A modified version of Moran's I
- Monica C Jackson^{1}Email author,
- Lan Huang^{2},
- Qian Xie^{1} and
- Ram C Tiwari^{2}
https://doi.org/10.1186/1476-072X-9-33
© Jackson et al; licensee BioMed Central Ltd. 2010
Received: 3 February 2010
Accepted: 29 June 2010
Published: 29 June 2010
Abstract
Background
Investigation of global clustering patterns across regions is very important in spatial data analysis. Moran's I is a widely used spatial statistic for detecting global spatial patterns such as an east-west trend or an unusually large cluster. Here, we intend to improve Moran's I for evaluating global clustering patterns by including the weight function in the variance, introducing a population density (PD) weight function in the statistics, and conducting Monte Carlo simulation for testing. We compare our modified Moran's I with Oden's I*_{ pop }for simulated data with homogeneous populations. The proposed method is applied to a census tract data set.
Methods
We present a modified version of Moran's I which includes information about the strength of the neighboring association when estimating the variance for the statistic. We provide a power analysis on Moran's I, a modified version of Moran's I, and I*_{ pop }in a simulation study. Data were simulated under two common spatial correlation scenarios of local and global clustering.
Results
For simulated data with a large cluster pattern, the modified Moran's I has the highest power (43.4%) compared to Moran's I (39.9%) and I*_{ pop }(12.4%) when the adjacent weight function is used with 5%, 10%, 15%, 20%, or 30% of the total population as the geographic range for the cluster.
For two global clustering patterns, the modified Moran's I (power > 25.3%) performed better than both Moran's I (> 24.6%) and I*_{ pop }(> 7.9%) with the adjacent weight function. With the population density weight function, all methods performed equally well.
In the real data example, all statistics indicate the existence of a global clustering pattern in a leukemia data set. The modified Moran's I has the lowest p-value (.0014) followed by Moran's I (.0156) and I*_{ pop }(.011).
Conclusions
Our power analysis and simulation study show that the modified Moran's I achieved higher power than Moran's I and I*_{ pop }for evaluating global and local clustering patterns on geographic data with homogeneous populations. The inclusion of the PD weight function which in turn redefines the neighbors seems to have a large impact on the power of detecting global clustering patterns. Our methods to improve the original version of Moran's I for homogeneous populations can also be extended to some alternative versions of Moran's I methods developed for heterogeneous populations.
Keywords
Background
Global indices of spatial autocorrelation have been used to evaluate the degree to which similar observations tend to occur near each other [1–4]. Spatial autocorrelation among disease counts or incidence proportions may reflect real association between cases due to infection, or perceived association based on a spatial aggregation of similar values. Moran's I [5] is a widely used global index that measures the similarity for values in neighboring places from an overall mean value and reflects a spatially weighted form of Pearson's correlation coefficient. The traditional calculation of Moran's I for disease cases does not account for population heterogeneity, so that, its application to disease rates or proportions may result in indication of spatial correlation that is completely due to the spatial proximity of population sizes, but not due to the similarity among the disease rates. Several alternative versions of Moran's I have been proposed to account for heterogeneous populations, for example by Oden [6], Waldhor [7], Walter [2], Assuncao and Reis [8], and Waller et al. [9].
In this article, we intend to improve the original version of Moran's I [5] that tests the similarity of numbers (e.g., cases) in neighboring geographic units, by doing the following: incorporating a weight function in the variance computation, introducing the population density (PD) weight function, and conducting Monte Carlo simulation for testing global clustering pattern. The weight function is not only included in the differences of the geographic unit's cases (e.g., county cases) from the overall mean, but also in the calculation of the variance. We also expand the definition of neighbors to a broader concept in the construction of Moran's I (e.g., all geographic units included in a pre-specified geographic range will be considered to be neighbors of the geographic unit in the center). Statistical inference is conducted assuming a null hypothesis of constant risk instead of a normal distribution for the statistic. The proposed approaches for improving the original Moran's I can also be applied to some alternative Moran's I methods developed for heterogeneous population data, such as the rate version of Moran's I and the normalized Moran's I [10].
A simulation study is performed to evaluate the power of both Moran's I and modified Moran's I for data generated with varying local and global clustering patterns. The statistic I*_{ pop }is also selected to be evaluated in the simulation study because I*_{ pop }was developed specifically to improve the original Moran's I for heterogeneous population data, and I*_{ pop }achieved better power than Moran's I as discussed in Jackson et al. [11]. We also compared these statistics with a traditional weight function definition that assigns a 0/1 to each neighboring location, representing a non-neighbor/neighbor and expanding the weight function definition that includes more information such as the population density of the surrounding neighbors [3].
The outline of this article is as follows: We describe the original Moran's I and construct the Modified Moran's I. We also describe Oden's I*_{ pop }. An extensive simulation study was carried out to compare the power of these statistics in identifying the global spatial heterogeneity. We illustrate the application of the methods using leukemia incidence data at the census tract-level from upstate New York. Results and a discussion conclude this article.
Methods
Observations and Locations
Define y_{ i }as the number of cases and n_{ i }as the population at risk at geographic unit i where i = 1, ..., N with N being the total number of geographic units (e.g. census tracts or counties). Let w_{ ij }be the weight assigned to the pair of geographic units i and j (i ≠ j), which reflects the strength of the relationship between geographic units i and j.
Moran's I
The weight w_{ ij }in Moran's I and its extensions are usually defined as in equation (2) (neighbor matrix). However, the weight function, w_{ ij }, can be defined in many other ways (see Song and Kulldorff [12] and Griffith [13]).
The value of I usually ranges between -1 and 1 and the expected value is . However, the range of I depends on the values of the weight function [3]. Positive values of I are associated with strong geographic patterns of spatial clustering, negative values of I are associated with a regular pattern, and a value close to zero represents complete spatial randomness. Note that areas with different population sizes were given the same weight (0 or 1) in Moran's I. The measure y_{ i }is the geographic unit's count, which does not include the geographic unit's population information. If a dataset has spatial correlations or clustering patterns due to the heterogeneous population sizes, the original Moran's I using counts will identify the clustering pattern which may be due to the spatial similarity of the population and not the spatial clustering pattern as desired.
Modified Moran's I
Note that . Therefore, I_{ w }≤ I if the term, . Since I_{ w }depends on the choice of the weight function, its mathematical properties such as moments and asymptotic distribution can possibly be obtained using the functional central limit theorem for sums of weighted random variables; however, we have not explored these issues here. The range of I_{ w }may be outside the interval [-1,1]. The expected value of the modified Moran's I depends on the weight function and the distribution of I_{ w }is not tractable, therefore, we use Monte Carlo simulation procedure to obtain the empirical distribution for statistical inferences.
where d_{ ij }is the distance between geographic units i and j, and m_{ i }= max{j:u_{j(i)}≤ λ}. The population density across geographic unit i and its j nearest neighbors is defined as u_{j(i)}. In this case, u_{j(i)}represents the total population count of the larger region comprised of geographic unit i and all of its neighbors j. The parameter λ is chosen by the user and allows the user to view the population density as a measure of spatial clustering, where large (small) λ is more sensitive to larger (smaller) clustering patterns. We define the parameter λ to be 50% of total population in this study, where we wanted a significant amount of neighbors included in the analysis. Note that k_{ i }will be larger in areas that are sparsely populated (allowing for greater distances to be incorporated in the weight function) and smaller in areas that are densely populated (allowing smaller distances to be incorporated in the weight function). Therefore, the population density of the geographic region is incorporated in this weight function.
Note that since both S_{ yy,w }and S_{ y,w }depend on the weights, W_{ ij }, I_{ w }is more sensitive to the variability in the weights than I. And, there is more variability in the weights Adj in (2) as compared to the weights PD in (5).
Oden's I*_{ pop }
where , v_{ i }= n_{ i }/n_{+}, v_{ j }= n_{ j }/n_{+}, e_{ i }= y_{ i }/y_{+}, e_{ j }= y_{ j }/y_{+}, and . Oden noted that symmetry is not required for I*_{ pop }and w_{ ii }≠ 0 (but can be fixed at any specified value). In order to capture the variability present in a region, Oden includes the first term in the numerator which is used to model the spatial variation in a manner similar to the conventional chi-squared test for heterogeneity of rates.
Power study
where (y_{1}, y_{2}, ..., y_{3109}) are regional counts generated from the multinomial distribution. The total number of cases y_{ + }here is always the same as the total number of cases in the corresponding data simulated under the alternative hypothesis.
where , r_{ i }is the relative risk at geographic unit i, which is not constant under the alternative hypothesis. The various value used for r_{ i }are based on the type of spatial pattern simulated as described in the following sections.
Local cluster pattern
Our power study is based on homogeneous populations in order to reduce any confounding effects that occur when using heterogeneous populations with varying relative risk. Waller et al. [9] state there is an impact of local geography (in particular, population density) on power comparisons between statistical tests of spatial pattern. Therefore we study homogenous populations in the simulation study to remove the effect of the population.
Global spiral clustering pattern
Global linear clustering pattern
where a is an integer from 0-99. The longitudinal coordinates of the continental United States ranges from -124.161 to -67.623. We divided this large interval into 100 sub-intervals of equal length where each interval corresponds to a value of a. Figure 2B shows the simulated data set.
Calculating power
We used Monte Carlo simulation methods to assess the power of rejecting the null hypothesis by simulating 10,000 data sets using the multinomial distribution defined in equation (7) under the hypothesis of equal relative risk. The statistics (Moran's I, Modified Moran's I, and I*_{ pop }) obtained from the simulated null data are used to construct the empirical distribution of the statistics. The 95^{th} percentile for each of the statistics is defined as the critical value.
Then, Moran's I, Modified Moran's I, and I*_{ pop }were calculated for each of the 1000 data sets simulated under the alternative hypothesis using the multinomial distribution defined in equation (8) with the clustering pattern shown in Figures 1 and 2. Power is calculated as the percentage of values out of the 1000 replicated data sets that exceed the critical point obtained from data under the null hypothesis.
Results
Powers (%) for Modified Moran's I, Moran's I, and I*_{ pop }for local and global spatial patterns with population density (PD) and adjacent (Adj) weight function.
Modified Moran's I | Modified Moran's I | Moran's I | Moran's I | I ^{ * } _{ pop } | I ^{ * } _{ pop } | ||
---|---|---|---|---|---|---|---|
Adj | PD | Adj | PD | Adj | PD | ||
Percent of population in cluster | |||||||
Local | 5 | 43.4 | 67.3 | 39.9 | 67.1 | 12.4 | 59.7 |
10 | 80.3 | 99.3 | 76.7 | 99.4 | 27.8 | 99.2 | |
15 | 96.7 | 100 | 94.9 | 100 | 46.5 | 100 | |
20 | 99.2 | 100 | 98.6 | 100 | 61.8 | 100 | |
30 | 99.9 | 100 | 99.8 | 100 | 74.5 | 100 | |
Global | Spiral | 25.3 | 99.9 | 24.6 | 99.9 | 7.9 | 99.6 |
Linear | 27.0 | 99.6 | 24.3 | 99.6 | 7.4 | 99.0 |
Results for the local cluster pattern
We experimented with cluster patterns that contained between 5% to 30% of the total population. We found that the larger the percent of the population within the cluster, the greater the power for all methods. Modified Moran's I has a higher power for all local cluster patterns. When the Adj weight function is used with 5% of the total population, we find that Modified Moran's I has a power of 43.4%, which is higher than the power for both the Moran's I (39.9%) and I*_{ pop }(12.4%). The same is true when for patterns with single cluster including 10%, 15%, 20%, and 30% of the total population. When the cluster reached 30% of the population, we obtained the highest power of 99.9% for Modified Moran's I, 99.8% for the Moran's I and 74.5% for I*_{ pop }.
When the PD weight function is used for cluster with 5% of the population we obtained powers of 67.3, 67.1, and 59.7% for Modified Moran's I, Moran's I and I*_{ pop }, respectively. Modified Moran's I performed slightly better in this scenario and similar for cluster with 10% of the population. When clusters with 15%, 20%, and 30% of the total population were used, all methods with the PD weight function performed equally with a power of 100% since the spatial pattern is very strong.
The statistics with the PD weight function obtained higher powers (compared to the Adj weight function) in all cases in our simulation study. Recall that the PD weight function incorporates a larger number of spatial neighbors compared to the Adj weight function; therefore, it includes more information for global clustering evaluation and has better powers for global spatial patterns. In general, there is no "best" definition of weights and weights can be based on distances or on other influences (e.g., how far a location is away from a contaminated water source)[14]. For a spatial analysis, weight functions are often chosen to have a spatial scale equivalent in size to the hypothesized cluster [15].
Results for the global spiral clustering pattern
For the global clustering pattern (as shown in Figure 2A), Modified Moran's I (power = 25.3%) performed better than both Moran's I (24.6%) and I*_{ pop }(7.9%) with the Adj weight function. With the PD weight function, all methods performed equally well with a power very close to 100%.
Results for the global monotone clustering pattern
Finally for the monotone pattern simulated as shown in Figure 2B, Modified Moran's I yielded the highest power of 27.0% compared to 24.3% for Moran's I and 7.4% for I*_{ pop }for the Adj weight function. For the PD weight function, Modified Moran's I had a power of 99.6% which is equal to the power of Moran's I and I*_{ pop }.
Application
Using the New York leukemia data set, we obtained p-values for Moran's I, Modified Moran's I and I*_{ pop }with the PD weight function. We find that Modified Moran's I has the lowest p-value (.0014) followed by Moran's I (.0156) and I*_{ pop }(.011). All three methods detected a global trend in this data (with significance level of α = 0.05), however, Modified Moran's I had the most significant p-value. This result is consistent with the finding from Waller and Gotway [3] that there exist global clustering pattern in the Leukemia rates in upstate New York.
Discussion and Conclusion
In conclusion, we improved the original Moran's I, and conducted a simulation study to evaluate the performance of the proposed method. We considered various simulated regional patterns in data that involved local cluster patterns and global clustering patterns. The five local cluster patterns used formed a single cluster in the eastern part of the U.S with either 5%, 10%, 15%, 20%, or 30% of the population included in the cluster with a relative risk to all other regions of 1.5. The two global patterns involved simulating a west to east linear trend, and a pattern resembling a cluster in the center with 2% of population with the spatial correlation slowly decreasing until you reach the east and west coast with a relative risk of 1.5. We also applied the proposed method to a census tract dataset, which has a more homogeneous population than state or county level data (however, census tract data still has heterogeneous population [18].) The proposed approach for improving the original Moran's I (for homogeneous population data) can be applied to the rate version and the normalized version of Moran's I, which may be more suitable for analyzing data with heterogeneous populations. Future research may be conducted to explore the performance of those methods for data with heterogeneous populations. Similar idea can also be applied to local indicators of spatial association (LISA) [19] for cluster detection in future work.
Modified Moran's I with the adjacent weight function (Adj) achieved higher power for the simulated local and global cluster patterns than Moran's I, and the modified Moran's I with PD has similar performance compared with Moran's I. We compared the modified Moran's I with I*_{ pop }as well, since the latter I*_{ pop }was developed to be an alternative of the original Moran's I for data with heterogeneous populations. However, I*_{ pop }does not always perform well on homogeneously populated data as shown in our simulation study.
For the local clustering patterns, the power for Modified Moran's I increased as the percentage of the population included in the cluster (cluster size) increased for both weight functions. For the global clustering patterns, modified Moran's I achieved higher power than both of the other statistics. The power for the global patterns are lower than for the local cluster patterns, since the spatial size of the area with the largest relative risk (1.5) is much larger in the local cluster patterns compared to the global patterns. There are other issues that affect power of identifying a global clustering pattern. For example, Lindsey [20] states that there are problems with building models based on nearest neighbors due to edge effects. We evaluate the effect of the cluster size in the simulation study, but not the edge effect. A cluster on the edge may lead to a lower power of identifying the clustering pattern.
Note that there is no inflated type 1 error since the proposed test is for a global clustering pattern evaluation (not for cluster detection) and Monte Carlo procedure is used for statistical inference. The type I error, which reflects the chance of the method identifying a spatial pattern when there is none, is controlled at the alpha level (0.05). Also, when there is a spatial pattern, the proposed Moran's I can identify the pattern with reasonable power as shown in the simulation study.
It turns out that the weight function played a large role in the method performance. The adjacent weight function only has two values (0 for a non-adjacent-neighbor and 1 for an adjacent neighbor). If the weight is 0 for a pair geographic units i and j, the difference between the geographic unit i and j is not evaluated in the formulation of the statistics (e.g. Moran's I and I_{ pop }). Since the number of adjacent neighbors for all the counties in the continental U.S. ranges from 0-14 (see Jackson et al. 2009), there is only a limited number of pairs of geographic units that are evaluated in the statistics when the Adj weight function is used. The PD weight function considers both geographic unit population information and the geographic distance of cell i and j. A higher weight is given to pairs with a shorter distance. The parameter λ in the PD function is chosen by the user and allows the user to view the population as a measure of spatial clustering, where large (small) λ is more sensitive to larger (smaller) clustering pattern. In this paper, we used λ as 50% of the total population, which allows for a large number of geographic neighbors for evaluation. For global clustering patterns (e.g. spiral or linear), many geographical units have a spatial correlation even if the distance between them is large and they are not adjacent. Therefore, the PD function with a large λ (i.e. which evaluates many geographic units) performs much better than the adjacent weight function.
Note that only different versions of Moran's I and I*_{ pop }were compared in this paper because the major purpose was to explore a way to improve Moran's I. We included I*_{ pop }for comparison because I*_{ pop }has not been studied by many (see Jackson et al. [11]), and it has been well known as an alternative method for Moran's I developed for data with heterogeneous populations. We did not include other methods, such as Tango's MEET [12], for global clustering evaluation in the comparison. Tango's MEET [12] has been shown to be the most powerful method for identifying global clustering patterns (see Jackson et al [11], Song and Kulldorff [21], and Huang et al. [10]). However, with the PD weight function, Moran's I and I*_{ pop }may perform as well as Tango's MEET. This issue will be explored further in future work.
Few spatial studies exist that explore data with homogenous populations. Spatial studies with homogenous populations allow for stronger power studies since confounding effects due to heterogeneous populations are removed [9]. For example, when performing a spatial study in California (with counties as the geographic unit), spatial statistics tend to detect clusters where there are two counties with large populations in close proximity (e.g., Los Angeles county and San Bernardino county). Counts or rates from areas with small populations are more unstable than those with large populations and they can be masked by areas with large populations[22, 23]. Also, for most applied studies involving real data, researchers are more interested in the pattern of the response variable (e.g. disease rate) rather than the population pattern [15, 24, 25]. Therefore, collecting and analyzing data with well defined regions (homogeneously populated) will be very useful.
Authors' information
MCJ is a Statistics Professor at American University. LH is a Mathematical Statistician at the Food and Drug Administration (FDA) in the Office of Biostatistics. QX is a Statistician at American University. RCT is the Associate Director in the Office of Biostatistics at the FDA. The work was conducted while LH was a contractor and RCT was a Program Director at the National Cancer Institute (NCI), NIH, Bethesda, MD. The views expressed by the authors are not necessarily of those of FDA or NCI.
Declarations
Acknowledgements
The authors would like to thank Eric J. Feuer of the National Cancer Institute for his support of this project. We are grateful for programming assistance from Jun Luo and Mark Hachey of Information Management Science (IMS). We also thank James Cucinelli and Jeremy Lyman of IMS for their assistance with GIS. The authors also thank the referees and the associate editor for their valuable comments.
Authors’ Affiliations
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