### 统计代写|贝叶斯统计代写Bayesian statistics代考|Measures of spatial association for areal data

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## 统计代写|贝叶斯统计代写beyesian statistics代考|Measures of spatial association for areal data

Exploration of areal spatial data requires definition of a sense of spatial distance between all the constituting areal units within the data set. This measure of distance is parallel to the distance $d$ between any two point referenced spatial locations discussed previously in this chapter. A blank choropleth map, e.g. Figure $1.12$ without the color gradients, provides a quick visual measure of spatial distance, e.g. California, Nevada and Oregon in the west coast are spatial neighbors but they are quite a long distance away from Pennsylvania, New York and Connecticut in the east coast. More formally, the concept of spatial distance for areal data is captured by what is called a neighborhood, or a proximity, or an adjacency, matrix. This is essentially a matrix where each of its entry is used to provide information on the spatial relationship between each possible pair of the areal units in the data set.

The proximity matrix, denoted by $W$, consists of weights which are used to represent the strength of spatial association between the different areal units. Assuming that there are $n$ areal units, the matrix $W$ is of the order $n \times n$ where each of its entry $w_{i j}$ contains the strength of spatial association between the units $i$ and $j$, for $i, j=1, \ldots, n$. Customarily, wii is set to 0 for each $i=1, \ldots, n$. Commonly, the weights $w_{i j}$ for $i \neq j$ are chosen to be binary where it is assigned the value 1 if units $i$ and $j$ share a common boundary and 0 otherwise. This proximity matrix can readily be formed just by inspecting a choropleth map, such as the one in Figure 1.12. However, the weighting function can instead be designed so as to incorporate other spatial information, such as the distances between the areal units. If required, additional proximity matrices can be defined for different orders, whereby the order dictates the proximity of the areal units. For instance we may have a first order proximity matrix representing the direct neighbors for an areal unit,

a second order proximity matrix representing the neighbors of the first order areal units and so on. These considerations will render a proximity matrix, which is symmetric, i.e. $w_{i j}=w_{j i}$ for all $i$ and $j$.

The weighting function $w_{i j}$ can be standardized by calculating a new proximity matrix given by $\tilde{w}{i j}=w{i j} / w_{i+}$ where $w_{i+}=\sum_{j=1}^{n} w_{i j}$, so that each areal unit is given a sense of “equality” in any statistical analysis. However, in this case the new proximity matrix may not remain symmetric, i.e. $\tilde{w}{i j}$ may or may not equal $\tilde{w}{j i}$ for all $i$ and $j$.

When working with grid based areal data, where the proximity matrix is defined based on touching areal units, it is useful to specify whether “queen” or “rook”, in a game of chess, based neighbors are being used. In the $R$ package spdep, “queen” based neighbors refer to any touching areal units, whereas “rook” based neighbors use the stricter criteria that both areal units must share an edge (Bivand, 2020).

There are two popular measures of spatial association for areal data which together serve as parallel to the concept of the covariance function, and equivalently variogram, defined earlier in this chapter. The first of these two measures is the Moran’s $I$ (Moran, 1950) which acts as an adaptation of Pearson’s correlation coefficient and summarizes the level of spatial autocorrelation present in the data. The measure $I$ is calculated by comparing each observed area $i$ to its neighboring areas using the weights, $w_{i j}$, from the proximity matrix for all $j=1, \ldots, n$. The formula for Moran’s $I$ is written as:
$$I=\frac{n}{\sum_{i \neq j} w_{i j}} \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} w_{i j}\left(Y_{i}-\bar{Y}\right)\left(Y_{j}-\bar{Y}\right)}{\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}},$$
where $Y_{i}, i=1, \ldots, n$ is the random sample from the $n$ areal units and $\bar{Y}$ is the sample mean. It can be shown that $I$ lies in the interval $[-1,1]$, and its sampling variance can be found, see e.g. Section $4.1$ in Banerjee et al. (2015) so that an asymptotic test can be performed by appealing to the central limit theorem. For small values of $n$ there are permutation tests which compares the observed value of $I$ to a null distribution of the test statistic $I$ obtained by simulation. We shall illustrate these with a real data example in Section 3.4.
An alternative to the Moran’s $I$ is the Geary’s $C$ (Geary, 1954) which also measures spatial autocorrelation present in the data. The Geary’s $C$ is given by
$$C=\frac{(n-1)}{2 \sum_{i \neq j} w_{i j}} \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} w_{i j}\left(Y_{i}-Y_{j}\right)^{2}}{\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}} .$$
The measure $C$ being the ratio of two weighted sum of squares is never negative. It can be shown that $E(C)=1$ under the assumption of no spatial association. Small values of $C$ away from the mean 1 indicate positive spatial association. An asymptotic test can be performed but the speed of convergence to the limiting null distribution is expected to be very slow since it is a ratio of weighted sum of squares. Monte Carlo permutation tests can be performed and those will be illustrated in Section $3.4$ with a real data example.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Internal and external standardization for areal data

Internal and external standardization are two oft-quoted keywords in areal data modeling, especially in disease mapping where rates of a disease over different geographical (areal) units are compared. These two are now defined along with other relevant key words. To facilitate the comparison often we aim to understand what would have happened if all the areal units had the same uniform rate. This uniform rate scenario serves as a kind of a null hypothesis of “no spatial clustering or association”. Disease incidence rates in excess or in deficit relative to the uniform rate is called the relative risk. Relative risk is often expressed as a ratio where the denominator corresponds to the standard dictated by the above null hypothesis. Thus, a relative risk of $1.2$ will imply $20 \%$ increased risk relative to the prevailing standard rate. The relative risk can be associated with a particular geographical areal unit or even for the whole study domain when the standard may refer to an absence of the disease. Statistical models are often postulated for the relative risk for the ease of interpretation.

Return to the issue of comparison of disease rates relative to the uniform rate. Often in practical data modeling situation, the counts of number of individuals over different geographies and other categories, e.g. sex and ethnicity, are available. Standardization, internal and external, is a process by which we obtain the corresponding counts of diseased individuals under the assumption of the null hypothesis of uniform disease rates being true. We now introduce the notation $n_{i}$, for $i=1, \ldots, k$ being the total number of individuals in region $i$ and $y_{i}$ being the observed number of individuals with the disease, often called cases, in region $i$. Under the null hypothesis
$$\bar{r}=\frac{\sum_{i=1}^{k} y_{i}}{\sum_{i=1}^{k} n_{i}}$$
will be an estimate of the uniform disease rate. As a result,
$$E_{i}=n_{i} \bar{r}$$
will be the expected number of individuals with the disease in region $i$ if the null hypothesis of uniform disease rate is true. Note that $\sum_{i=1}^{k} E_{i}=\sum_{i=1}^{k} y_{i}$ so that the total number of observed and expected cases are same. Note that to find $E_{i}$ we used the observations $y_{i}, i=1, \ldots, k$. This process of finding the $E_{i}$ ‘s is called internal standardization. The word internal highlights the use of the data itself to perform the standardization.

The technique of internal standardization is appealing to the analysts since no new external data are needed for the purposes of modeling and analysis. However, this technique is often criticized since in the modeling process $E_{i}$ ‘s are treated as fixed values when in reality these are functions of the random observations $y_{i}$ ‘s of the associated random variables $Y_{i}$ ‘s. Modeling of the $Y_{i}$ ‘s while treating the $E_{i}$ s as fixed is the unsatisfactory aspect of this strategy. To overcome this drawback the concept of external standardization is often used and this is what is discussed next.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Spatial smoothers

Observed spatially referenced data will not be smooth in general due to the presence of noise and many other factors, such as data being observed at a

coarse irregular spatial resolution where observation locations are not on a regular grid. Such irregular variations hinder making inference regarding any dominant spatial pattern that may be present in the data. Hence researchers often feel the need to smooth the data to discover important discernible spatial trend from the data. Statistical modeling, as proposed in this book, based on formal coherent methods for fitting and prediction, is perhaps the best formal method for such smoothing needs. However, researchers often use many non-rigorous off-the shelf methods for spatial smoothing either as exploratory tools demonstrating some key features of the data or more dangerously for making inference just by “eye estimation” methods. Our view in this book is that we welcome those techniques primarily as exploratory data analysis tools but not as inference making tools. Model based approaches are to be used for smoothing and inference so that the associated uncertainties of any final inferential product may be quantified fully.

For spatially point referenced data we briefly discuss the inverse distance weighting (IDW) method as an example method for spatial smoothing. There are many other methods based on Thiessen polygons and crude application of Kriging (using ad-hoc estimation methods for the unknown parameters). These, however, will not be discussed here due to their limitations in facilitating rigorous model based inference.

To perform spatial smoothing, the IDW method first prepares a fine grid of locations covering the study region. The IDW method then performs interpolation at each of those grid locations separately. The formula for interpolation is a weighted linear function of the observed data points where the weight for each observation is inversely proportional to the distance between the observation and interpolation locations. Thus to predict $Y\left(\mathrm{~s}{0}\right)$ at location $\mathbf{s}{0}$ the IDW method first calculates the distance $d_{i 0}=\left|\mathbf{s}{i}-\mathbf{s}{0}\right|$ for $i=1, \ldots, n$. The prediction is now given by:
$$\hat{Y}\left(\mathbf{s}{0}\right)=\frac{1}{\sum{i=1}^{n} \frac{1}{d_{i 0}}} \sum_{i=1}^{n} \frac{y\left(\mathbf{s}{i}\right)}{d{i 0}}$$
Variations in the basic IDW methods are introduced by replacing $d_{i 0}$ by the $p$ th power, $d_{i 0}^{p}$ for some values of $p>0$. The higher the value of $p$, the quicker the rate of decay of influence of the distant observations in the interpolation. Note that it is not possible to attach any uncertainty measure to the individual predictions $Y\left(\mathbf{s}{0}\right)$ since a joint model has not been specified for the random vector $Y\left(\mathbf{s}{0}\right), Y\left(\mathbf{s}{1}\right), \ldots, Y\left(\mathbf{s}{n}\right)$. However, in practice, an overall error rate such as the root mean square prediction error can be calculated for set aside validation data sets. Such an overall error rate will fail to ascertain uncertainty for prediction at an individual location.

There are many methods for smoothing areal data as well. One such method is inspired by what are known as conditionally auto-regressive (CAR) models which will be discussed more formally later in Section 2.14. In implementing this method we first need to define a neighborhood structure.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Measures of spatial association for areal data

C=(n−1)2∑一世≠j在一世j∑一世=1n∑j=1n在一世j(是一世−是j)2∑一世=1n(是一世−是¯)2.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Internal and external standardization for areal data

r¯=∑一世=1ķ是一世∑一世=1ķn一世

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