The Knox test presents a particular limitation, namely the choice of critical values is entirely arbitrary. This test uses a single set of critical values for defining close proximity in space and time (e.g. "close in space", denoted s = 1 km, and "close in time", denoted t = 12 months). Selection of a number of different critical values and subsequent repetition of the Knox analysis would result in multiple testing. A simplification of the K-function method has been used to partially avoid the arbitrary choice of critical values and therefore avoid multiple testing [23]. This approach involved a simultaneous set of 225 calculations similar to the single Knox calculations to obtain values of R. Critical values changed over a pre-specified set of close values in time (t = 0.1, 0.2,...,1.5 years) and close values in space (s = 0.5, 1.0, 1.5,..., 7.5 km). The observed value of the K-function, K_O, was obtained by summing the 225 calculated values of R(s,t), i.e. K_O = ∑_s,tR and the distribution of the K-function was simulated using 999 random permutations of time. At each simulation, dates of birth (or dates of diagnosis) were randomly reallocated to each of the cases in the data set, creating a simulated value of the K-function. Note that the Knox test corresponds to a single dimension K-function where there is only one set of critical values. Statistical significance was assessed by comparing the observed value of the K-function with the simulated distribution.

Unlike the Knox test, the K-function does not give a readily available measure of the size of the clustering effect. Hence S (obtained from the Knox test, with critical spatial values s = 0.5,...,7.5 km and critical temporal values t = 0.1,...,1.5 years) was used to describe the magnitude of the clustering effects for a given pair of critical values. Additionally, the nominal statistical significance of each value of S was assessed using the Poisson distribution. To enable comparisons to be made between the geographical distance and nearest neighbour (NN) metrics (see below), an overall indicator of the strength of clustering was obtained using (where i refers to the i^th combination of s and t).

If clustering has arisen due to a geostationary exposure, then this could lead to detection only by the fixed geographical distance threshold. Alternatively, if clustering has arisen due to an infective process, then this could lead to detection only by the variable NN threshold. If clustering is due to an infective process, then it must be noted that analysis based on a NN metric is likely to be more appropriate when both urban and rural areas are included. Any specified distance between two cases will have different meanings in urban and rural locations. For example, the size of school catchment areas will differ greatly. Using the NN metric the specification of critical values for "close in space" is not fixed, but determined empirically by the local density of the spatially heterogeneous underlying population. Using the n^th NN, two cases were close in space if the locations of one (or both) of the cases was nearer than the other's n^th NN in the total data set (of all birth and diagnosis addresses). Thus the number of these pairs of cases observed to be in close proximity was counted. To adjust for variations in population densities, we repeated the K-function analyses by replacing fixed geographical distances with variable distances to the (N-7)^th,...,(N + 7)^th NNs if N ≥ 8 and with variable distances to the 1st,..., 15th NNs if N ≥ 7. N was chosen so that the mean distance was around 5 km, thus N = 3 for birth addresses (the fixed geographical distances were replaced by variable distances to the 1st,...,15th NNs) and N = 12 for diagnosis addresses (the fixed geographical distances were replaced by variable distances to the 5th,...,19th NN). The use of a single threshold NN approach was originally proposed by Jacquez [27].

The distributions of distances between the 3rd NNs for births and the 12th NNs for diagnoses were highly skewed, with median distances of 1.2 km and 2.9 km respectively. An exact geographically based match to the underlying population distribution was not available. Thus we used the case distribution as a proxy for the underlying population distribution to test whether population density was associated with space-time clustering. Cases were divided into two groups: 50% in a "more densely populated" group and 50% in a "less densely populated" group, according to whether the 3rd NN (for births) or 12th NN (for diagnoses) was closer or further away than the median distance. There are then three possible ways in which pairs of cases may be in close proximity: (i) a case from a "more densely populated" area may be in close proximity to another case from a "more densely populated" area; (ii) a case from a "less densely populated" area may be in close proximity to another case from a "less densely populated" area; or (iii) a case from a "more densely populated" area may be in close proximity to a case from a "less densely populated" area. Therefore, if we are interested in whether cases from a "more densely populated" area show a tendency to cluster, it does not matter whether partner cases are from either a more or less densely populated area. Thus, population density analyses proceeded by analysing pairs of cases that included at least one case from a "more densely populated" area (i.e. "more densely populated: any" case pairs) and pairs of cases that included at least one case from a "less densely populated" area (i.e. "less densely populated: any" case pairs).

It has been argued that population shifts may cause artificial space-time clustering [28,29]. We were not able to analyse population shifts, because this would require data on small area population estimates for short time intervals, which are not available. If population shifts led to space-time clustering we would predict that this would only occur within a specific sub-period. Thus, we also analysed space-time clustering within two shorter time periods (1974-1990 and 1991-2006).

As a supplementary analysis, Kulldorff's scan statistic based on a space-time permutation model was used to identify individual clusters [30] and examine geographical and spatial patterning between covariates (and thus this method is distinct from the Knox and K-function methods which analyse overall space-time clustering patterns). The complete study region and time span was scanned by construction of a three-dimensional cylindrical moving window. The base of the cylinder represents two-dimensional geographical space and the height represents time. The base and height of this cylinder vary so that they include at most 10% of the entire time span and at most 10% of the entire geographical area. The variable base is centred on the postcode centroid of each case [31]. This method has been used previously in an analysis of childhood leukaemia data [32]. The scan statistic was applied to test for differences in the propensity to cluster between gender and levels of population density, using a Bernoulli-based model [33]. This method is a case-control approach where one stratum (e.g. males) is treated as the case group and the other stratum (e.g. females) is treated as the control group. Thus the test assesses differences between the spatio-temporal distributions of the two groups. These scan statistics were calculated using the geographical locations of the addresses (OS grid references of residence at birth or diagnosis) and temporal reference (date of birth or date of diagnosis).

Four possible space-time interactions were analysed: those between (i) times and places of birth; (ii) time of diagnosis and place of birth; (iii) time of birth and place of diagnosis; and (iv) times and places of diagnosis. The interpretation of these interactions depends on the extent of residential movement between birth and diagnosis among the cases. If there was no residential movement then there would only be two interactions (time of birth or diagnosis with place of domicile). An interaction based on birth would indicate that cases who resided close to one another were also born at close points in time, indicating that they shared a similar environment at birth. An interaction based on diagnosis would indicate that cases who resided close to one another were also diagnosed at similar times, suggesting that they shared a similar environment at diagnosis. However, more than approximately 60% of children moved between birth and diagnosis, indicating that residential movements need to be taken into account. Thus a time of birth/place of birth interaction would suggest a transient environmental exposure affecting children in-utero or shortly after birth and that there is a variable latent period between exposure and diagnosis. A time of diagnosis/place of diagnosis interaction would suggest an exposure around diagnosis place and close to diagnosis time with a short latent period. A time of diagnosis/place of birth interaction would indicate an exposure at a heterogeneous time after birth, with a constant latent period. A time of birth/place of diagnosis interaction would suggest an exposure around residence at diagnosis, affecting those born at similar times with a short latent period (for a more detailed description see Birch and colleagues [34]). K-function and Knox analyses were done using programs written in FORTRAN 90 [35] and Kulldorff's scan statistic was performed using SaTScan v7.0 [36].