Beyond Moran's I: Testing for Spatial Dependence Based on the Spatial Autoregressive Model

Abstract

The statistic known as Moran's I is widely used to test for the presence of spatial dependence in observations taken on a lattice. Under the null hypothesis that the data are independent and identically distributed normal random variates, the distribution of Moran's I is known, and hypothesis tests based on this statistic have been shown in the literature to have various optimality properties. Given its simplicity, Moran's I is also frequently used outside of the formal hypothesis‐testing setting in exploratory analyses of spatially referenced data; however, its limitations are not very well understood. To illustrate these limitations, we show that, for data generated according to the spatial autoregressive (SAR) model, Moran's I is only a good estimator of the SAR model's spatial‐dependence parameter when the parameter is close to 0. In this research, we develop an alternative closed‐form measure of spatial autocorrelation, which we call APLE, because it is an approximate profile‐likelihood estimator (APLE) of the SAR model's spatial‐dependence parameter. We show that APLE can be used as a test statistic for, and an estimator of, the strength of spatial autocorrelation. We include both theoretical and simulation‐based motivations (including comparison with the maximum‐likelihood estimator), for using APLE as an estimator. In conjunction, we propose the APLE scatterplot, an exploratory graphical tool that is analogous to the Moran scatterplot, and we demonstrate that the APLE scatterplot is a better visual tool for assessing the strength of spatial autocorrelation in the data than the Moran scatterplot. In addition, Monte Carlo tests based on both APLE and Moran's I are introduced and compared. Finally, we include an analysis of the well‐known Mercer and Hall wheat‐yield data to illustrate the difference between APLE and Moran's I when they are used in exploratory spatial data analysis.

Motivation

The spatial autoregressive (SAR) model is commonly used to analyze spatial processes on a lattice. Following the notation of Ord (1975), we specify a SAR model for Z≡(Z(s₁), …, Z(s_n))′, a vector of observations on (possibly irregular) lattice locations {s_i: i=1, …, n}, by letting

(1)

The matrix W≡{w_ij} is a known spatial‐neighborhood matrix with elements w_ii=0 for i=1, …, n, and ɛ≡(ɛ(s₁), …, ɛ(s_n))′ is a vector of independently and identically distributed normal random variables, each with mean zero and variance σ². As Z=(I−ρW)⁻¹ɛ, it is clear that E(Z)=0. In practice, data will almost always have to undergo some detrending. For the rest of the article, we shall assume that this detrending has already taken place, and hence it is appropriate for Z to have mean 0.

In exploratory analyses of spatial data, a formal statistical model may not be explicitly assumed. However, we argue that in these situations the informal notion of spatial dependence, or spatial autocorrelation, is often implicitly based on a SAR framework where the goal is to assess the predictive ability of neighboring values of the data. In order for informal assessments of the strength of spatial autocorrelation to translate into this implied formal statistical modeling framework, exploratory spatial data analysis (ESDA) tools should be based on estimators of ρ, the spatial autocorrelation parameter in the SAR model (equation [1]). However, obvious estimators of ρ, such as the maximum‐likelihood estimator (MLE), cannot be written in a closed form; consequently, they are impractical to calculate in exploratory analysis. Therefore, instead of computing an estimate of ρ, the statistic known as Moran's I, Z′WZ/Z′Z, is commonly used as a measure of spatial autocorrelation in exploratory analyses (Moran 1950; see also “Background”). Due to its simplicity and widespread use in exploratory analyses of spatially referenced data, it is tempting to interpret Moran's I as an estimator of ρ, although the early literature (Ord 1975; Anselin 1988) makes it clear that this interpretation is inappropriate. Still, Moran's I is often referred to as a coefficient of spatial autocorrelation making it tempting, in practice, to misuse it as an estimator of ρ. Thus, there is a need for a comparable (closed‐form and easy‐to‐compute) measure of spatial autocorrelation that can also be used to assess, or efficiently estimate, the strength of spatial autocorrelation in a process on a lattice.

In this article, we derive a closed‐form likelihood‐based estimator of the spatial‐autocorrelation parameter ρ in the SAR model (1). We use the notation APLE below, as it is an approximate profile‐likelihood estimator (APLE) of ρ. We show by simulation that, for data generated according to the SAR model, APLE provides a better assessment of the strength of spatial dependence in the process, especially when the true value of ρ is not close to 0, than both Moran's I and another closed‐form estimator of ρ, Ord's statistic. We also provide a simulation‐based comparison of APLE and the MLE of ρ to motivate further the use of APLE as an estimator of ρ. In addition to exploring the properties of APLE itself, we also propose an APLE scatterplot, which is analogous to the Moran scatterplot, a commonly used spatial exploratory analysis tool (Anselin 1996). We show that the APLE scatterplot is a more informative spatial exploratory tool than the Moran scatterplot.

Background

In this section, we provide a brief historical overview of the development of Moran's I and of previous work on the estimation of ρ in the SAR model. In addition, we illustrate the consequence of incorrectly interpreting Moran's I as an estimator of ρ. While it is named after P. A. P. Moran, in its current form the statistic known as Moran's I was not developed until a full 20 years after Moran's (1950) initial article. Moran (1950) first proposed a test statistic to assess the degree of spatial autocorrelation between adjacent locations. In order to construct this statistic for n independent variates {X_i: i=1, …, n} on lattice locations {s_i: i=1, …, n}, we define δ_ij to be an indicator such that δ_ij=1 if the ith and jth locations are “adjacent” (defined a priori), and δ_ij=0 otherwise. Assuming that the observations have constant mean, the obvious way to detrend the data is to define . Originally, Moran's (1950) test statistic was defined as

(2)

Subsequently, Cliff and Ord (1981) proposed a statistic to test for a more general form of spatial dependence in the residuals from a linear regression model. Their statistic was defined by analogy with an approach first advanced by Durbin and Watson (1950) in the context of testing for serial correlation in time series, where {s_i=i: i=1, …, n}. If Z is the column vector of residuals from a linear regression model, and A is a real symmetric matrix, the Durbin–Watson test statistic is d≡Z′AZ/Z′Z. A special case of this statistic is approximately equal to , which is 2(1−I⁰) in (2), for δ_ij=1 if j=i+1 and 0 otherwise. For testing the null hypothesis that ρ=0 in the SAR model (1), Cliff and Ord (1981) proposed the test statistic

(3)

which they called Moran's I. Recall that W is the spatial‐neighborhood matrix in (1) and notice that Z′WZ=Z′[(W+W′)/2]Z; hence Moran's I is often defined with a symmetrized version of the matrix W. Statistic (3) has the same form as the Durbin–Watson statistic d, but it is developed in a spatial context. Cliff and Ord informally argued that, for values of ρ in the neighborhood of 0, their test coincides with the likelihood ratio test of the null hypothesis that ρ=0 versus the alternative hypothesis that ρ is some given (nonzero) value. (However, in their argument they did not consider the composite alternative hypothesis that ρ≠0.) In addition, they showed that, given certain regularity assumptions, the distribution of I under the null hypothesis is asymptotically normal. Other articles have demonstrated additional properties of Moran's I. Burridge (1980) showed that the test based on I is identical to the Lagrange multiplier test and is therefore asymptotically equivalent to the likelihood ratio test. Furthermore, King (1981) showed I to be locally best invariant (LBI) in the neighborhood of ρ=0.

By analogy to the calculation of the exact distribution of the Durbin–Watson statistic d for serial autocorrelation of regression residuals, Tiefelsdorf and Boots (1995) calculated the exact small‐sample distribution of Moran's I under the spatial independence assumption (i.e., ρ=0 in equation [1]) by numerical integration. For ρ not necessarily equal to 0 in (1), Tiefelsdorf (2000) developed the distribution of Moran's I using the distribution theory for the ratio of quadratic forms. Moreover, Tiefelsdorf (2002) showed that the saddlepoint method applied to the ratio of quadratic forms in normal random variables can be used to obtain an accurate and computationally efficient approximation to the sampling distribution of Moran's I.

We now review briefly previous work on the estimation of ρ in the SAR model (1). Ord (1975) derived a least squares estimator of ρ, namely Z′WZ/Z′W′WZ. Although Ord's statistic is not a consistent estimator of ρ, we consider it as a potential alternative to Moran's I in exploratory analyses of spatial data as it has a closed form. In the simulation study introduced below and considered further in “Comparison of APLE with other statistics via simulation,” we demonstrate that our proposed statistic, APLE, is superior to Ord's statistic. Because this first statistic is not consistent, Ord then suggested a modified least squares estimator, which is the solution to the following quadratic equation in ρ:

and therefore not available in the closed form. Although this estimator is consistent, Ord illustrates that its efficiency relative to the MLE declines drastically as ρ increases. Unfortunately, the MLE of ρ can only be obtained by numerically determining the maximum value of equation (5) (see “The APLE statistic”), making it impractical to calculate when performing exploratory analyses. Finally, Kelejian and Prucha (1999) proposed a generalized moments estimator of ρ; however, like the MLE, this estimator does not have a closed form.

In contrast to the MLE of ρ, Moran's I is straightforward to evaluate. As a result, it is commonly used in exploratory analyses of spatial data. Despite its widespread use, Moran's I may only be interpreted as a measure of spatial dependence and is not a good estimator of ρ, especially when ρ is not near 0. While this limitation of Moran's I is known, for illustrative purposes, we briefly present results from a simulation study (discussed in further detail in “Comparison of APLE with other statistics via simulation”) that demonstrates how Moran's I performs as an estimator of ρ. This simulation study consists of generating data according to the SAR model (1) for known (or true) values of ρ=. We specify the spatial neighborhood matrix W to be row standardized (i.e., the rows of W sum to 1) and to be defined according to a second‐order (four first‐order and four second‐order neighbors) neighborhood structure on a 10 × 10 toroidal lattice. Hence, each location has an equal number of neighbors (eight neighbors for the second‐order neighborhood structure used), which removes the potential for bias due to boundary effects and implies that W is symmetric. When generating data from the SAR model, we consider values of 0≤<1. Although the parameter space of ρ is (−∞, ∞), this restriction guarantees that the variance matrix of the SAR model, {(I−ρW′)(I−ρW)}⁻¹, is invertible. This sufficient condition can be shown by considering the inverse of the variance matrix, (I−ρW′)(I−ρW), which has eigenvalues {|1−ρλ_i | ²: i=1, …, n}, where {λ_i, i=1, …, n} are the eigenvalues of W. Therefore, invertibility is guaranteed if ρ∉{1/λ_i, i=1, …, n}. As W is row standardized and symmetric, we know that {λ_i}≤1. This condition implies that the variance matrix is invertible for all 0≤ρ<1. As we are typically interested in positive spatial dependence, we note that we do not consider values of data generated according to a SAR model with <0. Returning to the description of our simulation study, we performed analyses for equal to 0, 0.1, 0.5, and 0.9 and took σ² to be 1. For each of these values of , we generated 5000 data sets consisting of samples from the corresponding SAR model. For each of these data sets, we then calculated Moran's I and compared it with . The third column of Fig. 1 provides histograms of the distribution of Moran's I for the data sets generated for each value of , which is represented by the vertical line. Clearly, the distribution of Moran's I is centered around for near 0. However, for the other values of (including 0.1), the distribution is shifted away from . These simulation results demonstrate the potential for erroneous conclusions in statistical procedures where the strength of spatial autocorrelation is being explored through Moran's I.

Figure 1

A comparison of the sampling distributions of APLE (first column), the MLE (second column), Moran's I (third column), and Ord's statistic (fourth column) obtained from data generated according to the SAR model (1) on a 10 × 10 toroidal lattice with (vertical bold line) equal to 0, 0.1, 0.5 and 0.9, based on 5000 simulations. APLE, approximate profile‐likelihood estimator; MLE, maximum‐likelihood estimator; SAR, spatial autoregressive.

The APLE statistic

Recall the SAR model (1) for data Z≡(Z(s₁), …, Z(s_n))′ on a lattice {s_i: i=1, …, n}. Assuming that I−ρW is invertible, it is straightforward to observe that

Consequently, the likelihood function of ρ (and σ²) can be easily obtained. In this section, using the likelihood function, we derive an APLE of ρ, which is

(4)

where λ is the vector of eigenvalues of the spatial neighborhood matrix W. Comparing APLE to Moran's I, both statistics can be written as the ratio of quadratic forms. As Z′[(W+W′)/2]Z=Z′WZ, it is clear that both statistics have the same numerator. However, their denominators differ; the denominator of APLE has a weighted form, , with weights {a_ij} obtained from the entries of the matrix [W′W+λ′λI/n].

As under the SAR model, data Z are distributed as N(0, (I−ρW)⁻¹ (I−ρW′)⁻¹σ²), the likelihood function of ρ and σ² is given by

(5)

Maximization of (5) with respect to ρ and σ² yields the MLEs of these parameters. However, a closed‐form solution to this maximization problem is not available (Ord 1975). Consequently, the MLE is usually approximated numerically using either the Newton–Raphson or Fisher's scoring algorithm. Rather than maximizing the likelihood function directly, we use the profile likelihood to derive APLE. For fixed ρ, the MLE of σ² is ; we can substitute this estimate into (5), yielding the profile‐likelihood function of ρ. After making this substitution, we define the negative profile log‐likelihood function of ρ as

(6)

By minimizing (6) with respect to ρ, we can obtain the maximum profile‐likelihood estimator of ρ. However, this estimator again is not available in closed form. In order to derive an estimate of ρ in closed form, we consider the profile‐likelihood estimating equation obtained by setting the first derivative of (6) with respect to ρ equal to 0. As , where {λ_i} are the eigenvalues of W, this profile‐likelihood estimating equation for ρ can be written as

(7)

Now, we approximate the first term on the left‐hand side of equation (7) using a Taylor series expansion, that is, , because . Similar approximations to the second term allow us to discard terms of order ρ² and higher in (7), yielding the following approximation to the profile‐likelihood estimating equation:

Solving this equation for ρ, we obtain the APLE for ρ, given by (4). While we could expect this approximation to break down for large |ρ|, our simulations show that it does quite well for ρ positive and away from 0.

Comparison of APLE with other statistics via simulation

In this section, we compare our proposed statistic, APLE, with both Moran's I and Ord's statistic (because they are alternative closed‐form measures of spatial autocorrelation) using a simulation study. We use the MLE of ρ as a basis for assessing the properties of these three closed‐form statistics because the MLE has various asymptotic optimality properties. However, we do not consider the MLE to be an acceptable alternative exploratory tool because it is not available in closed form, but rather compute it for purposes of comparison. In the simulation study, the data are generated in the same manner as described at the end of “Background.”Fig. 1 contains histograms representing the sampling distributions of APLE, the MLE, Moran's I, and Ord's statistic for equal to 0, 0.1, 0.5, and 0.9. The sampling distribution of APLE is centered around , regardless of whether is near 0 or not. This is in contrast to the behavior of both Moran's I and Ord's statistic; their sampling distributions are only centered around when is near 0. Therefore, compared with Moran's I and Ord's statistic, APLE appears to provide a much better estimate of ρ, especially when the true value of ρ is not close to 0. The sampling distributions of the MLE and APLE are nearly identical; both are centered near , and they have similar shapes.

To compare the sampling distributions of APLE and the MLE further, we provide Q–Q plots (scatterplots of the corresponding empirical quantiles of the two distributions) in Fig. 2. The Q–Q plots show that the profile‐likelihood approximation used to define APLE is quite accurate when the level of spatial dependence is small or moderate. For larger values of , the quantiles of the sampling distributions of APLE and the MLE do not line up as well, as expected.

Figure 2

Q–Q plot to compare the sampling distribution of the MLE and APLE obtained from the SAR model (1) on a 10 × 10 toroidal lattice with equal to 0, 0.1, 0.5 and 0.9, based on 5000 simulations. APLE, approximate profile‐likelihood estimator; MLE, maximum‐likelihood estimator; SAR, spatial autoregressive.

Finally, we compare sampling distributions of APLE and the MLE over a finer resolution of values of 0≤<1, over different grid sizes (10 × 10 or n=100, 30 × 30 or n=900, and 50 × 50 or n=2500), and for both first‐ and second‐order neighborhood structures (i.e., four nearest neighbors and eight nearest neighbors, respectively). Fig. 3 compares the mean and 0.025 and 0.975 quantiles of the sampling distributions of both statistics. Clearly, these distributions are not identical. However, for purposes of exploratory assessments of the strength of spatial autocorrelation, APLE appears to perform reasonably well. In addition, for both estimators, Fig. 4 displays the mean squared error (MSE), a commonly used measure of the performance of an estimator, which can be calculated via

Figure 3

Comparison of the means (black lines) and 0.025 and 0.975 quantiles (gray lines) of the sampling distributions of APLE (solid lines) and the MLE (dashed lines) for data generated from the SAR model with various values of . The data sets were generated on 10 × 10 (left column), 30 × 30 (middle column), and 50 × 50 (right column) toroidal lattices. The plots in the top row correspond to a first‐order spatial‐neighborhood structure, and the plots in the bottom row correspond to a second‐order spatial‐neighborhood structure. The dotted 45° line represents the ideal result where . APLE, approximate profile‐likelihood estimator; MLE, maximum‐likelihood estimator; SAR, spatial autoregressive.

Figure 4

Comparison of the MSE of APLE (solid line) and the MLE (dashed line) for different data generated from the SAR model with various values of . The data sets were generated on 10 × 10 (left column), 30 × 30 (middle column), and 50 × 50 (right column) toroidal lattices. The plots in the top row correspond to a first‐order spatial‐neighborhood structure, and the plots in the bottom row correspond to a second‐order spatial‐neighborhood structure. APLE, approximate profile‐likelihood estimator; MLE, maximum‐likelihood estimator; SAR, spatial autoregressive; MSE, mean squared error.

Again, we see that the performance of the two statistics is nearly identical using this measure, especially for values of <0.9.

The APLE scatterplot

Proposing an ESDA tool, Anselin (1996) interpreted Moran's I as a regression coefficient in a regression of WZ on Z. This interpretation provides a way to visualize the linear association between Z and WZ in the form of a bivariate scatterplot of WZ against Z. Anselin referred to this plot as the Moran scatterplot. He also pointed out that the least squares slope in a regression through the origin is equal to Moran's I, although we would like to add that its significance (using the standard t test for linear regression) is not appropriate.

Based on the same idea, APLE, given by (4), can be visualized as a least squares regression coefficient. Let

(8)

and

(9)

Then APLE can be thought of as a regression coefficient in a regression of Y on X through the origin, that is,

We propose a visualization tool based on this decomposition of APLE, which is analogous to the Moran scatterplot and which we call the APLE scatterplot. The APLE scatterplot consists of plotting points {(X_i, Y_i): i=1, …, n}, where X≡(X₁, …, X_n)′is given by (8) and Y≡(Y₁, …, Y_n)′ is given by (9). Superimposed on the scatterplot of X–Y points is the regression line through the origin whose slope is given by APLE. To illustrate the APLE scatterplot, we simulated data Z from , which is the SAR model (1) with ρ=, on a 10 × 10 square lattice with and σ²=1. For this simulated data set Z, Moran's I equals 0.207, while APLE equals 0.483, which is much closer to . Fig. 5 illustrates both the APLE and the Moran scatterplots corresponding to these simulated data. As we would expect, the least squares line corresponding to APLE (dashed line) is virtually indistinguishable from the line with slope , represented by the solid line. However, there is a substantial difference between the dashed line corresponding to Moran's I and the solid line with slope (the true value of ρ).

Figure 5

APLE and Moran scatterplots for simulated data generated from the SAR model (1) with equal to 0.5. The solid line is a line through the origin with slope 0.5; the dashed line is a line through the origin with slope given by the statistic (APLE on the left plot and Moran's I on the right plot). APLE, approximate profile‐likelihood estimator; SAR, spatial autoregressive.

APLE as a test statistic

While our underlying motivation for the APLE statistic is the need for an easy‐to‐compute estimator of ρ, APLE can also be used as a test statistic. To illustrate this use of APLE, consider testing the hypothesis H₀: ρ=ρ₀ versus H₁: ρ≠ρ₀ for data generated according to the SAR model (1). We show in Appendix A that the test based on APLE can be derived from the Lagrange multiplier test statistic, up to first‐order terms in a Taylor‐series expansion. A related approach has been used to motivate the use of Moran's I as a test statistic (e.g., Anselin 1988), where the null hypothesis is H₀: ρ=0. Moreover, a score statistic derived in the case where σ² is a nuisance parameter can also be shown to yield, up to first‐order terms, the APLE statistic (see Appendix B).

Rather than comparing exact or large‐sample tests based on APLE and Moran's I, we compare the two test statistics within the Monte Carlo testing framework. To set up a Monte Carlo hypothesis test, we first define the null and alternative hypotheses H₀: ρ=ρ₀ and H₁: ρ≠ρ₀, respectively. We let denote K values of the statistic (either APLE or Moran's I) generated by independently simulating the spatial data set K times under the null hypothesis ρ=ρ₀. When simulating the data set, we can assume, without loss of generality, that σ²=1 because both APLE and Moran's I do not depend on σ². For large K, we obtain the acceptance region (e.g., Hope 1968; Kornak, Irwin, and Cressie 2006) at the α significance level, where denotes the smallest integer number that is larger than x and U_(i) denotes the ith‐order statistic. In order to compare the Monte Carlo tests based on APLE and Moran's I, we consider each test for a range of values of ρ under the null hypothesis. For ρ₀∈{−0.9, −0.8, …, 0.9}, we perform the Monte Carlo test of H₀: ρ=ρ₀, that is, we obtain the acceptance region based on simulating K=5000 data sets on a 10 × 10 lattice, assuming that ρ=ρ₀ and σ²=1. Then, for each value of ρ₀, we determine the upper and lower bounds of the acceptance region for both Monte Carlo tests (i.e., based on I and on APLE) with α=0.05 and plot them as a function of ρ₀; see Fig. 6. The solid curves in the figure are obtained by linearly interpolating the upper and lower bounds of the acceptance regions from the APLE‐based Monte Carlo test for values for ρ₀ between −1 and 1, and the dashed curves correspond to linear interpolations of the upper and lower bounds of the acceptance regions from the Moran's I–based Monte Carlo tests for values for ρ₀ between −1 and 1.

Figure 6

Acceptance regions for (APLE and Moran's I). The two solid curves represent the upper and lower bounds of the acceptance region based on APLE; the two dashed curves represent the upper and lower bounds of the acceptance region based on Moran's I. For illustration, intervals A and B are the acceptance regions corresponding to the APLE‐ and Moran's I–based Monte Carlo tests, respectively, when ρ₀=0.5. APLE, approximate profile‐likelihood estimator.

Using Fig. 6, we can approximate the acceptance region for both the APLE‐ and Moran's I–based Monte Carlo tests. To illustrate, consider the null hypothesis ρ₀=0.5. The intersections of the vertical line at ρ₀=0.5 with the solid and dashed curves indicate the acceptance regions (on the vertical axis) for both the APLE and Moran's I–based tests, respectively. These acceptance regions are represented by arrows to the left of the vertical axis: the acceptance region for the test based on APLE is represented as interval A and the acceptance region for the test based on Moran's I is represented as interval B. Comparing these two acceptance regions, we see that the true value of ρ falls inside A, but outside B. As a result, it is difficult to interpret the test based on Moran's I for ρ₀=0.5. By including a gray 45° line on the plot, it is apparent that only for values of ρ₀ near 0 will the gray line pass through the acceptance region based on Moran's I, that is, only for these values of ρ₀ near 0 will the acceptance region for the Moran's I–based test include . For other values of ρ₀, Moran's I falls outside the acceptance region, and therefore the test is difficult to interpret. This undesirable property of the Moran's I–based Monte Carlo test is not surprising because Moran's I is not a good estimator of ρ for values of ρ away from 0, as illustrated by the simulation study in “Background” and “Comparison of APLE with other statistics via simulation.”

By inverting the acceptance regions obtained from the two Monte Carlo tests, we can obtain confidence intervals for ρ. Fig. 7 illustrates this procedure. Suppose we have a statistic , where can be APLE or Moran's I. If we draw a horizontal line at , then the intersections of this line with the acceptance region curves (illustrated by the two squares in Fig. 7) define the corresponding (1−α) × 100=95% confidence interval for ρ. From Fig. 7, we can determine that for values of near 0, both the APLE‐ and Moran's I–based Monte Carlo tests provide nearly identical confidence intervals, which both include 0. However, for , say, the confidence interval based on APLE includes the true value ρ=0.5, while the confidence interval based on Moran's I does not include the true value ρ=0.5. As with the previous figure, the 45° gray line is included in Fig. 7 in order to identify the range of values of where the confidence interval based on Moran's I does not cover the true value, . The fact that the confidence intervals based on APLE include for all values of between −1 and 1 provides further evidence that APLE is a better statistic upon which to base inference than Moran's I, whenever is not close to 0.

Figure 7

Illustration of the procedure for obtaining confidence intervals for ρ using APLE‐ and Moran's I–based Monte Carlo tests. Confidence interval A corresponds to both tests when . Confidence interval B is constructed from the APLE‐based Monte Carlo test with (the corresponding confidence interval from the Moran's I–based Monte Carlo test does not include and is not shown for clarity). APLE, approximate profile‐likelihood estimator.

Illustrative example

In this section, we illustrate the difference between the values of APLE, the MLE, and Moran's I calculated for a real data set. We test for the presence of spatial dependence in the famous wheat‐yield data of Mercer and Hall (1911) using all three statistics. These data were used by both Whittle (1954) and Besag (1974) to illustrate their models for spatial dependence. The wheat yields are from 10.82 × 8.50 feet plots and were collected in the summer of 1910. There were a total of 500 of these plots arranged in a lattice with 20 rows (8.50 feet) running east to west and 25 (10.82 feet) columns running north to south. Cressie (1993, p. 250) showed the presence of an east–west trend in the data. Consequently, we subtracted the column median to remove this trend and then subtracted the overall mean. We then assumed that the residuals follow the zero‐mean SAR model given by (1). Based on a variogram analysis in Cressie (1985, 1993), we chose a row‐standardized spatial neighborhood matrix W defined using a third‐order (i.e., 12 nearest neighbors) neighborhood structure. For these data, Moran's I= 0.194, MLE=0.603, and APLE=0.661. Fig. 8 provides an APLE scatterplot for the data. In addition, in order to display the uncertainty in the slope, a 95% confidence cone is included on the APLE scatterplot. This confidence cone is constructed by shading the area between the lines through the origin with slopes equal to the upper and lower confidence bounds, derived from the APLE‐based Monte Carlo test described in “APLE as a test statistic” for a sample size of n=20 × 25=500, a third‐order neighborhood structure, and using K=5000 simulated data sets. For comparison, we also provide the confidence cone of the MLE, which is calculated using the information matrix (Cressie 1993, p. 483). We see that the confidence cone based on the MLE is nearly identical to the confidence cone based on APLE. By examining this plot, we can visualize the estimate of ρ given by APLE and assess our confidence in this estimate. Given that the APLE‐based confidence cone does not include the horizontal line ρ=0, we reject the null hypothesis H₀: ρ=0.

Figure 8

APLE scatterplot for the Mercer and Hall wheat‐yield data. The value of APLE is represented by the slope of the middle solid line. A confidence cone (area inside the two outer solid lines), derived by inverting the 95% acceptance region of an APLE‐based Monte Carlo test, is also included; see the text for details. For comparison, the MLE is included, represented by the slope of the middle dashed line. The 95% confidence cone for the MLE is almost identical to the 95% confidence cone for APLE and therefore is omitted. APLE, approximate profile‐likelihood estimator; MLE, maximum‐likelihood estimator.

Conclusion

In this article, we develop APLE, a closed‐form measure of spatial dependence. We assess the performance of this statistic as an estimator of the spatial dependence parameter, ρ, in the SAR model and in terms of testing the null hypothesis ρ=ρ₀. We show by simulation that APLE provides a better assessment of the strength of spatial autocorrelation for data generated according to the SAR model, especially when the spatial dependence parameter ρ is not near 0, than alternative measures of spatial dependence such as Moran's I and Ord's statistic. To further assess the properties of APLE as an estimator of ρ, we include a simulation‐based comparison of APLE with the MLE of ρ. As the MLE does not have a closed form, we do not consider it to be a comparable statistic for our purposes. However, we include it in our simulation study for purposes of assessing APLE as an estimator and we see that APLE and MLE are very similar. In addition, to formally test the strength the hypothesis H₀: ρ=ρ₀, where ρ₀ may not equal 0, we derived a Monte Carlo test based on APLE. Properties of this test were obtained from simulation and shown to be superior to the corresponding test based on Moran's I. Finally, our analysis of the Mercer and Hall (1911) wheat‐yield data demonstrated the use of APLE as an ESDA tool. In this work, we have assumed that E(Z)=0, but acknowledge that, in practice, data will almost always have to undergo some detrending before APLE is computed. After initially detrending the data and calculating APLE, it is possible then to obtain a generalized least squares estimator of unknown parameters in the mean structure by substituting APLE into the data's covariance matrix. While this sequential approach is commonly used in spatial statistics, we plan to consider extensions of our methods for estimating ρ in the regression setting. In addition to relaxing the assumption that E(Z)=0, in future work, we shall look at the effect of keeping higher‐order terms in the Taylor‐series approximation that gives rise to APLE, as well as exploring computational issues related to computing APLE for large data sets. Finally, we also plan to generalize APLE for other models of spatial dependence.

Acknowledgements

We would like to thank the editor and the referees for their comments on the original submission, which improved the motivation for the use of APLE in exploratory analyses of spatial data. This research was supported by the Office of Naval Research under grant number N00014‐05‐1‐0133.

Appendix A: Lagrange multiplier test

Burridge (1980) showed that Moran's I is identical to the Lagrange multiplier test statistic for the test with null hypothesis ρ=0 versus the alternative hypothesis ρ≠0. In this section, we show that APLE, up to first order in ρ, can be derived from the Lagrange multiplier test under a general null hypothesis ρ=ρ₀, where ρ₀ may not be 0. For the test H₀: ρ=ρ₀ versus H₁: ρ≠ρ₀, the log‐likelihood function of ρ and σ² is given by

(10)

Under the null hypothesis, imposing the restriction that ρ=ρ₀ yields the restricted log‐likelihood:

where the Lagrange multiplier λ is to be evaluated. The estimating equations are given by

(11)

(12)

(13)

Equations (11) and (13) imply that

and

From equations (12) and (13), the restricted MLE of λ is

(14)

which is the Lagrange multiplier test statistic. If differs significantly from 0, this indicates that the data do not support the null hypothesis. To estimate ρ using the Lagrange multiplier framework, in equation (14), we replace ρ₀ with ρ, set , and solve for ρ. Using a Taylor‐series expansion, we derive , as before. Then equation (14) implies

(15)

Substituting , obtained from equation (11), into equation (15) and keeping only first‐order terms in ρ, we obtain an estimator of ρ,

which is APLE, given by (4).

Appendix B: Score test

In this section, we show that APLE, up to first order in ρ, can be derived from the score statistic. Recall that the log‐likelihood function of ρ and σ² is given in equation (10). For the test H₀: ρ=ρ₀ versus H₁: ρ≠ρ₀, ρ is the parameter of interest and σ is the nuisance parameter. In our context, the score statistic, as defined by Cox and Hinkley (1974), is

(16)

where is the MLE of σ under ρ=ρ₀. In equation (16),

where J is the 2 × 2 information matrix with diagonal elements J_.ρρ, J_.σσ, and off‐diagonal element J_.ρσ. It is clear that ; hence, in this special case, the score statistic does not depend on b., a function of both the parameter of interest under the null hypothesis and the MLE of the nuisance parameter (see Cox and Hinkley 1974, p. 324, for the exact form of b.). Consequently, using the score statistic (16) simplifies to

(17)

We know Q_S is asymptotically distributed as (Cox and Hinkley 1974). Therefore, for the null hypothesis H₀: ρ=0 versus the alternative hypothesis H₁: ρ≠0, we accept H₀ if is close to 0. Using the log‐likelihood of ρ and σ² as given by equation (10), it can be shown that

(18)

We note in passing that, upon substituting ρ₀=0 and into equation (18), we obtain the test statistic

which is equivalent to Moran's I, up to a constant. For testing the general null hypothesis H₀: ρ=ρ₀ versus the alternative H₁: ρ≠ρ₀, we accept H₀ if is close to 0, where . To get an estimate of ρ, we apply the same strategy as in Appendix A and set . This yields the estimating equation, equation (7), whose solution, up to first order in ρ, is the statistic APLE.