MED. INFORM.

(1978) VOL. 3 , NO. 1, 51-68

Analysis of the occurrence of anencephalic stillbirths in the Fylde Peninsula, 1956-1967 B. McA. SAYERS, S. D. SEGAL, W. R. HENSHALL, Inform Health Soc Care Downloaded from informahealthcare.com by University of Alberta on 12/12/14 For personal use only.

Engineering in Medicine Laboratory, Imperial College, London SW7, UK

J. P. BOUND, Department of Paediatrics, Victoria Hospital, Blackpool FY3 8 S R , Lancashire,

UK

and P. W. HARVEY Department of Pathology, Royal Lancaster Infirmary, Lancaster LA1 4RP, Lancashire, UK

(Received 18 December 1977) Keywords: Anencephalus, Pattern analysis, Epidemiology, Spectral analysis, Simula tion,

Computer-based statistical signal methods have been applied to the analysis of case-occurrence time and location data from a 12-year complete series of anencephalic stillbirths in the Fylde Peninsula of Lancashire. In the absence of any certainty about causative or contributory factors in these occurrences, the primary questions are if the underlying process is Poisson, or if there is evidence of a seasonal influence, and if any regional differences or communicable factors can be identified. With the limited case numbers (124) most small-sample statistical tests were not sufficiently informative. But using pattern and point-process analysis methods and simulation techniques, further elucidation is possible ; for example, it is shown that a seasonally rate-modulated Poisson process is a reasonable model for case occurrences in the southern region, but not in the north, for which a different process must apply and where some evidence exists of clustering of preferred inter-event intervals. It is concluded in particular, that there is a regional dependence, with seasonal factors, of the process underlying case occurrences, and in general, that signal analysis methods are useful in the interpretation of small-sample spatio-temporal epidemiological data. Des mCthodes statistiques de traitement des signaux sont utilisCes pour I’analyse spatiot-temporelle d’observations de nouveaux-nks anencCphaliques. Ces observations ont CtC recueillies sur une pCriode de 12 ans dans la pkninsule du Lancashire. En l’absence de toute certitude sur les causes ou les facteurs favorisants de cette effection, les recherches se sont orientkes vers les toxiques, les influences saisonnihres, les diffkrences entre rCgions, les facteurs de contagion. Etant donnC le petit nombre d’observations (1 24), les tests statistiques adapt& aux petits Cchantillons n’apportent que peu d’informations. En revanche, les

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mithodes de reconnaissance de forrnes et de simulation perrnettent de progresser. Dans la rCgion sud par exernple, un processus saisonnier rnodulC par ]’intervention de toxiques constitue un modkle acceptable. En conclusion, 1’Ctude rnontre qu’il existe une dCpendance locale et des facteurs saisonniers, et qu’au plan mCthodologiques, les techniques d’analyse des signaux s’avkrent utiles pour les interprktations CpidCrniologiques spatiot-temporelles des petits Cchantillons.

1. Introduction This paper concerns a spatio-temporal epidemiological problem-the occurrence of anencephalic stillbirths-and the techniques suitable for its study. The aim of the paper is to report the results of statistical signal analysis carried out on a specific distribution of anencephalic stillbirths, and to outline the computer-based pattern analysis methodology proposed and used for studying this and related kinds of data. A seasonal variation for anencephalus was reported by McKeown and Record [l], who found the incidence to be significantly higher in children born in the half-year October to March. Pleydell [2] noted that the incidence was doubled in urban districts and that cases occurred in groups in time and space. These findings suggested that environmental factors, for example infections, played a part in causing the malformation. However, attempts to identify causative agents have been unsuccessful [3], although the incidence of anencephalus showed a progressive fall to the end of the 1960s [4]. Therefore it was considered valuable to re-examine the features of anencephalus in a p o p lation, using a pattern analysis approach. The occurrence of anencephalic stillbirths exhibits appreciable randomness in space and time; the underlying process is therefore stochastic. Thus the observed sequence of case occurrences must be envisaged as a single statistical sample generated by an underlying conceptual process, and the task of analysis is partly to estimate the statistics of this process, and so of any underlying patterns. Spatial and temporal smoothing of the data is one possible starting point for improving the estimate of the underlying probability density of case occurrences, and is the basis of the present approach. 2.

Materials and methods Anencephalus was studied among babies born to mothers who lived in the major part of the Fylde of Lancashire, the district served by Victoria Hospital, Blackpool. T h e Fylde is a well-demarcated area (very approximately 20 km by 7 km) bounded on three sides by water barriers. T h e fourth, landward, border of the district studied is in sparsely populated country. T h e population is concentrated around the coast and has increased from about 250 000 to 300 000 during the period of study. Blackpool is situated centrally on the west coast. In the north the main urban areas are Thornton-Cleveleys, Fleetwood and Pculton le Fylde, and in the south, Lytham-St. Annes and Kirkham. T h e remainder of the district is comprised of rich farming land. There is an appreciable shifting population in Blackpool but there has been little change in the total. Growth in population has occurred in theother North and South Fylde main centres. I n Lytham-St. Annes, Blackpool, and Thornton-Cleveleys the proportion of people over the age of 65 years is higher

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than the national averages, but in Fleetwood it is only slightly raised. A century ago all centres of population were very small. T h e number of live births, stillbirths, and neonatal deaths was notified to us by the Medical Officers of Health. T h e case histories of babies dying in the perinatal period were studied and the date of the mother's last menstrual period (LMP) recorded. 93 per cent of the babies were examined at necropsy, irrespective of place of birth. This paper concerns the years 1956-1967. I n the early years studied the incidence of anencephalics in the Fylde was 3.2 per thousand births. T h e addresses of mothers who gave birth to anencephalic babies were visited and their location plotted on a street map. T h e Ordnance Survey map grid references (to 0.1 km) were then determined. T h e data, consisting of the date of the maternal LMP and the grid references, were then entered through punched cards into the IBM 1800 computer used for all analyses. T h e aggregate of case occurrences is shown on an outline map in figure 1 (note scaling). Additionally, demographic data were obtained from figures produced by the Bureau of Population Census and Surveys.

LO

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Figure 1.

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Outline map of the Fylde Peninsula showing 124 case-occurrence locations in relation to main population centres.

Both statistical and pattern analysis methcds were used in the study.

The

LMP dates were treated as point-events, distinguished only by their place and time of occurrence (figures 1 and 2). Smoothing, convolution, and correlation analysis of the point-event signals were carried out by single- or two-dimensional frequency-domain methods, using either the fast Fourier transform or the Chirp-z transform [5, 6, 71. T h e other statistical tests employed were used in the form described by Bendat and Piersol [8] or Siege1 [9].

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Figure 2. (a)Sequence of time intervals between successive LMP dates. (b) Time interval occurrences as a function of time in days following 1 January 1956. (c) Mean of successive non-overlapping groups of 6 intervals (last 4) through the sequence. (d) Standard deviation in the same groups.

3. Results and analysis Both temporal and geographical aspects of the distributions of anencephalic stillbirths are potentially of interest so the distribution of occurrences in sequential time, and seasonally, must be assessed; further, possible interrelations (e.g. do seasonal patterns alter with geographical location?) should be explored. Both statistical and pattern analysis methods need to be applied.

3.1. Statistical analysis Statistical measures require that the data meet appropriate standards of homogeneity and stationarity. Thus preliminary testing is needed. The curves of figure 2 ( c ) and 2 ( d ) indicate that no special alterations occur in the mean or standard deviation of sub-samples of inter-LMP intervals throughout the data set. The trend test and the run test [8] were both negative at the 5 per cent level. Hence, over the period studied there is no overt indication in these data of any inhomogeneity that would warrant closer investigation, or of non-stationarity . 3.1.1. Generalfeatures. The distribution of case occurrences in time is compactly represented by the inter-(LMP) event interval histogram figure 3 . This has an approximately negative exponential shape ; also, the first serial correlation coefficient of intervals is - 0.04 indicating serial independence. The hypothesis of a negative exponential distribution is accepted by a Chi2 (x2) test (x2=8.21 at D F = 16; P=0.95), so a Poisson process is indicated. For a Poisson process, the mean interval should equal the SD : here the figures are 36.6 and 40.2 days respectively. The matter was further tested by the study of the interval Fourier power spectrum which, for a Poisson process, should generate a flat spectrum. Testing

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the hanned power spectrum of the sample of 124 intervals, with the KolmogorovSmirnov one-sample (two-tailed) test [9] applied to the 61 harmonics (excluding DC and Nyquist terms), produces a value of the test-parameter 0=0-1155 which shows a non-significant difference (2" 0-10) from a flat power spectrum. However the scaled interval histograms do not entirely match expectations for a Poisson process. T h e K-scaled interval is formed from the K-scaled intervals : kTi= Tt+ Ti+l+ . . . Tifk-1 as an estimate of the k-scaled probability density of intervals p k ( t ) . Scaling a Poisson process to order k generates Gamma distributions of order ( K - 1) altering progressively with order from the original negative exponential distribution towards the Gaussian form [ll]. T h e first four scaled-interval

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Figure 3. ( a ) Time inter-event interval histogram (all data). (b) As 3 ( a ) , but fcr intervals following Blackpool cases only. ( c ) Inter-event histogram of distances between successive events in time (all data). ( d ) As (3 ( b ) , but for pairs of successive cases, both in Blackpool. ( e ) Interval histograms for cases occurring north of 370 N. (f) As 3 ( e ) , north of 360 N. ( g ) As (e), north of 350 N. (h) As ( e ) , north of 340 N.

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histograms are shown in figure 4. The expected two-scaled distribution, for example, does not match the observed two-scaled histogram ; this observation raises doubts about the underlying process. It is known that when the point-event records from several independent non-Poisson processes are superposed the resultant record tends towards the Poisson [ll]. So it is necessary to consider if any natural sub-divisions of the data are non-Poisson, such that the over-all Poisson character stems from superposition. The cases can be separated for this purpose according to population regions. 3 - scaled

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3.1.2. Regional variations. Figure 3 ( a ) shows the time-interval histogram for all 123 intervals; this is accepted as negative exponential with a parameter h=0.028 cases/day. Figure 3 ( b ) shows the same histogram restricted to the 62 time intervals (A = 0.014) between successive cases occurring in Blackpool (located between grid references 300 and 335 E., 320 and 390 N.); figure 3 ( d ) shows the histogram restricted further to case-pairs that both occur in the defined area and are successive. The null hypothesis of a negative exponential distribution is accepted ( P > 0.20) by x2 for the over-all case sequence, accepted (P> 0.10) for the 47 cases in Thornton-Cleveleys and Fleetwood, but rejected (P 0.10). However, X2-testing gives a different result. M.I.

E

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T h e x2 test was carried out on the observed seasonal distribution by dividing the 365 days into N-day groups (the end group was combined with the penultimate) counting from 1 January; N was varied from 20 to 40. As group size N was changed, substantial fluctuation of the x2 value occurred, from nonsignificance ( P >0.2) to significance (at P < 0.005). Varying the starting point by up to 10 days, but still us!’ng the whole 365-day data by treating the full period as cyclic (because the data from several years have been summated) so that 3 1 December is contiguous with 1 January, affects the value of x2 appreciably. Changes in the ratio of 2-5 : 1 have been seen. All group size values iY= 20-40 produce maximum x2 values (when the starting point is rotated through the record) that are significant at P=0*05and usually at P < 0-01. These effects originate in the short-term clustering of a few groups of LMP dates, but do not reflect any significant longer-run non-uniformity of L M P date distributions throughout the year, considered on a monthly basis or thereabouts. T h e seasonal distribution of cases by 20-day groups is shown in figure 6 ( b ) which suggests that a gross segmentation might produce different results. The case occurrences divide unequally between two halves of the year; the maximal difference occurs when one half is counted from the second group after 1 January. Then the actual case numbers are 47 and 77, which (a X2-test on 1 DF is significant at P=O.Ol) offers some confirmation of a gross seasonal fluctuation. 3.1.4. Space-time clustering. The inter-event distance histogram (figure 3 ( c ) ) is bimodal because successive cases may occur close together (say within one conurbation) or spaced (for example between two conurbations) ; the second mode has a value of about 12 km, matching the approximate centre-to-centre distance (12.5 km) between the main population centres in Blackpool and Fleetwood. (The distance-interval histogram for cases within Blackpool alone, not shown, is not appreciably different in shape from the first modal component of figure 3 ( c ) . ) Any possible clustering of cases could be evidenced by spatial grouping or by the close spacing in distance of cases that occur closely in time. The only evident spatial grouping of cases observed is associated with the locations of population centres (see figure 1). T h e space-time aspect (figure 5) is assessed at the simplest level by cross correlating the two 124-interval sequences; the correlation coefficient found (at zero relative shift) = + 0.047, a non-significant value showing that cases that are closely spaced in time do not tend to be closely spaced in location. This finding supports the view that a Poisson process is involved. These results appear to exclude a simple communicable origin for anencephalus, but we could not rule out an appreciably more complex mechanism that involved a substantial time-delay fxtor. Following the scheme of Knox [lo], the question of possible space-time clustering was also studied by forming a two-dimensional array of data so as to collect all intervals in 25-day groups up to a maximum of 1000 days) between pairs of events (consecutive or not) located within 0.25 km from 0.25-0.5 km apart and so on. No consistent alterations across the array were evident.

3.2. Pattern analysis A pattern analysis approach to the study of seasonal distributions by region is also possible. This is based on an initial two-dimensional smoothing of the

Anencephalic stillbirths in Lancashire (6)Spatial Interval Sequence

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Figure 5. ( a ) Time interval sequence. (b) Spatial intervals between time-successive ( d ) Cross-correlation events. (c) Scatter diagram of time-and spatial-intervals. coefficient of the two interval sequences as a function of relative shift.

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Figure 6 ( a ) . Seasonal distribution of events from January-December, regardless of year. ( b ) Seasonal occurrence histogram of events in 20-day intervals throughout the year (last interval is 25 days).

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Figure 7.

Northing in kilometres referred to 265 N. (at the bottom of the map in figure 1) for case occurrences as a function of time after 1 January 1956.

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data. The region was divided into 64 x 3.2 km segments according to northing grid reference, and all case occurrences in the individual northing segment stored in a single row of an array, by week of occurrence. Each 52-point row was smoothed by a circular frequency-domain method (using the Chirp-x transform) effectively to about the 4th harmonic ; the result is shown isometrically in figure 8.

0 Weeks from 1 January

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Figure 8. Isometric plot of the data of figure 7 low-pass filtered in two-dimensions to show the estimated underlying seasonal pattern of case-occurrences as a function of northing position.

The pattern of individual case occurrences can be used to indicate the underlying probability density of cases provided that each can be generalized in some plausible way. A specific (e.g. Gaussian) distribution of case-occurrence probability is assumed: on the grounds that while an individual case might have occurred at some different time, the estimated local probability contributed by the evidence of that case must be presumed to be a maximum a t the observed date, while decreasing symmetrically at times away from that date. Summating the result of assigning a Gaussian curve to each case in this way leads to the total estimated case-occurrence probability : the entire process corresponds to a convolution of case occurrences with the Gaussian curve, and is equivalent to a filtering operation in which no ' ripples ' are generated, because of the nature of the filter impulse response. For this purpose the data were stored in a 205-row array, corresponding to northing grid values in units of 0.1 km (minimum 276 in the south, maximum 481 in the north), each row of dimension 365 representing days throughout the year. The data in each northing row was convoluted with a zero-mean Gaussian curve (choosing o = 36-6 days : the mean inter-event interval of the raw data: produced satisfactory smoothing) and sub-sampled to a reduced, 32-point array. The 205-point columns of the array were extended with zeros to 256-points, and convoluted with a spatial Gaussian curve (zero-mean, with u = 1 km,or 10 northing rows; again selected empirically). The convolutions were performed by frequency-domain operations using the fast Fourier algorithm. The resulting array was contoured and the relevant 205-row region of the result is shown in figure 9. Both figures 8 and 9 show that in the Blackpool region there is a local maximum of LMP dates towards week 21 (late May); the most northerly (Fleetwood)

Anencephalic stillbirths in Lancashire

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region shows a maximum near the end of the year, evidenced in NovemberFebruary because the records are inherently periodic. Thus there is some suggestion of a seasonal pattern that may be different according to northing location, but it is important to consider how the statistical significance of the pattern might be assessed. Two methods can be suggested. The first depends on a suitable method for quantitative description of the pattern. T h e second depends on evaluating the expected power spectral variations of the pattern at various northing locations. Inform Health Soc Care Downloaded from informahealthcare.com by University of Alberta on 12/12/14 For personal use only.

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Contours of underlying seasonal case-occurrence probability as a function of northing position.

The main features of a one-dimensional pattern can often be described in frequency-domain terms very largely by the phase spectrum of its low-order Fourier harmonics. The extent to which this can be justified can be illustrated by the effects of interchanging amplitude and phase spectra between two signals, as in figure 10 (u). The patterns (i) and (ii) are the seasonal rates of case occurrences for the Blackpool region and the Fleetwood region respectively ;patterns

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(b) Figure 10. (a)Two related seasonal case-occurrence rate patterns for cases located in (i) Blackpool and (ii) Fleetwood. Pattern (iv) is the signal reconstructed from the amplitude spectrum ABP of (i) and the phase spectrum +FW of (ii), and pattern (iii) vice-versa. Four harmonics (of 1 cycle per 365 days) have been used. This illustrates the extent to which phase spectra are sufficient to describe the main pattern features. (b) Phase spectrum (harmonics 1 4 of 1 cycle per 365 days) of total seasonal case-occurrence pattern. T indicates the location of the phase values observed. Also shown is the ensemble of phase spectral values derived from an ensemble of simulations, drawn from a rate-modulated Poisson process, to represent annual case occurrences. The values of phase for actual seasonal case-occurrence patterns from 3 different northing regions in the Blackpool vicinity (BP1, BP2, BP3) and another from the Fleetwood region (FW) are also shown; in each case about 14 cases were involved, drawn from 1 k m northing bands.

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(iii) and (iv) are obtained by reconstructing signals having the phase spectrum of pattern (i) and the amplitude spectrum of (ii), or vice versa, respectively. T h e similarity of patterns (i) with (iii), and of (ii) with (iv), shows the role of the phase spectrum in specifying pattern features. A Chirp-z transform [7] of the 365-day case occurrences (each event is treated as a Dirac pulse) allows the phase spectrum to be determined for the present data. T h e complex Chirp-z transform produces the complex Fourier coefficients which are converted to polar form in order to obtain the harmonic phase angles. Figure 10 ( b ) shows the Hl-H4 phase spectrum found for the total seasonal occurrence data (shown in figure 6 ( a ) ) ,marked by the positions T . A control pattern can be set up by use of a numerical simulation that models either a uniform Poisson process or a Poisson process rate-modulated in accordance with the pattern of real data, low-pass filtered as in figures 8 or 9. T h e computer simulation is required to produce an ensemble of signals, each representing a sequence of point-event occurrences throughout a 365-day period. T h e phase spectra of the ensemble can then be determined for control purposes. Each Poisson set of inter-event intervals was generated by forming intervals Ti until their running sum exceeded 365 days;

where RNg is the ith random number drawn from a uniform distribution (0.1) and X is the observed case-occurrence rate, here =0*35 per day. T h e ratemodulated version was produced by modifying X in accordance with the time of the year, assessed by the running sum of intervals in the current sample; the modifying pattern was taken from the low-pass filtered version (Hl-H4) of the observed total daily occurrences over 365 days (the modulation depth was adjusted empirically so that the average low-pass filtered pattern [Hl-H4]of an ensemble of simulated records had approximately the same amplitude as the control signal). Figure l O ( b ) shows an ensemble of 30 phase spectra derived from such simulated records, illustrating the scatter observed ; the distribution is approximately Gaussian. (Th e uniform Poisson case is not shown since the phase values in each harmonic are uniformly distributed over the 360 degree range.) A clear aggregation of phase values occurs only for the first few harmonics (Hl-H4) so only these can be used in comparing actual and simulated processes. (Note that the positive phase values in the H3 and H4 ensembles are occasioned by phase wrap-around due to the calculation of phase within the principal-angle range [-180, +180 deg].) T h e mean values of three ensembles of phase spectra obtained from actual data by dividing the most populous (Blackpool) area into 3 contiguous northing regions, and one from the Fleetwood area, are also shown. T h e simulated data, not surprisingly, fit the actual total observations (T) well, in these terms. But the Fleetwood (FW) pattern is both distinguishable from the Blackpool patterns (BP) and falls outside the 97.5 per cent scatter of simulated data. I t is therefore concluded that the phase spectral approach justifies a distinction between the Fleetwood and the remainder, in respect of the pattern of seasonal occurrences. This procedure does not, however, distinguish any of the patterns from a uniform distribution since the control

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situation produces uniformly distributed phase spectral values thus making any distinction impossible. Discussion Several basic questions can be addressed by the detailed results of this study of occurrences of anencephalic stillbirths. The first question concerns the randomness or otherwise of case occurrences ; non-randomness may exist in a spatial sense, temporally, or both, and there may be a seasonal fluctuation of case occurrences. Second, do regional differences exist in any description of the case occurrences? If the case occurrences are random, and then presumably Poisson, little further progress can be made in studying the origins and precursors of this abnormality except in specifying the estimated case-occurrence rate for the population studied. If the case occurrences are non-random, or if there are any regional differences, opportunity for further insight is presented. Hence these specific basic questions are particularly important and, given the unusually precise data that have been obtained, warrant careful study. The introduction of a method for quantifying pattern features in records (by their phase spectra) has justified the use of a simulation to attempt to identify any separable seasonal dependence of cases between regions. The simulation approach, using a rate-modulated Poisson process, indicates that the seasonal pattern of case occurrences in the Fleetwood area is unlikely to be generated by the same process as generates the Blackpool cases, presuming that a rate-modulated Poisson process is involved. But it does not establish that the process is rate-modulated rather than constant parameter Poisson. On the other hand, taking the cases occurring in several different regions, all in the vicinity of the main population centre, does lead to similar seasonal patterns (as described by their phase spectra). The more such similarities then the less plausible the uniform distribution model. In the present data, however, there are insufficient different neighbouring regional segments containing a reasonable number of cases to permit a fully significant distinction in this way. In the presence of seasonal patterns in the data, the sensitivity of simple non-parametric statistical tests is also uncertain; nevertheless, even a x2 test is able to confirm the gross seasonal fluctuation of total case occurrences. On the other hand, all three separate measures (proportion of anencephalic to normal births, interval distributions, and the regional pattern analysis by phase spectrum) point in the one direction. A 2 x 2 contingency table shows a N.-S. difference in proportion of anencephalic births. The interval distribution alters shape substantially, and consistently towards the negative exponential, as the population of cases is increased by adding more of the Blackpool region to that of Fleetwood and Thornton-Cleveleys. The regional pattern analysis of Blackpool cases produces results consistent with a rate-modulated model. Thus the probability that a set of regional cases would produce phase spectral values within the range of simulation values (l2Oo, 180°, 180" for H1, H2, H3 respectively) is, assuming a uniform Poisson model, approximately

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Anencephalic stillbirths in Lancashire

taking Hl-H3 into account; the probability of this result repeated twice more with separate case occurrences, as in figure 10, will be 0*08/3or about 0.03. Taken together, these findings constitute sufficient reason to suspect that the difference between the northerly region of the Fylde and the remainder, in the proportion of anencephalic stillbirths, can be associated with a difference in timing pattern of case occurrences. But there is another aspect: the interval histogram of occurrences in Blackpool and the southern Fylde does not fit a simple negative exponential distribution; this is due to an excess of short intervals and a deficit of medium length intervals. Such effects could be explained if the generating process draws its intervals from more than one Poisson mechanism,

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Days after 1 January 1956

(4 Figure 11. (a)Interval sequence of case occurrences in the North Fylde region located north of O.S. reference 340 N. Note the number of very short intervals. (b) Interval sequence as figure 11 (a) for cases located north of 370 N. Note the preponderance of 20-50 day intervaIs. ( c ) The selected 20-50 day intervals for case occurrences located north of O.S. reference 340 N., 350 N., 360 N., 370 N., showing their actual occurrence (terminating) times after 1 January 1956. Note the apparent clustering of cases and the fact that the bulk of the cases occur above 370 N.

having different parameters. This follows because the attempt to fit a single negative exponential function to the sum of several different negative exponentials can result in just such an unexplained excess of short intervals and deficit of longer intervals. Precisely the same can be shown to apply if t h e process comprised a continuously rate-modulated Poisson instead of a discrete set of constant-parameter mechanisms. Thus the observed interval distribution of cases in Blackpool and the southern Fylde is consistent with a rate-modulated Poisson process, and is difficult otherwise to explain. The constant-parameter Poisson model fits the over-all interval distribution quite satisfactorily, and this is attributed to the effects of superposition. Also, taking into account the change in distribution as cases are removed by moving northwards the lower boundary of the

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Anencephalic stillbivths in Lancashire

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catchment area, it is inferred that Fleetwood cases are distributed non-exponentially with a 20-50-day preferred interval. I n this respect, this region is distinctive, and the rate-modulated Poisson model is not applicable. Figures 11 ( a ) and (b) show the interval sequences due to all cases located north of 340 N. and 370 N . respectively. T h e preponderance of short intervals in the former, and of the preferred-modal intervals in the latter, is clear. Figure 11 (c) shows the same data selectively limited to the 20-50-day intervals, plotted to indicate the terminating time of each selected interval, in days after 1 January 1956. T h e clustering of 20-50 day intervals is demonstrated in each of the diagrams for 370, . . . 340 N. but this is apparently due to cases located north of 370 N. ; little is contributed by cases in the south. At a rate of 28 selected intervals per 4383 days = 0.0064 per day the expected number of intervals (the selected events) that are followed by one other interval within say, 33 days is, on the basis of independent occurrences, 4.79 compared with 14 observed. Finally, considerable problems of interpretation exist when case numbers are small, as in this study; it cannot be claimed that these have been circumvented. Nevertheless, two extensions of classical small-sample statistical tests seem to be of value. T h e first is the detailed examination of regionally-selected interval sequences of L M P dates, in the light of the properties of point-process signals. T h e second is the study of seasonal patterns of the events, based on two-dimensional smoothing, on phase-spectral analysis, and on the use of digital simulations of hypothesized generating processes. T h e results in each case consistently supplement the indication of a regional contingency table (table 1) and distinguish the process underlying case occurrences in the upper-northern Fylde from that applicable elsewhere in the Fylde, attributing the latter to a rat--modulated, rather than simple, Poisson process. There is no way to extend the rate-modulation scheme based on a Poisson process so as to create the modal distribution apparently generated by North Fylde. But an inhibitory mechanism would serve. If events of a Poisson process are selectively inhibited by event Occurrences due to a second, independent, Poisson process (say), the resulting interval distribution approaches the required form-in the simplest model, the difference between two negative exponentials, rather like a first-order Gamma distribution [12]. If such a mechanism were to apply here, it would be implied that the generation of the anencephalic foetus is pre-empted by the occurrence of another randomlyoccurring event. Since the near-modal intervals in North Fylde tend to cluster somewhat, periods can be identified in which expected events fail to materialize, and further aspects of the hypothesized inhibitory mechanism could thus be described. This speculative model of the North Fylde process must, however, be considered further elsewhere.

ACKNOWLEDGMENT This work was supported by the Medical Research Council. REFERENCES 1. MCKEOWN, T. and RECORD,R. G. (1951) Seasonal incidence of congenital malformations of the central nervous system, Lancet, 1, pp. 192-6.

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Anencephalic stillbirths in Lancashire

M. J. (1960) Anencephaly and other congenital abnormalities. An epi2. PLEYDELL, demiological study in Northamptonshire, British Medical Journal, 4, pp. 309-31 5. 3. Leading article (1975) End of the potato avoidance hypothesis, British Medical Journal, 4, p. 308-9. 4. Leading article (1976) Epidemiology of anencephalus, spina bifida, and congenital hydrocephalus, British Medical Journal, 2, p. 1156. 5. MONRO,D. M. (1975) Complex discrete fast Fourier transform, algorithm AS53, Applied Statistics, 24, p. 153-160. 6. MONRO, D. M. (1976) Real discrete fast Fourier transform, algorithm AS83, Applied Statistics, 25, pp. 166-172. J. L. (1977) The Chirp discrete Fourier transform of 7. MONRO,D. M. and BRANCH, general length algorithm AS117, Applied Statistics 26, pp. 351-361. A. G. (1971) Random data: analysis and measurement 8. BENDAT,J. S. and PIERSOL, procedures (Wiley-Interscience). 9. SIEGEL, S. (1956) Non-parametric statistics for the behavioural sciences (McGraw-Hill). 10. KNOX,G. (1963) Detection of low intensity epidemicity, BritishJ. prev. SOC. Medicine, 17, p. 121-127. 11. SAYERS, B. McA. (1970) Inferring significance from biological signals, in Biomedical Engineering Systems. Eds. Clynes, M. and Milsum, J. H. (McGraw-Hill) pp. 84164. 12. TENHOOPEN,M. and REUVER,H. A. (1965) Selective interaction of two recurrent processes, Journal of Applied Probability, 2, pp. 286-292.

Analysis of the occurrence of anencephalic stillbirths in the Fylde Peninsula, 1956--1967.

MED. INFORM. (1978) VOL. 3 , NO. 1, 51-68 Analysis of the occurrence of anencephalic stillbirths in the Fylde Peninsula, 1956-1967 B. McA. SAYERS, S...
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