- Research
- Open Access
Comparing adaptive and fixed bandwidth-based kernel density estimates in spatial cancer epidemiology
- Dorothea Lemke^{1, 2}Email author,
- Volkmar Mattauch^{3},
- Oliver Heidinger^{3},
- Edzer Pebesma^{2} and
- Hans-Werner Hense^{1, 3}
https://doi.org/10.1186/s12942-015-0005-9
© Lemke et al.; licensee BioMed Central. 2015
- Received: 16 December 2014
- Accepted: 21 February 2015
- Published: 31 March 2015
Abstract
Background
Monitoring spatial disease risk (e.g. identifying risk areas) is of great relevance in public health research, especially in cancer epidemiology. A common strategy uses case-control studies and estimates a spatial relative risk function (sRRF) via kernel density estimation (KDE). This study was set up to evaluate the sRRF estimation methods, comparing fixed with adaptive bandwidth-based KDE, and how they were able to detect ‘risk areas’ with case data from a population-based cancer registry.
Methods
The sRRF were estimated within a defined area, using locational information on incident cancer cases and on a spatial sample of controls, drawn from a high-resolution population grid recognized as underestimating the resident population in urban centers. The spatial extensions of these areas with underestimated resident population were quantified with population reference data and used in this study as ‘true risk areas’. Sensitivity and specificity analyses were conducted by spatial overlay of the ‘true risk areas’ and the significant (α=.05) p-contour lines obtained from the sRRF.
Results
We observed that the fixed bandwidth-based sRRF was distinguished by a conservative behavior in identifying these urban ‘risk areas’, that is, a reduced sensitivity but increased specificity due to oversmoothing as compared to the adaptive risk estimator. In contrast, the latter appeared more competitive through variance stabilization, resulting in a higher sensitivity, while the specificity was equal as compared to the fixed risk estimator. Halving the originally determined bandwidths led to a simultaneous improvement of sensitivity and specificity of the adaptive sRRF, while the specificity was reduced for the fixed estimator.
Conclusion
The fixed risk estimator contrasts with an oversmoothing tendency in urban areas, while overestimating the risk in rural areas. The use of an adaptive bandwidth regime attenuated this pattern, but led in general to a higher false positive rate, because, in our study design, the majority of true risk areas were located in urban areas. However, there is a strong need for further optimizing the bandwidth selection methods, especially for the adaptive sRRF.
Keywords
- Risk Area
- Kernel Density Estimation
- Risk Estimator
- Fixed Estimator
- Optimal Bandwidth
Introduction
Investigating the spatial distribution of diseases has a long tradition in epidemiology: it provides important insights into the spatial processes of disease development and disease etiologies. Thus, the demand for spatial disease surveillance programs is rising, both for chronic diseases such as cancer or multiple sclerosis, and for infectious diseases, as the recently implemented project for contagious disease surveillance (http://www.healthmap.org) indicates. A common methodological strategy in monitoring spatial disease risks is to use regional count data as a reference. In most countries, administrative population counts are freely available only at a spatial level that does not provide sufficiently high resolutions for the demands and applications of a small-area disease surveillance [1]. Alternatively, case–control designs have been developed and employed for research purposes. This design has certain advantages through its ability to distinctly identify localized risk elevations which tend to be blurred in aggregated data [1-4]. A common risk measure used in a spatial case–control design is the spatial relative risk function (sRRF) computed via kernel density estimation (KDE), usually employing odds ratios [5-10]. The estimation of the sRRF has successfully been applied in many different areas of spatial epidemiology [1,11-14]. However, the critical point with any KDE is the optimal choice of the smoothing parameter, i.e., the bandwidth that is applied: if the bandwidth is too large the data will be over-smoothed and local extremes might be missed; on the other hand, if the bandwidth is chosen too small the data will be under-smoothed and the estimator appears “hairy” (with many modes). Many statistical methods have been suggested in order to find the optimal bandwidth and typical bandwidth selection methods are Likelihood Cross Validation (LCV [15,16], Least Squares Cross Validation (LSCV [17,18], Biased Cross Validation (BCV [19], Smoothed Cross Validation(SCV [20], and the direct-plug-in method [21]. Detailed descriptions of these bandwidth selection methods have been provided [22-25]. Nevertheless, and despite progress in finding the optimal bandwidth, KDE may still reflect the true density insufficiently when the data are very skewed or show a multimodal structure [26].
Commonly, sRRF are computed using a fixed bandwidth, that is, with the same amount of smoothing being applied for cases and controls over the entire study area. However, considering that the source population in reality is not homogeneously distributed across space, it appears more reasonable to assign less smoothing to areas with a high population density and more smoothing to regions with a lower population density. This is known as the adaptive bandwidth approach [27]. Davies and Hazelton [28] found theoretical and practical advantages in the application of the adaptive bandwidth to clinical data and implemented it in the R package sparr [29].
In this study we evaluate different bandwidth selection methods and the resulting fixed and adaptive bandwidth-based risk estimator using the R package sparr. To date, the presumed advantages of certain approaches have been evaluated only in simulation studies using theoretical data and scenarios [28]. However, many disease surveillance systems, such as cancer registries, are faced with a demand for a regional monitoring of small-area disease risk based on individual cases of disease. To address the general lack of comparison of these two methods under real-world conditions, we create a risk surface employing methods that use real-world data and that artificially produce urban ‘risk areas’. Finally, we assess the performance of the sRRF, using fixed and adaptive bandwidth-based kernel density estimation, to detect these urban ‘risk areas’.
Methods
Incident cancer cases
Estimated bandwidths [m] and nearest neighborhood ratios (NN-ratio)
N | Fixed bandwidth | Adaptive bandwidth | Nearest Neighbour ratio | ||||
---|---|---|---|---|---|---|---|
Cancer type | N _{ f } = N _{ g } | OS h(pooled) | LSCV h(f) | LSCV h(g) | OS’ h(pooled) | f(NN_ratio) | g(NN_ratio) |
Lung cancer (male) | 4826 | 6802.04 | 918.08 | 637.53 | 7635.04 | 0.36 | 0.47 |
Lung cancer (female) | 1189 | 8753.08 | 1008.07 | 873.61 | 9825.00 | 0.41 | 0.55 |
Breast cancer (female) | 5628 | 6881.42 | 840.50 | 566.05 | 7724.13 | 0.35 | 0.46 |
Prostate cancer | 2926 | 7839.94 | 713.21 | 713.50 | 8800.03 | 0.40 | 0.39 |
Cancer all (male) | 20213 | 5438.46 | 522.82 | 530.88 | 6104.47 | 0.32 | 0.42 |
Cancer all (female) | 18019 | 5642.10 | 564.36 | 529.21 | 6333.01 | 0.30 | 0.42 |
Sampling of spatial controls and risk areas
In order to create a surface of ‘risk areas’, a high-resolution (HR) population grid (grid size 20 meters) (Figure 1b) was constructed for the Regierungsbezirk Münster (RB-MS). We used ancillary land cover data that were based on remote sensing information about the degree of sealing (0 - 100%) [32] and population at community level (age group 40–79 years; averaged for the period 1994–1998). This approach is known to produce systematic errors, that is, population sizes in urban areas were grossly underestimated while population sizes in rural areas were overestimated, mostly due to a lack of knowledge of the height of buildings or floor area ratio [33-35].
with ŷ _{b} the estimated number of inhabitants and y _{b} the true number of inhabitants in census tract b.
In this study, those census tracts where the relative error for the population density estimate was below −50%, denoting the third quartile (0.75) of the error distribution, were subsequently assigned as ‘risk areas’. These ‘risk areas’ were spatially buffered with a 500 m radius in order to obtain more smoothed borders of the risk areas.
Samples of spatial controls (which represent the spatial distribution of the disease-free population) were proportionally drawn from this biased HR population grid as spatial point coordinates in a 1:1 sampling design using the function “genrandompts” in the freely available Geospatial Modeling Environment (GME) (version 0.7.2.0) [36].
Predicting spatial relative risk function (sRRF) via adaptive and fixed kernel density estimation
where K is the Gaussian kernel function with a radially symmetrical probability density function, x_{1}, …, x_{n} the bivariate coordinates of the case and, respectively, control locations, and h_{i} the bandwidth for the i-th observation.
where h_{0} refers to the global bandwidth, which is scaled by the product of the inverse square-root of the pilot density (f(X_{i})^{-0.5}) and the geometric mean (γ) of this term. Thus, two bandwidths must be selected: a pilot and a global bandwidth. The pilot bandwidth is needed to replace the unknown density with an estimate, which is itself a fixed-bandwidth kernel estimate. For estimating pilot densities, a fixed bandwidth was applied using the least-squares cross validation (LSCV) approach [38]. The pilot bandwidths were calculated separately for the case and control data in order to assist in preserving the spatial heterogeneity in pilot densities [28]. For the global bandwidth, which is adjusted by the pilot density estimate, a common value for both data was used, implementing again the OS principle [37]. The relative risk function is then expressed as the ratio of the case and control densities f and g, respectively [5,6]. In order to symmetrize the treatment of the two density estimates and to stabilize the numerical results, the ratio was log-transformed [5,6,28,29]. To avoid boundary effects, the correction method of Diggle [39] was used for the fixed sRRF and for correcting the adaptive sRRF we applied the methods described in [40].
Different bandwidth sizes were applied for the adaptive and fixed risk estimator in sensitivity analyses (see Additional files 1, 2 and 3). For the fixed sRRF, two scenarios were implemented, based on halving (S1_fix) and doubling (S4_fix) the OS bandwidth. For the adaptive sRRF, four scenarios were set up: in the first scenario (S1_ad) the pilot (LSCV for cases and controls) and the global (OS) bandwidth were halved, and in the fourth scenario (S4_ad) both bandwidth were doubled. For the second scenario (S2_ad) the pilot bandwidth was halved and the global bandwidth was doubled and vice versa for the third scenario (S3_ad).
Statistical analyses
To extract significant areas of risk elevations (H_{0}: r(x) = 0; H_{1}: r(x) > 0), based on the test statistic r(x), tolerance contours (at a significance level of α = 0.05) were calculated using the z-statistic-based asymptotic normality test [28]. This approach approximates the variances of the fixed and adaptive sRRF to construct test statistic z(x) at each location x, using the asymptotic theory of the kernel estimator z(x) ~ N(0,1), where N(0,1) denotes the standard normal distribution [28,41].
The spatial relative risk functions with the p-value contours were estimated using the R package sparr version 0.3-3 from [29]. This package is freely available on the Comprehensive R Archive Network [42]. All maps were projected in the ETRS89/ UTM 32N coordinate reference system [43].
Overlay analyses of ‘risk areas’
Based on the spatial overlay of the ‘true risk areas’ with the areas of significant tolerance contours (α-level =0.05), the area portions of correctly detected risk regions (True Positives, TP) were calculated. Also, the area portions that were falsely detected (False Positives, FP) and of non-detected risk areas (False Negatives, FN) as well as that of correctly assigned non-risk areas (True Negatives, TN) were determined. We calculated for each cancer type and bandwidth method the sensitivity as Se = TP/(TP + FN), i.e. the proportion of correctly detected ‘risk areas’, and the specificity as Sp = TN/(FP + TN), i.e. the proportion of correctly assigned non-risk areas. Furthermore, the likelihood ratio was calculated as LR+ = Se/(1-Sp) = (TP/(TP + FP))/(FP/(FP + TN)), indicating how many times more likely a significant p-contour was in a risk area as compared to a non-risk areas.
Ethics statement
The data used in this study were stored by the Epidemiological Cancer Registry North Rhine-Westphalia with doubly encrypted personal identifiers. The procedure is precisely defined by state legislation and does not require personal consent of cancer patients. Data were transferred to the investigators in an anonymized form. Use of anonymized data for research purposes does not require a vote by ethics committee or an institutional review board.
Results
Interestingly, the sRRF for different cancer types showed contrasting spatial patterns. Thus, the significant risk areas for ‘all cancers’ were mainly located around urban centers while the risk areas for ‘lung cancer’ were mainly located in the southern study area which covers the northern Ruhr district, the largest urban-industrial agglomeration in Germany. By contrast, sRFF analyses for ‘prostate cancer’ and ‘breast cancer’ revealed few or no risk areas in the south. ‘Prostate cancers’ produced a large pronounced risk area in the north of the study region, while the significant risk areas for ‘breast cancer’ clustered around urban areas.
Summary of the spatial overlay analysis of the ‘risk areas’ with the significant tolerance contours (α = .05): Sensitivity, specificity, and the positive likelihood ratio (LR+) are presented as area ratios
Bandwidth | Cancer type | Sensitivity | Specificity | LR+ |
---|---|---|---|---|
Adaptive | Lung cancer (male) | 0.33 | 0.93 | 4.71 |
Lung cancer (female) | 0.38 | 0.90 | 3.80 | |
Breast cancer (female) | 0.21 | 0.83 | 1.24 | |
Prostate cancer | 0.24 | 0.78 | 1.09 | |
Cancer all (male) | 0.34 | 0.85 | 2.27 | |
Cancer all (female) | 0.34 | 0.85 | 2.27 | |
Fixed | Lung cancer (male) | 0.30 | 0.92 | 3.75 |
Lung cancer (female) | 0.33 | 0.92 | 4.13 | |
Breast cancer (female) | 0.20 | 0.87 | 1.54 | |
Prostate cancer | 0.17 | 0.78 | 0.77 | |
Cancer all (male) | 0.30 | 0.91 | 3.33 | |
Cancer all (female) | 0.26 | 0.89 | 2.36 |
The impact of applying different bandwidth sizes was assessed for ‘all cancers’ in men and women. The results of the overlay analyses were summarized in Additional file 1 and the sRRF maps are displayed in Additional files 2 and 3. Halving the bandwidths (S1), both risk estimators showed an increase in sensitivity, while the specificity, and thus the LR+ were increased only for the adaptive estimator. Halving the pilot and doubling the global bandwidths (S2) led to a similar sensitivity as in S1, but it reduced the specificity and thus lowered the LR+. By contrast, scenario 3 (S3) markedly reduced the sensitivity while raising the specificity; the LR+ was slightly lower than in S1. Doubling both bandwidths in the adaptive approach (S4) resulted in a sensitivity comparable to that in Table 2, but specificity and LR+ dropped to their lowest values. This contrasted with the fixed estimators, where despite a marked drop in sensitivity the increased specificity resulted in the highest LR+.
Discussion
Existing studies that compare the fixed against the adaptive bandwidth-based estimation of the sRRF use either a simulation design, where the data arise from a standard parametric distribution with an a priori known risk surface [28,45], or real-world data, where the data rarely follow a standard distribution and the risk surface is commonly unknown [1,11,14,46,47]. The novelty of this study is that it combines the advantages of both study types: we use real-world data and a known risk surface to evaluate the performance of both bandwidth types in estimating the sRRF. We artificially created ‘risk areas’ for cancer by application of a spatial sampling method for the controls that is recognized as being biased to underestimating the population density in urban areas.
When employing in our study the default setting for automatic bandwidth selection – as implemented in sparr –, the optimal bandwidth was inversely related to sample size for both methods (OS principle and LSCV score). This is a known property of the optimal bandwidth [48]: the larger the sample size the less smoothing is required. The OS bandwidths were also consistently wider than for the LSCV score, because the OS method applies ordinary scale estimators such as the standard deviation or the interquartile range and these are inclined to place upper bounds on the smoothing parameter and hence produce conservative density estimates which tend to eliminate “accidental features” [37]. The differences in the two OS bandwidths (OS h(pooled) and OS* h(pooled) in Table 1) are caused by the different definitions of the overall sample size used in the denominator for calculating this bandwidth [28,37]. For the fixed risk estimator, the overall sample size (n_{f} + n_{g}) is used as described by Terrell [38]. However, Davies and Hazelton [28] suggest to use \( \sqrt{n_f*{n}_g} \) (=geometric mean) instead in an attempt to define an ‘effective’ sample size for the adaptive sRRF. Since ((n_{f} + n_{g}) > (\( \sqrt{n_f*{n}_g}\left)\right), \) the OS bandwidth for the fixed is smaller than that for the adaptive risk estimator. The LSCV bandwidth selector aims to find an optimal bandwidth in minimizing the mean integrated squared error (MISE) between the kernel estimator (\( \widehat{f} \)) and the true density (f) [37] which implies a known tendency of undersmoothing. It has also been recognized, that the LSCV score tends to show multiple local minima which are occasionally minimized at h = 0 [24,26,49]. This has been particularly observed when the sample sizes increase and clusters of observations at the same or nearly the same location are likely [48,50-52]. This could be one possible explanation for the “counterintuitive” observation of larger bandwidths for the spatial case patterns and may indicate a possible failure of the LSCV method to find a minimum bandwidth which minimizes the MISE.
The OS principle was elected due to its potential to control excess variability in the estimated densities [28]. However, if the risk is constant across the study area, the fixed estimator naturally overestimates the relative risk in rural areas due to a too small bandwidth. In turn, in urban regions, the risk is underestimated due to too large bandwidths relative to the underlying population density. In our study, the risk was assumed to be a function of the underlying biased population density, that is, the relative risk was artificially increased (due underestimation of the population density) in urban areas and decreased in rural areas. This explains much of the generally lower sensitivity of the fixed estimator: the main portion of the risk areas was located in urban areas where the fixed estimator has a priori a reduced sensibility to detect the risk areas. On the other hand, this led to reduced FPs and a higher LR+ as compared to the adaptive estimator. The adaptive estimator was introduced to reduce this bias by an adjustment of the bandwidth inversely to the underlying case resp. control density. Therefore, the adaptive risk estimator had a greater variability in urban areas, leading to more detected risk areas, but at the cost of a reduced specificity and therefore a lower LR+.
These patterns were most clearly observable in ‘all cancer’, because the case density reflects (more or less) the true population density: areas of high case densities were therefore strongly linked to areas of the most pronounced underestimation (due to the sampling with bias) of the control densities, thus producing the largest relative risk differences. When regarding the other cancer types, sensitivity and specificity were additionally influenced by cancer-specific processes, such as regional over- or underreporting or the presence or absence of screening bias. For example, the case density for ‘lung cancer’ was particularly high in the south of the study region, a district with large agglomerations of heavy industries and coal mines. Thus, the high lung cancer incidence attributable to smoking among the industrial workers may have been compounded by additional occupational and environmental exposures to high doses of dusts, foams and other industrial emissions. Therefore, the sensitivity of both estimators was fairly similar to that for ‘all cancers’ because the majority of risk areas are located in the south of the study area. But compared to ‘all cancer’ the specificity was also similar, because the fixed estimator detected an unusual large risk area north of the lung cancer agglomeration, probably due to a too small bandwidth in this rural setting (higher FP rate). This difference in specificity regarding the rural areas emerged more clearly in the ‘prostate cancer’ data. Here, the highest case density was found in the north, a less populated region in the study areas possibly related to overdetection of ‘prostate cancer’ by an active prostate cancer center which has a northern catchment area. The generally low sensitivity for this tumor type is attributable to the fact that only few risk areas were a priori located in this region due to an overestimation of the source population. While differences in sensitivity between the two estimators were largest, the specificity was exactly the same. And again too small bandwidths might have caused these enlarged risk areas as compared to the adaptive sRRF. Expectedly, the spatial distribution of ‘breast cancer’ was more concentrated in urban areas because participation in - at this time - an opportunistic screening (e.g. mammography offered at gynecologist contacts) was more prevalent here. On the other hand, several regions, in particular in the south of the study area, showed low screening participation rates which have been shown to be associated with reduced socioeconomic resources at individuals and regions level [53,54]. Indeed, the risk surface resembles that for ‘all cancer’ but with a reduced overall sensitivity due to the reduced case density. Here, again the concentration of the case density mostly over urban (‘risk’) areas explains the sensitivity and specificity pattern: The fixed estimator oversmoothed the relative risk in urban areas due to larger bandwidth, but produced simultaneously fewer FPs, resulting in an increased LR+.
In summary, we observed that the fixed risk estimator was characterized by a conservative behavior in the urban (‘risk’) areas i.e., less sensitive while avoiding FP alarms. In contrast, when the risk differences coincided with sparse data (= rural areas), especially seen for ‘prostate cancer’, the adaptive risk estimator profits from a variance stabilization due to a larger bandwidth, resulting in higher sensitivity and comparable specificity. These findings correspond well with the results of Davies and Hazelton [28]. Zhang et al. [14] applied the adaptive and fixed bandwidth-based sRRF to 80 cases and controls of schistosomiasis in the Guichi region (China) and found that the adaptive sRRF had a better ability to depict the spatial heterogeneity of sparsely populated risk areas.
Varying the originally derived bandwidth sizes for ‘all cancers’ accentuated the bias-variance tradeoff of the fixed sRRF: A reduction of the bandwidth (S1) reduced the difference between the density estimate and true density (bias), but at the expense of a higher variability of the estimate (variance). Therefore, a higher sensitivity was offset by a reduced specificity (mainly in rural areas). A doubling of the bandwidth (S4) led to the inverse process: the higher specificity due to a reduced variance of the risk estimate came at the cost of a higher bias (reduced sensitivity). Varying the adaptive bandwidths is more complicated, because at least two different bandwidths must be selected. Here, we observed that the global bandwidth affected the specificity/precision of the sRRF: when it was large (S2 &S4), specificity was a generally lower due to a reduced variability of the sRRF (oversmoothing) as compared to a smaller global bandwidth (S1 & S3). The pilot bandwidth determines the degree of preserved spatial heterogeneity present in the distribution of cases resp. controls, and it seems that this pilot bandwidth influenced the accuracy/sensitivity of the risk estimator. Therefore, halving both bandwidths led to an increase in sensitivity and specificity due to a more precise and accurate estimation of the densities, especially in the urban risk areas. In contrast, doubling the both bandwidths led to a loss of spatial precision due to oversmoothing, resulting in the lowest specificity.
Our results highlight the relative importance of choosing the optimal global bandwidth, because it essentially controls the specificity of the sRRF. Therefore, other bandwidth selection methods should be approved for selecting the global bandwidth. For instance, Davies [45] found promising advantage(s) in using the asymptotic mean integrated squared error (AMISE) criterion for selecting the global bandwidth. In another study, Davies [55] suggested to use a triangle dimension as scale estimator in the OS principle. However, these selectors have not yet been investigated for the adaptive sRRF. A further point to consider is the use of HR population grids for selecting a spatial sample of disease-free controls. This is a novel aspect that will become more popular with a better availability of an increased spatial resolution of these grids, because it is a time and cost effective way to obtain these controls. At present, freely available populations grids with a resolution of 100m exist for different countries of the world [56,57], but these grids are still too coarse to permit a regionalized monitoring of disease risks in small areas.
Strengths and limitations
This study has several strengths and limitations. In the first place, the use of real world cancer data with a sufficiently large sample size as well as the use of a known risk surface, which in practice is always unknown, increased the precision and the credibility of the study results. A further advantage with regard to implementation aspects is the restriction to the exclusive use of open-source data and software environments given the often limited financial resources to conduct regional monitoring. Limitations arise from the fact that we explicitly searched only for elevated risk areas while there are also areas with decreased disease risks. However, generally the detection of risk areas is of greater public health interest than the identification of low risk areas or cold spots. Another limitation may have resulted from the assumption of a constant cancer risk across the underlying, but biased, population density, because additional processes may violate this lead to misclassifications of the sRRF. This should, however, have had no impact of the comparison of the two methods because the same sample of cases is used in both methods. Another critical aspect is the statistical testing for obtaining the p-values. Here, only the z-test statistic-based asymptotic normality test was used because these tests are computationally far less expensive, and appear more stable in sparsely populated areas than tests based on Monte-Carlo (MC) randomizations. However, given the large sample size we regard this source of bias as relatively small.
Conclusion
In this study, we compared the fixed and the adaptive bandwidth-based sRRF in their ability to detect ‘risk areas’ for the occurrence of different cancer types. We observed that the fixed sRRF oversmoothes the risk in urban regions probably due to a too large bandwidth while the fixed estimator behaves conversely in rural settings. Because in our study design most risk areas were located in urban areas (due to a biased source population estimation), the fixed sRRF shows mostly a conservative behavior with reduced sensitivity but increased specificity, compared to the adaptive risk estimator that shows increased sensitivity in these urban areas at the cost of a lower specificity. Our results indicate that in situations where the risk was more concentrated in rural regions (e.g. due to sparse data) compared to the fixed sRRF, the adaptive risk estimator benefits from variance stabilization. Furthermore, the automatically selected bandwidth sizes used for the adaptive sRRF appear only suboptimal, which highlights the need for optimizing existing, and investigating new bandwidth selection methods, in particular for global bandwidth.
Declarations
Acknowledgements
We thank IT.NRW for geocoding the complete cancer registry data. In addition, we acknowledge support by Open Access Publication Fund of University of Münster.
Authors’ Affiliations
References
- Wheeler DC. A comparison of spatial clustering and cluster detection techniques for childhood leukemia incidence in Ohio, 1996–2003. Int J Health Geogr. 2007;6:13.View ArticlePubMed CentralPubMedGoogle Scholar
- Amaral S, Gavlak AA, Escada MIS, Monteiro AMV. Using remote sensing and census tract data to improve representation of population spatial distribution: case studies in the Brazilian Amazon. Popul Environ. 2012;34:142–70.View ArticleGoogle Scholar
- Lemke D, Mattauch V, Heidinger O, Pebesma E, Hense HW. Detecting cancer clusters in a regional population with local cluster tests and Bayesian smoothing methods: a simulation study. Int J Health Geogr. 2013;12:54.View ArticlePubMed CentralPubMedGoogle Scholar
- Tobler WR. Cellular geography. In: Gale S, Gale S, Olsson G, editors. Philosophy in geography. Dordrecht, Holland: Reidel; 1979. p. 379–86.View ArticleGoogle Scholar
- Bithell JF. An application of density-estimation to geographical epidemiology. Stat Med. 1990;9:691–701.View ArticlePubMedGoogle Scholar
- Bithell JF. Estimation of relative risk functions. Stat Med. 1991;10:1745–51.View ArticlePubMedGoogle Scholar
- Bithell JF. A classification of disease mapping methods. Stat Med. 2000;19:2203–15.View ArticlePubMedGoogle Scholar
- Gatrell AC, Bailey TC, Diggle PJ, Rowlingson BS. Spatial point pattern analysis and its application in geographical epidemiology. T I Brit Geogr. 1996;21:256–74.View ArticleGoogle Scholar
- Kelsall JE, Diggle P. Kernel estimation of relative risk. Bernoulli. 1995;1:3–16.View ArticleGoogle Scholar
- Kelsall JE, Diggle PJ. Nonparametric-estimation of spatial variation in relative risk. Stat Med. 1995;14:2335–42.View ArticlePubMedGoogle Scholar
- Benschop J, Hazelton ML, Stevenson MA, Dahl J, Morris RS, French NP. Descriptive spatial epidemiology of subclinical Salmonella infection in finisher pig herds: application of a novel method of spatially adaptive smoothing. Vet Res. 2008;39:2.View ArticlePubMedGoogle Scholar
- Han D, Rogerson PA, Bonner MR, Nie J, Vena JE, Muti P, et al. Assessing spatio-temporal variability of risk surfaces using residential history data in a case control study of breast cancer. Int J Health Geogr. 2005;4:9.View ArticlePubMed CentralPubMedGoogle Scholar
- Sabel CE, Gatrell AC, Loytonen M, Maasilta P, Jokelainen M. Modelling exposure opportunities: estimating relative risk for motor neurone disease in Finland. Soc Sci Med. 2000;50:1121–37.View ArticlePubMedGoogle Scholar
- Zhang ZJ, Davies TM, Gao J, Wang Z, Jiang QW. Identification of high-risk regions for schistosomiasis in the Guichi region of China: an adaptive kernel density estimation-based approach. Parasitology. 2013;140:868–75.View ArticlePubMedGoogle Scholar
- Duin RPW. Choice of smoothing parameters for Parzen estimators of probability density functions. Ieee T Comput. 1976;25:1175–9.View ArticleGoogle Scholar
- Habbema JDF, Hermans J, Van den Broek K. A stepwise discrimination analysis program using density estimation. In: Compstat 1974: Proceedings in Computational Statistics. Vienna: Physica Verlag; 1974.Google Scholar
- Bowman AW, Hall P, Titterington DM. Cross-validation in nonparametric-estimation of probabilities and probability densities. Biometrika. 1984;71:341–51.View ArticleGoogle Scholar
- Rudemo M. Empirical choice of histograms and kernel density estimators. Scand J Stat. 1982;9:65–78.Google Scholar
- Scott DW, Terrell GR. Biased and unbiased cross-validation in density-estimation. J Am Stat Assoc. 1987;82:1131–46.View ArticleGoogle Scholar
- Sheather SJ, Jones MC. A reliable data-based bandwidth selection method for kernel density-estimation. J Roy Stat Soc B Met. 1991;53:683–90.Google Scholar
- Hall P, Sheather SJ, Jones MC, Marron JS. On optimal data-based bandwidth selection in kernel density-estimation. Biometrika. 1991;78:263–9.View ArticleGoogle Scholar
- Duong T, University of Western A, School of M. Bandwidth selectors for multivariate kernel density estimation. 2004.Google Scholar
- Li Q, Racine JS. Nonparametric econometrics: theory and practice. Princeton, N.J.: Princeton University Press; 2007.Google Scholar
- Scott DW. Multivariate density estimation: theory, practice, and visualization. New York: Wiley; 1992.View ArticleGoogle Scholar
- Wand MP, Jones MC. Kernel smoothing. London; New York: Chapman & Hall; 1995.View ArticleGoogle Scholar
- Wang H. Multivariate Dichteschätzung in der explorativen Datenanalyse. Mathematik: Ausburg University; 2009.Google Scholar
- Abramson IS. On bandwidth variation in kernel estimates - a square root law. Ann Stat. 1982;10:1217–23.View ArticleGoogle Scholar
- Davies TM, Hazelton ML. Adaptive kernel estimation of spatial relative risk. Stat Med. 2010;29:2423–37.PubMedGoogle Scholar
- Davies TM, Hazelton ML, Marshall JC. sparr: analyzing spatial relative risk using fixed and adaptive kernel density estimation in r. J Stat Softw. 2011;39:1–14.Google Scholar
- Epidemiological cancer registry North Rhine-Westphalia. [http://www.krebsregister.nrw.de/].
- State office for information and technology NorthRhine-Westphalia (Germany) [http://www.it.nrw.de/].
- EEA Fast Track Service Precursor on Land Monitoring - Degree of soil sealing [http://www.eea.europa.eu/data-and-maps/data/eea-fast-track-service-precursor-on-land-monitoring-degree-of-soil-sealing].
- Aubrecht C, Ozceylan D, Steinnocher K, Freire S. Multi-level geospatial modeling of human exposure patterns and vulnerability indicators. Nat Hazards. 2013;68:147–63.View ArticleGoogle Scholar
- Steinnocher K, Köstl M, Weichselbaum J. Grid-based population and land take trend indicators - new approaches introduced by the geoland2 core information service for spatial planning. In: Book Grid-based population and land take trend indicators - new approaches introduced by the geoland2 core information service for spatial planning. City: NTTS; 2011.Google Scholar
- Steinnocher K, Weichselbaum J, Köstl M. Linking remote sensing and demographic analysis in urbanised areas. In: Book Linking remote sensing and demographic analysis in urbanised areas. 1st EARSel workshop of the SIG urban remote sensing (conference proceedings). Berlin. 2006.Google Scholar
- Beyer HL. Geospatial Modelling Environment (Version 0.7.2.0). (software). In: Book Geospatial Modelling Environment (Version 0.7.2.0). (software). 2012.Google Scholar
- Terrell GR. The maximal smoothing principle in density-estimation. J Am Stat Assoc. 1990;85:470–7.View ArticleGoogle Scholar
- Bowman AWAA. Applied smoothing techniques for data analysis: the kernel approach with S-Plus illustrations. Oxford; New York: Clarendon Press; Oxford University Press; 1997.Google Scholar
- Diggle P. A kernel-method for smoothing point process data. Appl Stat-J Roy St C. 1985;34:138–47.Google Scholar
- Marshall JC, Hazelton ML. Boundary kernels for adaptive density estimators on regions with irregular boundaries. J Multivariate Anal. 2010;101:949–63.View ArticleGoogle Scholar
- Hazelton ML, Davies TM. Inference based on kernel estimates of the relative risk function in geographical epidemiology. Biometrical J. 2009;51:98–109.View ArticleGoogle Scholar
- R Development Core Team: R: Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2013.Google Scholar
- Spatial Reference [spatialreference.org/ref/epsg/etrs89-utm-zone-32n/].Google Scholar
- Baddeley A, Turner R. spatstat: An R package for analyzing spatial point patterns. J Stat Softw. 2005;12:1–42.Google Scholar
- Davies TM. Jointly optimal bandwidth selection for the planar kernel-smoothed density-ratio. Spat Spatiotemporal Epidemiol. 2013;5:51–65.View ArticlePubMedGoogle Scholar
- Zhang Z, Carpenter TE, Chen Y, Clark AB, Lynn HS, Peng W, et al. Identifying high-risk regions for schistosomiasis in Guichi, China: a spatial analysis. Acta Trop. 2008;107:217–23.View ArticlePubMedGoogle Scholar
- Zhang Z, Clark AB, Bivand R, Chen Y, Carpenter TE, Peng W, et al. Nonparametric spatial analysis to detect high-risk regions for schistosomiasis in Guichi, China. Trans R Soc Trop Med Hyg. 2009;103:1045–52.View ArticlePubMedGoogle Scholar
- Silverman BW. Density estimation for statistics and data analysis. London: New York: Chapman and Hall; 1986.View ArticleGoogle Scholar
- Zhang J. Adaptive normal reference bandwidth based on quantile for kernel density estimation. J Appl Stat. 2011;38:2869–80.View ArticleGoogle Scholar
- Amstrup SC, McDonald TL, Durner GM. Using satellite radiotelemetry data to delineate and manage wildlife populations. Wildlife Soc B. 2004;32:661–79.View ArticleGoogle Scholar
- Gitzen RA, Millspaugh JJ, Kernohan BJ. Bandwidth selection for fixed-kernel analysis of animal utilization distributions. J Wildlife Manage. 2006;70:1334–44.View ArticleGoogle Scholar
- Seaman DE, Griffith B, Powell RA. KERNELHR: a program for estimating animal home ranges. Wildlife Soc B. 1998;26:95–100.Google Scholar
- Berens EM, Reder M, Kolip P, Spallek J. A cross-sectional study on informed choice in the mammography screening programme in Germany (InEMa): a study protocol. BMJ Open. 2014;4:e006145.View ArticlePubMed CentralPubMedGoogle Scholar
- Espinas JA, Aliste L, Fernandez E, Argimon JM, Tresserras R, Borras JM. Narrowing the equity gap: the impact of organized versus opportunistic cancer screening in Catalonia (Spain). J Med Screen. 2011;18:87–90.View ArticlePubMedGoogle Scholar
- Davies TM. Scaling oversmoothing factors for kernel estimation of spatial relative risk. Epidemiologic Methods. 2013;2:67-83.Google Scholar
- European population grid dataset for the reference year 2006 [http://www.efgs.info/data/GEOSTAT_Grid_POP_2006_1K].
- WorldPop. [http://www.worldpop.org.uk/].
Copyright
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.