chapter 13: correlation an introduction to statistical problem solving in geography as reviewed by:...
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Chapter 13:Correlation
An Introduction to Statistical Problem Solving in Geography
As Reviewed by:Michelle Guzdek
GEOG 3000
Prof. Sutton
2/27/2010
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Learning Objectives
The Nature of Correlation Association of Interval/Ratio Variables Association of Ordinal Variables Use of Correlation Indices in Map
Comparison Issues Regarding Correlation
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Introduction
One of the more important concerns in geographic analysis is the study of the relationships between spatial variables
Many geographic studies involve determining the degree of relationship between two or more map patterns
Using visual comparison to measure correspondence or association is subjective Two people can view the same maps and
interpret their association very differently
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Introduction (cont.)
Focus of geographic inquiry is often to establish the spatial association between two variables
Correlation analysis provides a more objective, quantitative means to measure the association between a pair of spatial variables Both direction and strength of association
between two variables can be determined statistically
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Nature of Correlation
Common tool for portraying the relationship or association between two variables is a two-dimensional graph called a scattergram, or scatterplot One variable plotted on each axis Provides an understanding of the nature of a
particular relationship Can determine direction (positive or negative) and
strength of association Any two variables can be correlated and the strength
and direction of relationship calculated IMPORTANT NOTE: A relationship or association
between variables does not necessarily imply the existence of a cause and effect relationship
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Scattergram or Scatterplot
Three examples of scattergrams
Neutral/
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Association of Interval/Ration Variables
Statisticians have defined various indices, called “correlation coefficients,” to measure the strength of relationshipsMinimum value of -1Maximum value of +1Value of 0 denotes no correlation or
association between variables
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Pearson’s correlation coefficient (r) Most powerful and widely used index to
measure the association or correlation between two variables is the Pearson’s product-moment correlation coefficient
To use this measure of association data must be of interval or ratio scale
Assumed variables have a linear relationship Relates closely to the statistical concept of
covariation The degree to which two variables vary
together or jointly
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Correlation Coefficient Examples
Image Source: http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient
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Pearson’s… (cont.)
Can be expresses mathematically in several different ways:
1. With deviations from the mean and standard deviations
2. With X and Y values transformed to Z-scores
3. With the original values of X and Y variables
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Pearson’s Scattergram
Image Source: http://userpages.umbc.edu/~nmiller/POLI300/%2311.SCATTERGRAMS.pdf
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Geographic Example
Image Source: http://www.uv.es/elopez/?21
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Association of Ordinal Variables In geographic problems with data in ranked
form, Spearman’s rank correlation coefficient (rs) is the most widely used measure of strength of association between variables
Statistical power of Spearman’s correlation has been shown to be nearly as strong as Pearson’s r
Appropriate when:1. Variables are measured on an ordinal
(ranked) scale2. Interval/ratio data are converted to ranks
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Spearman’s correlation index May be appropriate when if samples are
drawn from highly skewed or severly nonnormal populations
Applicable is situations where X and Y variables have a monotonic relationship
Spearman’s rank correlation coefficient does not distinguish between a linear relationship and a monotonic one
Values are the same as Pearson’s
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Geographic Example
Image Source: http://www.nhm.ac.uk/research-curation/research/projects/worldmap/diversity/c2.htm
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Use of Correlation Indices in Map Comparison How can a geographer measure the
association between two map patterns when the original data are not readily available? With the use of spatial sampling methods,
correlation indices can be applied to numerical data acquired from maps
Three type of maps:• Dot maps, isoline maps, and choropleth maps
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Dot Maps
A set of equal size quadrants are placed over the maps If the scale is not the same, the quadrant
size can be adjusted for the second map Each quadrant represents an observation
and the the frequency of points per quadrant from the two maps are the X and Y values
Using the data set created from the dot map, either Pearson or Spearman correlation indices can be calculated
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Isoline Maps
Method is analogous to that used for dot maps, however instead of placing quadrants over the maps a set of sample points are placed systematically on each isoline map
The value of the continuously distributed variable is recorded for each matching pair of points
The recorded values from the two isoline maps provide the corresponding matched X and Y values
A correlation coefficient is calculated that measures the strength of association between the two map variables
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Choropleth Maps
Measure the degree of association between two choropleth maps having the same internal subarea boundaries
Maps may show Classified data into a set of ordinal categories
Assign numerical values to each category suitable for correlation analysis
For this problem type, Spearman’s correlation index is a better choice than Pearson’s to show the generalized association between the two variables
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Map Comparison Example
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Issues Regarding Correlation When geographers apply statistical analysis to
spatial data, the level of aggregation of the observation units may influence the results
Concern is especially important when inferences are drawn from the results of geographic analyses
Significant findings at one level of aggregation may not occur at other levels
Example: level of income and amount of education may be highly correlated for individuals, but may not be at county or state levels
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Ecological Fallacy
Another critical geographic concern is the so called ecological fallacy concept Reversal of the problem of aggregation described in
the previous slide Researchers sometimes use highly aggregated
data and attempt to infer these results to lower levels of aggregation or to the individual level Example: Just because crime rates are statistically
correlated with percentage of persons under the poverty level at the state or census tract level, it does not imply that all persons under that poverty level are criminals
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References
I was hardpressed to find some good web examples for correlation!
McDonald, J.H. 2009. Handbook of Biological Statistics (2nd ed.). Sparky House Publishing, Baltimore, Maryland. pp. 221-223 http://udel.edu/~mcdonald/statspearman.html
Statistics Canada 2010. Scatterplots.http://www.statcan.gc.ca/edu/power-pouvoir/ch9/scatter-nuages/5214827-eng.htm
Wikipedia 2010. Pearson Product-Moment Correlation Coefficient. http://en.wikipedia.org/wiki/Pearson%27s_r