Spatial Data Analysis: Intro to Spatial Statistical Concepts Spatial Structures in the Social Sciences Spatial Data Analysis: Intro to Spatial Statistical Concepts Scott Bell GIS Institute Global Positioning Systems

Spatial Stats rely on Spatial Data Spatial Structures in the Social Sciences Spatial Stats rely on Spatial Data Traditional statistics are based on distributions of data along a single axis Spatial data by its nature exists on two axes (X and Y) I. E. the median in traditional statistics in the sum of all values divided by the number of observations Spatial mean is the X, Y coordinate result from calculating the means of X and Y Global Positioning Systems

Exploratory Spatial Data Analysis Spatial Structures in the Social Sciences Exploratory Spatial Data Analysis Used like descriptive statistics Potentially more options Related to Thematic Mapping and Geo-visualization Pattern identification/Hypothesis generation Global Positioning Systems

Traditional vs Spatial Spatial Structures in the Social Sciences Traditional vs Spatial “Independence of observations” Assumption Spatial Statistics operate on data that are assumed to be spatially dependent Spatial statistics (Spatial autocorrelation(SA)) have been developed to account for SA so distribution theory can be applied Global Positioning Systems

Traditional vs Spatial Spatial Structures in the Social Sciences Traditional vs Spatial “Replication” Assumption Spatial (and other systems) are complex and hard to replicate Precise Data Samples drawn from hypothetical universe In ability to replicate (and size and complexity of system) usually means our sample spatial data is the universe Distribution under null can be obtained by creating an experiment (environment) in which the null is true due to sample being universe it is virtually impossible to obtain the distribution under null hypothesis conditions First, they assume designed experiments that can be replicated. This notion of "replication" forms the philosophical basis of distribution theory. Usually, although not always, classical statistics assume precise data so that an observation's value could be known exactly if we employed sufficiently accurate and precise measuring instruments. This differs subtly from the notion of experimental error that is incorporated into most classical statistical models. Third, classical statistics assume the samples are drawn from a hypothetical universe or population. Again, this notion of a universe is part of the philosophical basis needed for distribution theory to apply. Fourth, classical statistics assume the distribution of the test statistic under the null hypothesis can be obtained by replicating a null experiment. By null experiment we mean the experiment conducted in a situation under which the null hypothesis is true. Examples of classical statistics include analysis of variance, t-test, regression, general linear models and so on. Spatial randomization tests are based on assumptions that differ substantially from those of classical tests. First, randomization tests assume observational data that cannot be replicated. Spatial systems are often large and complex, and it is usually difficult or impossible to conduct experiments on such systems. For this reason, we are often faced with only the data set at hand. This lack of replication means the sample is taken to be the universe -- we cannot sample from a larger population because the study itself cannot be replicated. These constraints mean the distribution under the null hypothesis cannot be obtained with any ease from distribution theory. Randomization tests side step this problem by randomizing the observations within the sample. A large body of literature has arisen around such randomization tests. For spatial randomization the text by Brian Manly is a good introduction. Global Positioning Systems

Spatial Autocorrelation Spatial Structures in the Social Sciences Spatial Autocorrelation What is it? Uses of spatial autocorrelation Types of spatial dependence Distance K-nearest neighbors Contiguity Rooks, bishops, and Kings cases “Everything is related to everything, but near things are more related.” (Tobler, 1976) Spatial autocorrelation is an assessment of the correlation of a variable in reference to spatial location of the variable. Assess if the values are interrelated, and if so is there a spatial pattern to the correlation, there is spatial autocorrelation. Spatial autocorrelation measures the level of interdependence between the variables, the nature and strength of the interdependence. Spatial autocorrelation may be classified as either positive or negative. Positive spatial autocorrelation has all similar values appearing together, while negative spatial autocorrelation has dissimilar values appearing in close association. Does the non-spatial variable (attribute) in adjacent spaces vary with the time or space separating them. Uses of assessment of spatial autocorrelation: - identification of patterns which may reveal an underlying process, - describe a spatial pattern and use as evidence, such as a diagnostic tool for the nature of residuals in a regression analysis, - as an inferential statistic to buttress assumptions about the data, - data interpolation technique. Global Positioning Systems

Spatial Autocorrelation Spatial Structures in the Social Sciences Spatial Autocorrelation Deal simultaneously with similarities in the location (space) of objects and their (non-spatial) attributes. (Goodchild, et. al. 2001) Similar location/Similar attribute = high spatial autocorrelation Similar location/dissimilar attributes = negative spatial autocorrelation Attributes are independent of location = zero/low correlation Global Positioning Systems

Spatial Structures in the Social Sciences Global Positioning Systems

Spatial Structures in the Social Sciences Correlation= -1.00 Correlation= -.393 Correlation= 0 Correlation= +.393 Correlation= +.857 Global Positioning Systems

Spatial Structures in the Social Sciences Global Positioning Systems

Spatial Regression (in GeoDa and ArcGIS) Allows for control of spatially auto-correlated error or DV (non-independent observations) Error: Unexplained variation in DV is related to nearby values of error Lag: spatial dependence in DV, additional IV term added to model