#### PORTAL USER GUIDE

# Local Moran's I

**Local Moran’s I** is a local spatial autocorrelation statistic that identifies local clusters or local outliers to understand their contribution to the ‘global’ clustering statistic. It was developed by *Anselin (1995)* as a class of local indicators called Local Indicators of Spatial Association (LISAs). The local Moran’s I statistic offers insight into the behaviour of data at local levels, by providing a decomposition of the Moran’s I global statistic into the degree of spatial association associated with each observation. LISAs serve two purposes in exploratory spatial data analysis (ESDA): they indicate local spatial clusters and they perform sensitivity analysis (identify outliers).

*Anselin (1995, p. 94)* defines LISA as:

a. the LISA for each observation gives an indication of the extent of significant spatial clustering of similar values around each observation; and

b. the sum of LISAs for all observations is proportional to a global indicator of spatial association.

As with global measures, LISAs test whether the observed spatial pattern of a variable of interest amongst areas is extreme or is likely or expected, given a random geographic distribution of the variable.

**Local Moran’s I** is calculated as:

\LARGE{I_i=z_i\sum_{j}w_{ij}z_j}

Where the z_i and z_j are the observations in deviations from the mean and w_{ij} is the spatial weight matrix element. Positive values of I_i suggest that there is a spatial cluster of similar values and negative values represent a spatial cluster of dissimilar values.

Significance testing of the local Moran statistics can be somewhat problematic. Unlike the G_i and G_i* statistics, the local Moran does not conform to a common distribution and so the test under a normality assumption should be treated with caution. Instead *Anselin (1995)* suggests a conditional randomisation or permutation approach to give so-called pseudo significance levels. *Tiefelsdorf & Boots (1998)* published the exact reference distribution of Moran’s I, but in a later paper recommends an application of the Saddlepoint approximation, as it “outperforms other approximation methods with respect to its accuracy and computational costs” *(Tiefelsdorf, 2002, p. 187)*. The Local Moran’s I tool in the AURIN Portal provides results for a normal distribution, a Saddlepoint approximation of the standard deviate and the exact standard deviate.

### SET UP

To demonstrate this tool in use, we will look at socio-economic disadvantage data in Greater Darwin to examine the extent of spatial-autocorrelation.

**Select** *Greater Darwin GCCSA* as your area.

**Select ***ABS – Socio-Economic Indexes for Areas (SEIFA) – The Index of Relative Socio-economic Disadvantage (SA1) 2016* as your dataset, select *IRSD Score *as the variable.

Use the **Spatialise Aggregated Dataset** tool to Spatialise the dataset.

Use the **Contiguous Spatial Weight Matrix** tool to build a Spatial Weights Matrix for the spatialised dataset, using* 1st order*, *row-standardised*, *Queen contiguity*.

### Inputs

Open the **Local Moran’s I** tool (*Tools → Spatial Autocorrelation → Local Moran’s I*) and enter the following parameters:

*Dataset Input:*The dataset that contains the variable to be tested.**Select**the*Spatialised Dataset*.*Spatial Weights Matrix:*The spatial weight matrix to be used.**Select**the*Contiguous Spatial Weight Matrix*.*Key Column:*Specify the unique codes for your areas.**Select***SA1 11-digit Code*.*Variable:*The variable to be tested.**Select***IRSD Score***.***Alternative Hypothesis:*Specifies the alternative hypothesis.**Select***two.sided*.*two.sided:*a priori assumption that the difference between*I*and the expected E[_{i}*I*] is not equal to zero (spatial autocorrelation)._{i}*greater:*a priori assumption that*I*is greater than the expected E[_{i}*I*] (positive spatial autocorrelation)._{i}*less:*a priori assumption that*I*is less than the expected E[_{i}*I*] (negative spatial autocorrelation)._{i}

*Spatial Weights Matrix Style:*The standardisation style used for the input spatial weight matrix.**Select***row.standardised*.*Test:*Indicates the shape of the distribution used to test for significance.**Select***saddlepoint*.*saddlepoint:*Implements the*Tiefelsdorf (2002)*application of the Saddlepoint approximation.*exact:*Implements the*Tiefelsdorf & Boots (1998)*application of the exact reference distribution.

*Significance Level:*Set the level of significance for your test.**Select***0.05*.

The input parameters are summarised in the image below, once complete click **Run** **Tool**.

### Outputs

Your output will be a dataset that can be mapped based on a number of the variables produced by the analysis. These are explained below:

*li:*The Local Moran’s I statistic.*Saddlepoint:*The standard deviate of the Saddlepoint Local Moran’s I statistic, this can also be interpreted as a Z-score (only output if “saddlepoint ” is selected).*Exact SD:*The standard deviate of the Exact Local Moran’s I statistic, this can also be interpreted as a Z-score (only output if “exact” is selected).*Pr_Sad:*The p-value of the saddlepoint Local Moran’s I statistic (only output if “saddlepoint” is selected).*Pr_Exact:*The p-value of the exact Local Moran’s I statistic (only output if “exact” is selected).*<input_variable>_Lagged:*the average value of the variable for the areas surrounding each area.*<input_variable>_Scaled:*the value of the variable for your area scaled (z-score).*<input_variable>_Lagged_Scaled:*the average value of the variable for the areas surrounding each area, scaled (z-score).*<input_variable>_map_group:*the number representing the group that your area belongs to: 0 = Not Significant, 1 = High surrounded by High; 2 = High surrounded by Low; 3 = Low surrounded by High; and 4 = Low surrounded by low.*<input_variable>_map_group_name:*the names of the above groups.

You can create a visualisation using the **Choropleth** tool on one of the above variables. For the image below, we have chosen to use the variable *irsd_score_map_group_name.* The tool produces five groups in this variable, listed above. Using a *Qualitative* and *Dark2* palette:

- Orange indicates areas of high IRSD index scores surrounded by other high scores (low disadvantage),
- Purple indicates high scores surrounded by low scores,
- Pink indicates low scores surrounded by high scores,
- Light-green indicates low areas surrounded by low scores, and
- Dark-green represents areas with no statistical significance.

Anselin, L. (1995). Local Indicators of Spatial Association – LISA. *Geographical Analysis*. *27*(2), 93-115.

Tiefelsdorf, M. & Boots, B. (1995). The Exact Distribution of Moran’s I. *Environment and Planning A: Economy and Space*. *27*(6), 985-999.

Tiefelsdorf, M. (2002). The Saddlepoint Approximation of Moran’s I’s and Local Moran’s I’s Reference Distributions and Their Numerical Evaluation. *Geographical Analysis*. *34*(3), 187-206.