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Transcript of Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s...
![Page 1: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/1.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
Luc Anselin
Spatial Clusters of Rates
http://spatial.uchicago.edu
![Page 2: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/2.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
• concepts
• EBI local Moran
• scan statistics
![Page 3: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/3.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
Concepts
![Page 4: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/4.jpg)
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• Rates as Risk
• from counts (spatially extensive) to rates (spatially intensive)
• rate = number of events / population
• rate as a measure of risk (a probability)
• crude rate: Oi / Pi
• relative: Oi / Ei observed relative to expected
![Page 5: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/5.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
• The Problem with Rates
• r = O / P
• O number of events
• P population (at risk)
• O is a random variable, P is not
• variance of r depends inversely on P
![Page 6: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/6.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
• Moments of the Binomial Variable
• mean: E [O] = π.P
• risk times population
• variance: V [O] = π (1 - π).P
• variance depends on population P
![Page 7: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/7.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
• Moments of the Rate
• P is just a constant
• E[r] = E[O]/P = π P / P = π
• crude rate is unbiased estimator for risk
• Var[r] = Var[O] / P2 = π (1 - π) P / P2 = π (1 - π) / P
![Page 8: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/8.jpg)
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• Non-Standard Features of Rate Variance
• variance depends on the mean (= risk)
• numerator π (1 - π) = π - π2 ≈ π
• higher risk implies greater variance
• variance depends inversely on population P
• P in the denominator
• smaller places (smaller P) have larger variance
![Page 9: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/9.jpg)
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crude rate map
Empirical Bayes (EB)smoothed map
effect of variance instability on outliers (schools/population)
![Page 10: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/10.jpg)
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• Approaches
• variance instability violates the basic assumption underlying spatial autocorrelation analysis of a constant variance
• solutions
• standardized local indicators of spatial autocorrelation (EBI LISA)
• scan statistics
![Page 11: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/11.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
EBI Local Moran
![Page 12: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/12.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
• Correcting Variance Instability
• NOT by smoothing rates and applying standard Moran’s I
• smoothing induces spatial correlation
• BUT by adjusting the Moran’s I statistic directly
• several proposals: constant risk hypothesis (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99)
![Page 13: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/13.jpg)
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• Empirical Bayes Index - EBI
• standardizing the rate variable using an Empirical Bayes (EB) logic
• zi = (ri - b) / siwith ri as the original rate (xi/pi), b as a mean and si as a standard deviation
• use local Moran with standardized rates zi
![Page 14: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/14.jpg)
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• EBI Adjustment
• mean b = Σi x
i / Σ
i p
i for i = 1,...,R
i.e., total sum of cases / total population, not the mean of the rates
• variancei = {[Σi pi(ri - b)2] / Ptot} - b/Pav
• Ptot = Σi pi and Pav = Ptot / m, average population by region
• si = square root of variance
![Page 15: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/15.jpg)
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local Moran for crude rate vs EBI local Moran(schools/population)
crude rate
EBI local Moran
![Page 16: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/16.jpg)
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Scan Statistics
![Page 17: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/17.jpg)
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• Scan Statistics
• count events within a given shape
• typically based on centroids and circle
• count until a given number of events is reached: Besag-Newell
• count until a given aggregate population is reached: Kulldorff
![Page 18: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/18.jpg)
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Besag-Newell
![Page 19: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/19.jpg)
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• Principle
• aggregate areal units until a chosen number of events has been reached
• then carry out a hypothesis test with the Poisson expected count as the null
• what is the probability that the observed count in the aggregate areal units is from a Poisson distribution with the average
• aggregate with highest significance (lowest p-value) is a cluster
![Page 20: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/20.jpg)
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• Implementation
• typically carried out using the centroids of areal units
• sort the neighbors in order of increasing distance
• add the number of events until the critical threshold (k) is exceeded
![Page 21: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/21.jpg)
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Besag-Newell clusters (schools/population)
cluster 1
cluster 2
![Page 22: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/22.jpg)
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• Interpretation
• care is needed to interpret the p-values
• multiple comparisons
• sequential tests
• clusters are overlapping
• same areal unit can appear in multiple clusters
![Page 23: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/23.jpg)
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Kulldorff Scan Statistic
![Page 24: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/24.jpg)
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• Principle
• aggregate areal units until a target population is reached
• likelihood ratio test of events within the “cluster” against events outside of the “cluster”
• null hypothesis is Poisson distribution with expected counts
• select cluster with max likelihood ratio
![Page 25: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/25.jpg)
Copyright © 2016 by Luc Anselin, All Rights Reserved
• Likelihood Ratio Test
• T = max (Oi/Ei)Oi (Oo/Eo)Oo
for Oi/Ei > Oo/Eo
• count within region (i) versus outside (o)
• Oi/o observed in/out, Ei/o expected in/out
• inference based on randomization
• Tr computed for simulation under constant risk
• compare reference distribution of Tr to observed T
• pseudo p-value = proportion of Tr that exceeds T
![Page 26: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/26.jpg)
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Kulldorff scan clusters (schools/population)
cluster 1
cluster 2
![Page 27: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/27.jpg)
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• Interpretation
• most likely cluster has highest log-likelihood ratio
• p-value based on Monte Carlo simulation
• other clusters ranked in order of log-likelihood ratio
• p-values suffer from multiple comparisons and sequential testing
![Page 28: Spatial Clusters of Rates - Spatial@UChicago | The ... · (Walter 92), Tango’s I (95), Oden’s Ipop (95) and Assuncao-Reis EBI (99) Copyright © 2016 by Luc Anselin, All Rights](https://reader031.fdocuments.in/reader031/viewer/2022022603/5b5bd9667f8b9a68368b842f/html5/thumbnails/28.jpg)