Diggle's Method: Example Case

Diggle and Rowlingson [1] considered cases of larynx cancers about an incinerator, using lung cancer patients in the same area as controls. The question was: is there evidence of elevated risk about the location of the incinerator?

The data are freely available from Peter Diggle's web site (case locations are labelled with 1, controls with a 0).

In the plot at left, the focus (location of the incinerator) is marked with the black "x" in the bottom: cases are red, controls blue).

Running GeoMed's Diggle routine with initial values of 

focusX:    354.5
focusY:    413.6
alpha :     23.67
beta  :       .91
yields the results 
Coordinates of the focus:
	(354.50,413.60)

Parameters of the fitted model:
	1+alpha*exp(-beta*x'x)

	alpha = 33.74344
	beta = 1.10214
	rho  = 0.05532

Model Fit:
	Maximized Likelihood = -219.214300
	Original  Likelihood = -223.540678
	Generalized Likelihood Ratio = 8.652756
	Nominal p-value = 0.013215
The probability p applies to the test of the null-hypothesis alpha=beta=0. We interpret these results as follows: the chance of getting a more extreme outcome for the model parameters is <= .013 (generally considered significant). Thus our model with two additional parameters is significant, and it appears that there is an elevation in the local risk about the toxic site.

GeoMed also produces a plot of the raised incidence model (which is isotropic - that is, the same in all directions) against the cases (in pink below) and controls (in light blue) by distance from the focus, which increased incidence plotted by bins (in black):

This gnuplot demo file shows the data and the fitted increased incidence function in 3-D.

The last word?
Elliot et al. [3] pursued Diggle's work further, increasing the scope of the investigation to include a large collection of similar incinerators, and they arrived at the conclusion that there was "...no evidence of association between cancer of the larynx or lung and incinerators of waste solvents and oils of the type found at Charnock Richard. The 'cluster' of cases of cancer of the larynx originally described near Charnock Richard was found following visual inspection of the data for a number of cancers. A statistically significant association with cancer of the larynx near Charnock Richard, in comparison with the geographical distribution of cancer of the lung, was found by Diggle (1990) [2] but not confirmed here using Stone's method.

"...as discussed by Diggle (1990) [2], his analysis could take no account of the post hoc nature of the original finding."

References:

  1. Diggle, P.J. and B.S. Rowlingson. A Conditional Approach to Point Process Modelling of Elevated Risk. Journal of the Royal Statistical Society, Series A (1994), 157, Part 3, pp. 433-40.
  2. Diggle, P.J. A point process modelling approach to raised incidence of a rare phenomenon in the vicinity of a prespecified point. Journal of the Royal Statistical Society, Series A (1990), 153, 349-62.
  3. Elliot, P., J. A. Beresford, D. J. Jolley, S. H. Pattenden, and M. Hills. Cancer of the larynix and lung near incinerators of waste solvents and oils in Britain. In: Geographical and Environmental Epidemiology: Methods for Small-Area Studies. Eds. P. Elliot, J. Cuzick, D. English, and R. Stern. Oxford University Press. 1992.
Website maintained by Andy Long. Comments appreciated.
longa@nku.edu