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

Copyright © 2016 by Luc Anselin, All Rights Reserved

Luc Anselin

Spatial Clusters of Rates

http://spatial.uchicago.edu


• concepts

• EBI local Moran

• scan statistics


Concepts


• 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


• 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


• 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


• 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


• 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


crude rate map

Empirical Bayes (EB)smoothed map

effect of variance instability on outliers (schools/population)


• 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


EBI Local Moran


• 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)


• 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


• 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


local Moran for crude rate vs EBI local Moran(schools/population)

crude rate

EBI local Moran


Scan Statistics


• 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


Besag-Newell


• 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


• 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


Besag-Newell clusters (schools/population)

cluster 1

cluster 2


• Interpretation

• care is needed to interpret the p-values

• multiple comparisons

• sequential tests

• clusters are overlapping

• same areal unit can appear in multiple clusters


Kulldorff Scan Statistic


• 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


• 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


Kulldorff scan clusters (schools/population)

cluster 1

cluster 2


• 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

Documents

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