a point process modelling approach to raised incidence of a rare phenomenon in the vicinity of a...

A Point Process Modelling Approach to Raised Incidence of a Rare Phenomenon in theVicinity of a Prespecified PointAuthor(s): Peter J. DiggleSource: Journal of the Royal Statistical Society. Series A (Statistics in Society), Vol. 153, No. 3(1990), pp. 349-362Published by: Wiley for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2982977 .

Accessed: 28/06/2014 11:53

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Wiley and Royal Statistical Society are collaborating with JSTOR to digitize, preserve and extend access toJournal of the Royal Statistical Society. Series A (Statistics in Society).

http://www.jstor.org

This content downloaded from 91.223.28.76 on Sat, 28 Jun 2014 11:53:39 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=black

http://www.jstor.org/action/showPublisher?publisherCode=rss

http://www.jstor.org/stable/2982977?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


J. R. Statist. Soc. A (1990) 153, Part 3, pp. 349-362

A Point Process Modelling Approach to Raised Incidence of a Rare Phenomenon in the Vicinity of a Prespecified Point

By PETER J. DIGGLEt

Lancaster University, UK

[Received September 1989. Revised January 1990]

SUMMARY Motivated by the current debate on possible raised incidence of certain types of cancers near nuclear installations, this paper develops a methodology for fitting a class of inhomogeneous Poisson point process models to data consisting of the locations of all occurrences of some phenomenon of interest within a designated planar region. The model is based on a multiplicative decomposition of the intensity function, with separate terms to describe natural spatial variation in intensity and possible raised incidence around a prespecified point. A nonparametric kernel smoothing approach, based on data from a related phenomenon, is used to describe natural spatial variation, while a parametric maximum likelihood approach is used to describe raised incidence near the prespecified point. The methodology is applied to data on the spatial distribution of cancers of the larynx and of the lung in the Chorley-Ribble area of Lancashire, England.

Keywords: EPIDEMIOLOGY; KERNEL ESTIMATION; POISSON POINT PROCESS; SPATIAL POINT PATTERN

1. INTRODUCTION

A topic of growing interest worldwide is the investigation of possible raised incidence of cases of a rare disease around a putative source of environmental pollution. For example, the recent identification of a 'cluster' of leukaemia cases in the village of Seascale, Cumbria, England, led to a government investigation into whether this unusually high incidence might be linked with the nearby nuclear installation at Sellafield (Black, 1984). The Black report concluded that, although the incidence of leukaemia in Seascale was unusually high, there was no evidence of a causal relationship between increased radioactivity and increased leukaemia. See also Wakeford et al. (1989).

In May 1989 the Royal Statistical Society held an Ordinary Discussion Meeting on the topic of 'Cancer near nuclear installations', the proceedings of which were published in the Society's journal, series A (Muirhead and Darby, 1989). At that meeting, I outlined a possible methodological contribution to this class of problems. The present paper is an elaboration of that contribution. My aim is to clarify and exemplify the methodology, rather than to provide definite answers to the substantive questions raised-which are both profoundly important and enormously complex.

In Section 2 I formulate a Poisson point process model for spatial variation in the intensity of a relatively rare phenomenon near a prespecified point source. The model

tAddress for correspondence: Department of Mathematics, Lancaster University, Bailrigg, Lancaster, LAI 4YL, UK.

? 1990 Royal Statistical Society 0035-9238/90/153349 $2.00



350 DIGGLE [Part 3,

incorporates collateral information on the nature of the spatial variation which would be expected in the absence of any association with the point source. For motivation, I consider the spatial distribution of cancers of the larynx in the Chorley and South Ribble Health Authority, Lancashire, England, where there is an apparent cluster of cases near a now-disused industrial incinerator and where the collateral information is provided by the spatial distribution of the much more common cancers of the lung. Section 3 gives the details of maximum likelihood estimation and goodness-of-fit assessment. Closed form expressions are given for the log-likelihood function and its derivatives in a useful special case. Section 4 describes the application of the methodology to the Chorley-Ribble cancer data. Section 5 discusses several open methodological issues.

2. THE MODEL

2. 1. Motivation We consider data in the form of a set of events xi in some planar region A, these

representing the locations of all occurrences within A of some phenomenon of interest. The phenomenon is known to exhibit substantial spatial variation in intensity over the region A. Additionally, there is a suspicion that the phenomenon may be associated with a prespecified point x0 in A, specifically that the intensity may be higher in the vicinity of x0. Our primary aim is to investigate this possible association.

For example, Fig. 1 shows the locations of all 58 cases of cancer of the larynx recorded in the Chorley-Ribble area during the years 1974-83 inclusive (data provided by Dr Tony Gatrell, Department of Geography, Lancaster University). Co-ordinates are Ordnance Survey map references. Also shown is the location of a

0

LO - o

C)

o

O

cm 0~~~~~~~~~~~~~~~~~~~~~~~~~~

cmJ

C)

34000 34500 35000 35500 36000 36500

Fig. 1. Spatial distribution of 58 cases of cancer of the larynx in the Chorley-Ribble area, and the location (+) of a now-disused industrial incinerator



1990] POINT PROCESS MODELLING APPROACH TO RAISED INCIDENCE 351

now-disused industrial incinerator. Note the apparent cluster of four cases near the incinerator. Is this sufficient to establish an association between the disease and the incinerator?

In seeking to answer this question, three important methodological issues are the following:

(a) because the phenomenon of interest is relatively rare, and the precise spatial scale of any possible association is not known a priori, it is desirable to preserve the continuous spatial setting of the data;

(b) even in the absence of any association with the prespecified point (here, the location of the incinerator), the natural forces of environmental heterogeneity will impart a substantial amount of spatial variation in the intensity of the phenomenon being studied;

(c) it is not clear how we should define what constitutes a cluster, and any arbitrariness in this definition should be avoided as far as possible.

The first issue leads us to adopt a point process model. This is in contrast with the usual approach of aggregating the data into counts in discrete spatial units, e.g. local government administrative units, and using the methods of generalized linear modelling (see, for example, Forman et al. (1987) and Cook-Mozaffari et al. (1989)).

The second issue leads us to look for collateral information on the kind of spatial variation in intensity which we would expect in the absence of any association with the prespecified point. One obvious form of such information would be census data on population figures within the region A. A possible disadvantage is that census data are available only in aggregated form. Also, crude population figures may not be the most appropriate form of information, nor may it be clear how best to adjust them. A less obvious, and perhaps more controversial, form of collateral information is the spatial distribution of all incidences of a more common phenomenon which is thought to reflect well the natural spatial distribution in the phenomenon of interest, but which can be assumed not to be associated with the prespecified point.

For example, Fig. 2 shows the locations of all 978 cases of cancer of the lung recorded in the Chorley-Ribble area during 1974-83. This map well reflects the spatial variation in population intensity over the area but also, presumably, the more subtle spatial variation in the relevant risk groups, e.g. in age distribution. In what follows, we shall assume that there is no association between cancer of the lung and the industrial incinerator. Whether this is reasonable is a matter for medical debate.

With regard to the third issue, namely the concept of a cluster, the weight of evidence for association with the prespecified point should presumably derive from a general excess of events around the point, rather than from a cluster of mutually close events per se. Furthermore, in general we have no basis a priori for determining the spatial scale on which the phenomenon of excess events can be expected to operate. Our approach is therefore to describe the possible excess by a continuous-valued function of location, with separate parameters to describe the magnitude and spatial scale of the excess.

2.2. Poisson Point Process Model We suppose the data {xi E A: i = 1, . . . , n} to be a partial realization of an inhomo-

geneous spatial Poisson point process with intensity function X(x), i.e. X(x) represents



352 DIGGLE [Part 3,

LO co

C-

coJ

I I ., . ;,

34000 34500 35000 35500 36000 36500

Fig. 2. Spatial distribution of 978 cases of cancer of the lung in the Chorley-Ribble area

the mean number of events per unit area in the immediate vicinity of x (Diggle, 1983) . We assume that X\(x) has a multiplicative decomposition into

A(x) = pX0o(x) f (x - x0; 0). (1 )

In equation (1), p reflects the overall number of events per unit area, X\0(.) represents the spatial variation in intensity in the absence of association with the prespecified point x0 and f (.) describes the change in intensity with position relative to x0. In particular, we shall parameterize f(.) so that f(x; 0) = 1 for all x; in other words 0 = 0 represents the hypothesis of no association. The model falls within the class of modulated Poisson processes introduced by Cox (1972).

2.3. Specification of X0(x) We suppose that a second set of data {y1 E A: j = 1 , .. ., m} is available as a partial

realization of an inhomogeneous Poisson point process with intensity function X0O(x). A natural way to use the y1 to estimate kO(x) is via a kernel estimator,

m

Xo(x; h)= h-2Z G{(x - y)/h} . (2) j=1

In equation (2) G(.) is a radially symmetric bivariate probability density function and h > 0 is a tuning constant, larger values of which impart greater smoothness to the estimate k<(x; h). For a general discussion of kernel estimation, see Silverman (1986).

Various approaches exist for choosing the value of h in equation (2). Diggle (1985) and Berman and Diggle (1989) develop a method which seems well suited to the present context. Diggle (1985) assumes that X\0(x) is itself a realization of a stationary random process and derives an expression for the mean-square error,




MSE(h) = E[{ Xo(x; h) - Xo(x)}2], which involves the second-moment properties of X0(x). These in turn can be estimated from the data {yj }, and the resulting estimate of MSE(h) can be minimized over h. Berman and Diggle (1989) give details. Diggle and Marron (1988) show that this method of choosing h is essentially equivalent to a least squares cross-validation technique introduced in the context of nonparametric probability density estimation by Bowman (1984).

The choice of kernel function G(.) in equation (2) is much less critical than the choice of h. In the present context, it will be convenient to use a Gaussian kernel,

G(x) = (2ir) exp { - 2 x'x}. (3)

Formulae in Diggle (1985) and Berman and Diggle (1989) apply specifically to a uniform kernel. Following Diggle (1985) we calibrate different kernels by equating the expected squared radial distance. Thus, if ho represents the optimum choice of h for a uniform kernel,

G(x) = fir (x'x 1) LO (x'x > 1),

the corresponding optimum for the Gaussian kernel (3) is h0/2.

2.4. Specification of f(x; 0) In principle, the function f(.) in equation (1) can be quite general. In practice, the

case of most interest will be when f(.) is unimodal, with a maximum at x = 0 and decaying smoothly towards a constant value as x moves away from 0. The value of this constant is arbitrary in view of the multiplicative constant p in equation (1), and we can assume that f(x; 0) 1-+ as x'x -+ oo.

In most applications, the dimensionality of 0 will be at least 2, unless we have good reason to prespecify either the magnitude or the spatial scale of the excess of events. For example, we might take 0 = (r, 3) and set

f(x; a, j) = 1 + a exp -3g(x'x)},

where ae >, 0, 3 0 and g(.) is monotone non-decreasing with g(O) = 0.

3. MAXIMUM LIKELIHOOD ESTIMATION AND GOODNESS-OF-FIT ASSESSMENT

3.1. Maximum Likelihood Estimation Maximum likelihood estimation for model (1) is straightforward in principle,

following Cox (1972). Given 0, the explicit maximum likelihood estimate of p is

pi(0) = n Xo(x) f(x - xo; 0) dx (4)

whereas the profile log-likelihood for 0, the set of parameters of primary interest, is

n L (0)=Zlog {f(xi -xo; 0)} -nlog~Xo (x) f(x -xo; 0)dxJ (5)



354 DIGGLE [Part 3,

In equations (4) and (5), integrations are over A or, equivalently, over R2 but setting XO(x) = 0 outside A.

The computational effort involved in maximizing equation (5) clearly depends on the particular specifications of X\0() and f(.). If we replace X0(.) by its Gaussian kernel estimate (2) as described in Section 2.3, then a convenient choice of f(.), with 0 (o, ), is

f(x; a,3)= 1 + a exp(- fx'x). (6)

This is unashamedly pragmatic, but is qualitatively reasonable. Using the Gaussian kernel estimate (2) for X0(.) and the specification (6) for f(.), we

obtain the profile log-likelihood for (cx, j), and its derivatives, as follows. Let ei denote the squared distance from xi to xo, and dj the squared distance from yj to xo. Define functions

- exp { dj/(23h2 + 1)} 23h 2 + 1

vj(0) = (dj + 2h2 + 4f3h4)/(213h2 + 1)2,

m

w(ax,13)= 1 + am- '3 uj(j3). j=1

Note that

uj(i3) = - u1(3) v1(3),

v3(0) = - 4h2(h2 +dj + 2fh4)/(2f3h2 + 1)3,

_9 (ag 4) = m 1 mu(A j=1

aw (c A) = -nm-' m

ao j~~~~=1

Then,

n

L(ax,3) E log{1 + a exp(-flei)} - n log{w(cx,j)}. (7) i=1

Expression (7) follows from writing

m

Xo(x) f(x - xo; oc, A) dx = 1 + mZ- ' EIj( ,), (8) j=1

where

Ij(x, ) =||(27rh 2)-'la exp [- 2 h -2(X2 + y2) - 0 {(x - dj/'2)2 + y2}] dx dy

exp { - fdj/(2f3h2 + 1)} (9) 2 +h(91




In the transition from equation (8) to equation (9) we take integrations over R2, with Xo(x) replaced by the kernel estimate defined at equation (2) for all x E [R2. Strictly, this is inconsistent with our earlier assumption that X0(x) = 0 outside A. However, provided that h is small, estimate (2) decays rapidly towards zero outside A. In these circumstances equation (9) should be regarded as a convenient and reasonable approximation which avoids the need for special consideration of edge effects arising from the non-observance of events outside A, or from the location of xo near the boundary of A.

Straightforward differentiation from equation (7) then leads to

aL n m

~ =Z exp ( - ei)/{1 + a exp ( - Oei)} - n Y. uj(0)/w((a, 3), i=l j=l

aL n m a=- E ei exp( - Oei)/{1 + a exp( - 1ei)} + nce Y. uj(1) vj(13)/w(ca, 1), d,3 i=l j=1

a2L n 2

a2=-Ej exp(-213e,)/{1 +axep 8e)2 t ui(13)}/|{w(a,13)}2,

d___ exp - Z ei exp(-- ee)/{21 + aj exp( - We2)}2

aa2 fl i=l

+ n {>] u>(13) vQ3)} {w(ax 13) - aE> ui(13)} /{w(a, 13)}2 a2L ~~~n

= e= expie(-ei)/{l + a exp(-1ei)}2

~12 i=1

+ na {w(a , 1) ( uw(13)[vc(1) - {Vi(1)}21)

{ u/fl) vi(13)}}/ {w(a ,13}2

Using these formulae, evaluation of the maximum likelihood estimates (&e, 13), and of their asymptotic variance matrix, is straightforward. Of particular interest is a test of the hypothesis that ae =13 = 0, i.e. f(.) a1. For this, we compare D = 2{L (&e, 13) - L (0, 0)} with critical values of x2. Although 13 is indeterminate when a = 0, the reference to X= iS initially appropriate because the alternative hypothesis involves two additional free parameters, ae and 13. We return to this point in Section 4.

3.2. Goodness-of-fit Assessment One method of assessing the goodness of fit of the model is the following. Order the

cases xi, i = 1, ... ., n, in increasing distance from x0. Call these ordered distances r2 <.. . ? r", let D, denote the disc with centre x0 and radius rV, and define

quantities



356 DIGGLE [Part 3,

ti = X (x) f(x - xo; 0) dx, i= 1,. . ., n. (10)

Under the model, the ti so defined are the points of a homogeneous one-dimensional Poisson process. In practice, we evaluate the ti using 0 in place of the unknown 0 in equation (10).

Further transformations of the ti provide the data for two complementary assess- ments of goodness of fit. Firstly, the quantities ui = ti/t", i = 1, ... , n - 1, are the order statistics of an independent random sample from the uniform distribution on (0, 1). Departure from uniformity is evidence of discrepancy between the observed and assumed spatial variation in the intensity of cases over the region A, indicating an incorrect formulation of either or both of X0(.) and f(.). Secondly, the intervals gi = ti+ 1 - t, i = 1, ... , n - I, are an independent random sample from an exponential distribution. Departure from exponentiality is evidence of interaction among cases. In particular, overdispersion of the intervals, with too many gi in both tails of the distribution, indicates clustering of cases.

An explicit form for transformation (10) is available in the special case of a Gaussian kernel (2) combined with specification (6) for f(.). We write

m

T(r) = m-l E {Aj(r) +Bj(r)}, (11) j=1

where

A1(r)= ||(27rh2)-y exp[- jh-2{(x- dj)2 +y2}] dx dy, (12)

Bj(r) =||(27rh2)- exp[- 1h-2{(x- dj)2 +y2} - $(X2 +y2)] dx dy, (13)

and the integrations are again over Di. Now, let F(a, b) denote the integral of a standard circular Gaussian probability density function over a disc with radius a and centre a distance b from the origin. Then, it is easy to show that equations (12) and (13) reduce to

Aj(r) = F(r/h, dj/h)

and

Bj(r) = {ao/(2fh2 + 1)} exp { - dj2/(23h2 + 1)} F(r/o, ,tj/o),

where Aj = dj/(2f3h2 + 1) and a2 = (23 + h -2)- i. By substituting these results in equation (11) we can evaluate the transformed quantities ti= T(ri) required for the goodness-of-fit assessment. A convenient approximation for F(a, b) is given in Abramowitz and Stegun (1965).

4. APPLICATION

We now use the methodology developed in Sections 2 and 3 to investigate a possible association between the spatial distribution of cancers of the larynx in the Chorley-Ribble area and the location of the now-disused industrial incinerator. A




TABLE 1 Estimates of M(h) = a + b MSE(h) for uniform kernel smoothing of the spatial distribution of cancers of the

lung

h M(h) h M(h)

0.1 -3.84 0.8 -5.09 0.2 -6.12 1.0 - 4.57 0.3 - 7.32 1.2 - 4.07 0.4 - 6.62 1.4 - 3.77 0.5 - 5.77 1.6 - 3.35 0.6 - 5.93 1.8 - 2.95 0.7 - 5.44 2.0 -2.66

more detailed account is in Diggle et al. (1990). The relevant data were shown in Figs 1 and 2. They consist of the location xo of the incinerator, locations xi, i = 1, ... , 58, of cancers of the larynx and locations yj, j = 1, ... , 978, of cancers of the lung. The parametric specification of the model is given by the combination of equations (1)-(3) and (6).

The first stage in the analysis is to construct the kernel estimate for X0(x). Table 1 lists estimates of M(h) = a + b MSE(h), where a and b are arbitrary constants, these estimates being obtained from the yj data using the method described in Berman and Diggle (1989). To avoid distortion due to the non-recording of cases outside A, this part of the analysis was confined to those lung cancer cases yj located in a square region contained within A. The estimate of the optimal h for a uniform kernel is 0.3 km (corresponding to h = 0.15 km for a Gaussian kernel), with a secondary local minimum at h = 0.6 km. Fig. 3 is a contour plot of the resulting estimate of X0(x). The extreme spatial variation is reasonable in a human geographical context, with very high concentrations of population in urban areas and, to a lesser extent, close to main roads.

The next stage is to maximize the log-likelihood function (7), using h = 0.15. The maximum likelihood estimates are (a^, 3) = (23.67,0.91), with estimated standard errors SE(^) = 24.69, SE(3) = 0.60 and estimated correlation corr(&^, 3) = 0.83. The maximized value of the log-likelihood is L(&^, O3) = - 394.59, whereas L(0, 0) = -399.36 giving a value of D = 9.54. Since P{X2 > 9.54} = 0.008, we find reasonably strong evidence for an association between the spatial distribution of cancers of the larynx and the location of the incinerator, in the form of a raised incidence near the incinerator.

More informatively, Fig. 4 is a contour plot of D(c, (a ) = 2{L(&, ( 3) - L(ce, (a )}. This confirms, as we would expect in view of the standard errors for &^ and 3, that the log-likelihood surface is relatively flat. The reason for this is not difficult to see. Most of the evidence for the raised incidence near the incinerator derives from the cluster of four cases, and many different (ci, 13) pairs which give the same value of f(x; a, 13) at the centroid of the cluster will give similar values of the log-likelihood. For this reason, we would not be justified in placing a detailed interpretation on the fitted



358 DIGGLE [Part 3,

43504 1 1 I

43004. -

42504. -

42004.-

41 504.-

41004. I l

34003 34503 35003 35504 36004 36504

Fig. 3. Contour map of the Gaussian kernel estimate of kO(x) for cancers of the lung, using h =0.15 km

4 .5-8.o-8.o 8.

4.05- @ 6.?- 6.0 6.0 3.5 - ro8. 6.

3.0 - 4 .

2.0 m .5-1

0 6.0

Aona

6 50 100 150 200 250

Fig. 4. Contour plot of D(oa, ,B) = 2{L(^, /3) - L(oa, ,B)}: contour heights are 1.0, 2.0, 3.0, 4.0, 6.0 and 8.0; values of (oa, /3) within the 6.0 contour constitute an approximate 95% confidence region



19901 POINT PROCESS MODELLING APPROACH TO RAISED INCIDENCE 359

ch-suae qunie cisurdunie

(a) ()) 0~

a, C >

0 2 4 6 8 10 0 2 4 6 8

chi-squared quantiles chi-squared quantiles (a) (b)

Fig. 5. Comparison between 99 simulated values of the D-statistic under the null hypothesis and its notional chi-squared sampling distribution: (a) Q-Q plot against x2; (b) Q-Q plot against X1

function f(x; oi, f3). Conversely, the qualitative conclusion does not depend critically on the particular parametric specification adopted for f(.).

Fig. 4 also shows that the log-likelihood surface is far from quadratic, which casts doubt on the usual likelihood asymptotics. We therefore computed the D-statistic for 99 independent simulations of the null hypothesis, i.e. using the combination of equations (1)-(3) together with f(x) = 1. Fig. 5 shows Q-Q plots of the 99 simulated values against a nominal chi-squared distribution on each of two and one degrees of freedom. The latter gives much the better fit, suggesting that the test with two degrees of freedom is conservative. Because the observed D = 9.54 is larger than all 99 simulated values, a Monte Carlo test formally rejects the null hypothesis at the 1 Wo level of significance. This Monte Carlo assessment is important in the absence of any compelling theoretical argument to support the x2 approximation to the null sampling distribution of D.

The primary purpose of introducing X0(x) into the model is to avoid a spurious conclusion of association based on the implausible assumption of spatial homogeneity. If we do assume spatial homogeneity for the Chorley-Ribble data, i.e. put XO(x) = constant, the generalized likelihood ratio statistic to test for association is D = 17.56, which is very highly significant. Although there is no guarantee that introducing spatial variation in XO(x) will make the inference more conservative, we would expect this to be the case whenever the model provides a reasonable fit to the data. Similarly, there is no guarantee of a monotone relationship between the value of the Gaussian kernel smoothing constant h and the significance of the generalized likelihood ratio test, since they will depend on the precise juxtaposition of the cases xi with the peaks and troughs of Xo(x; h). For the Chorley-Ribble data, the significance of the result persists for h up to about 2.0. Thereafter, values of h between about 2.0 and 6.0 give a non-significant result, while values larger than 6.0 give progressively greater significance.

The results of the goodness-of-fit assessment, using the methods described in Section 3.1, are summarized in Fig. 6. Fig. 6(a) shows the empirical distribution function of the ui, with 5 Wo critical values of the Kolmogorov-Smirnov statistic to test departure from uniformity shown as a pair of parallel lines. Clearly, there is no significant evidence of departure from uniformity. Fig. 6(b) is a Q-Q plot for the



360 DIGGLE [Part 3,

OD

CZ

0

CD~~~~~~~~~~~~-

0.0 0.4 0.8 0.0 0.04 0.08

u exponential quantiles (a) (b)

Fig. 6. Goodness-of-fit assessment: (a) empirical distribution function of ui, with 50/o critical values of the Kolmogorov-Smirnov statistic to test for departure from uniformity (--------); (b) Q-Q plot of intervals gi against a fitted exponential distribution

intervals gi, using the observed mean of the intervals to define the scale parameter of the theoretical exponential distribution. Neither plot gives cause to doubt the assumed exponential distribution of intervals.

As with any goodness-of-fit procedure, the sensitivity of this assessment is limited by the size of the data set, here n = 58 cases. Within this limitation, the results are encouraging.

5. DISCUSSION

The point process modelling approach advocated in this paper has three important ingredients: it avoids the need for arbitrary aggregation of individual events into counts in discrete regions; it incorporates a flexible description of natural spatial variation in the local intensity of events; it shifts the focus of the analysis away from the somewhat artificial definition of clusters of events and towards a quantitative description of variation in local intensity around a prespecified point. Although I believe that each of these ingredients is beneficial to the analysis, there remain many open issues. Some of these relate to possible extensions of the methodology as described in this paper. Others are more fundamental.

Two essentially straightforward extensions would be to widen the class of functions f(.) and X0(*) under consideration. With regard to f(.), perhaps the most obvious extension would be to incorporate directional dependence, e.g. to reflect prevailing weather patterns. Another would be to allow multiple sources of raised incidence. With regard to X)0(), several possibilities suggest themselves. Most obviously, the collateral information used to construct X0(.) need not consist of a set of events yj, j = 1, . . . , m. Any geographical or demographic information could be used, although the technical details of the implementation might then be quite different. For example, if X0(.) were thought to be determined by the values of a spatial variable z(y), observations of which were available at a discrete set of points yj, j = 1, . ... m, construction of X0(.) would entail spatial interpolation or smoothing of the z(yj)-values. This is the basic problem of geostatistics, or spatial prediction. See, for example, Ripley (1981), chapter 4 and references therein. Using census data from dis-




crete regions would amount to constructing a spatial variable z(y) which is piecewise constant. A further extension would be to incorporate several such spatial variables into a log-linear regression model of the form

y= exp { fk Zk(Y) (14)

We would be reluctant to estimate additional parameters from the (typically sparse) data xi, i = 1, . . . , n, but estimating a regression model of the form (14) from the more abundant data yj, i = 1, . . . , m, would be a viable alternative to the kernel smoothing approach. Evaluation of the log-likelihood for 0 might then require two-dimensional numerical quadrature in place of the closed form expression (7).

An alternative to the model-based approach to inference would be to condition on both sets of events xi, i = 1, .. ., n, and yj, j = 1, .. ., m, and to consider a test of the hypothesis that the xi are a random sample from the totality. A Monte Carlo implementation of a permutation test could be used, based on any sensible choice of test statistic, e.g. the sum of the squared distances from the xi-events to the prespecified point x0, or the number of xi-events within a given distance of x0. This apparently nonparametric approach involves no less restrictive assumptions than the model-based approach, since the permutation hypothesis of the former is equivalent to the non- homogeneous Poisson assumption of the latter. Is it then better to pick an intuitively sensible test statistic to measure association with x0, or to pick sensible forms for X0(.) and f(.) and to let these determine the test via the likelihood principle? The former approach is exemplified by Cuzick and Edwards (1990), who review the relevant literature.

None of the above deals with the issue of what to do when the inhomogeneous Poisson assumption is found wanting. Our application to the Chorley-Ribble data shows that the model can accommodate quite extreme spatial variation. But it assumes that the locations of the xi-events are conditionally independent given XO(o) and f(.). This would not be true if, for example, the events refer to cases of a disease with a strong genetic component. Given the strong duality between environmental heterogeneity and clustering, as pointed out explicitly by Bartlett (1964), it is difficult to resolve this question convincingly by purely data analytic arguments. In addition to the goodness-of-fit assessment carried out in Section 4, it may therefore be of interest to explore the sensitivity of the results of the analysis to deletion of events within a tight cluster. In the Chorley-Ribble application, arbitrarily deleting one event from the cluster of four gives a value D = 5.82 for the test of no association with the incinerator, corresponding to ap-value of 0.054, orp = 0.016 if we use x2 rather than X2 Further deletion of a second event from the cluster gives D = 2.96, removing any strong evidence of association. This is not to suggest that we should lightly discard scarce data, only that it is useful to know how sensitive are the results of the analysis to variations of this kind. However, we emphasize that the methodology does not require the existence of tight clusters to find evidence of association.

Finally, and perhaps most fundamentally, nothing in this paper addresses the problem of retrospective formulation of a hypothesis based on inspection of the data. To put it bluntly, I doubt whether I would ever have been shown the Chorley-Ribble data had they not included the visually striking cluster of cancers of the larynx near the incinerator. This problem pervades the whole of applied statistics but is seen in a



362 DIGGLE [Part 3,

particularly acute form here. The analysis of the Chorley-Ribble data does no more than to establish a prima facie case for a possible association between cancer of the larynx and industrial incineration. Even if all aspects of the modelling exercise are deemed to be satisfactory, confirmation or otherwise of the apparent association requires a systematic study of comparable regions chosen without prior reference to the detailed spatial distribution of cases.

REFERENCES

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions, formulae (26.3.24)-(26.3.27). New York: Dover Publications.

Bartlett, M. S. (1964) Spectral analysis of two-dimensional point processes. Biometrika, 51, 299-311. Berman, M. and Diggle, P. J. (1989) Estimating weighted integrals of the second-order intensity of a

spatial point process. J. R. Statist. Soc. B, 51, 81-92. Black, D. (1984) Investigation of the Possible Increased Incidence of Cancer in West Cumbria. London:

Her Majesty's Stationery Office. Bowman, A. (1984) An alternative method of cross-validation for the smoothing of density estimates.

Biometrika, 71, 353-360. Cook-Mozaffari, P. J., Darby, S. C., Doll, R., Forman, D., Hermon, C., Pike, M. C. and Vincent, T.

(1989) Geographical variation in mortality from leukaemia and other cancers in England and Wales in relation to proximity to nuclear installations, 1969-78. Br. J. Cancer, 59, 476-485.

Cox, D. R. (1972) The statistical analysis of dependencies in point processes. In Stochastic Point Pro- cesses (ed. P. A. W. Lewis), pp. 55-66. New York: Wiley.

Cuzick, J. and Edwards, R. (1990) Spatial clustering for inhomogeneous populations (with discussion). J. R. Statist. Soc. B, 52, 73-104.

Diggle, P. J. (1983) Statistical Analysis of Spatial Point Patterns, ch. 4. London: Academic Press. (1985) A kernel method for smoothing point process data. Appl. Statist., 34, 138-147.

Diggle, P. J., Gatrell, A. C. and Lovett, A. A. (1990) Modelling the prevalence of cancer of the larynx in part of Lancashire: a new methodology for spatial epidemiology. In Spatial Epidemiology (ed. R. W. Thomas). London: Pion.

Diggle, P. and Marron, 1. S. (1988) Equivalence of smoothing parameter selection in density and intensity estimation. J. Am. Statist. Ass., 83, 793-800.

Forman, D., Cook-Mozzafari, P., Darby, S. J., Davey, G., Stratton, I., Doll, R. and Pike, M. (1987) Cancer near nuclear installations. Nature, 329, 499-505.

Muirhead, C. and Darby, S. (eds) (1989) Royal Statistical Society Meeting on Cancer near Nuclear Installations. J. R. Statist. Soc. A, 152, 305-384.

Ripley, B. D. (1981) Spatial Statistics. New York: Wiley. Silverman, B. W. (1986) Density Estimation for Statistics and Data Analysis. London: Chapman and

Hall. Wakeford, R., Binks, K. and Wilkie, D. (1989) Childhood leukaemia and nuclear installations. J. R.

Statist. Soc. A, 152, 61-86.



a point process modelling approach to raised incidence of a rare phenomenon in the vicinity of a...

Documents