
Coking works, smoking	1159.4	5
Coking works, hay fever	1127.6	5
Hay fever only	1132.5	3

Conclusions:

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hay fever is biggest single risk factor
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proximity to coke works increases risk, with or without prior adjustment for hay fever
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no significant association with main roads, or with cigarette smoking

Diggle et al. (2000) give the adaptation of this methodology to matched case-control designs, and discuss associated classical and Bayesian methods of inference.