Theoretical motivation

For the expectation-based Poisson scan statistic, the null hypothesis of no anomaly states that at each location \(i\) and duration \(t\), the observed count is Poisson-distributed with expected value \(\mu_{it}\): \[ H_0 \! : Y_{it} \sim \textrm{Poisson}(\mu_{it}), \] for locations \(i=1,\ldots,m\) and durations \(t=1,\ldots,T\), with \(T\) being the maximum duration considered. Under the alternative hypothesis, there is a space-time cluster \(W\) consisting of a spatial zone \(Z \subset \{1,\ldots,m\}\) and a time window \(D = \{1, 2, \ldots, d\} \subseteq \{1,2,\ldots,T\}\) such that the counts in \(W\) have their expected values inflated by a factor \(q_W > 1\) compared to the null hypothesis: \[ H_1 \! : Y_{it} \sim \textrm{Poisson}(q_W \mu_{it}), ~~(i,t) \in W. \] For locations and durations outside of this window, counts are assumed to be distributed as under the null hypothesis. Calculating the scan statistic then involves three steps:

• For each space-time window \(W\), find the maximum likelihood estimate of \(q_W\), treating all \(\mu_{it}\)’s as constants.
• Plug the estimated \(q_W\) into (the logarithm of) a likelihood ratio with the likelihood of the alternative hypothesis in the numerator and the likelihood under the null hypothesis (in which \(q_W=1\)) in the denominator, again for each \(W\).
• Take the scan statistic as the maximum of these likelihood ratios, and the corresponding window \(W^*\) as the most likely cluster (MLC).

Using the Poisson scan statistic

The first argument to any of the scan statistics in this package should be a matrix (or array) of observed counts, whether they be integer counts or real-valued “counts”. In such a matrix, the columns should represent locations and the rows the time intervals, ordered chronologically from the earliest interval in the first row to the most recent in the last. In this example we would like to detect a potential cluster of brain cancer in the counties of New Mexico during the years 1986-1989, so we begin by retrieving the count and population data from that period and reshaping them to a matrix using the helper function df_to_matrix:

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

counts <- NM_popcas %>% 
  filter(year >= 1986 & year < 1990) %>%
  df_to_matrix(time_col = "year", location_col = "county", value_col = "count")

## Warning: `spread_()` was deprecated in tidyr 1.2.0.
## Please use `spread()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.

Spatial zones

The second argument to scan_eb_poisson should be a list of integer vectors, each such vector being a zone, which is the name for the spatial component of a potential outbreak cluster. Such a zone consists of one or more locations grouped together according to their similarity across features, and each location is numbered as the corresponding column index of the counts matrix above (indexing starts at 1).

In this example, the locations are the counties of New Mexico and the features are the coordinates of the county seats. These are made available in the data table NM_geo. Similarity will be measured using the geographical distance between the seats of the counties, taking into account the curvature of the earth. A distance matrix is calculated using the spDists function from the sp package, which is then passed to dist_to_knn (with \(k=15\) neighbors) and on to knn_zones:

library(sp)
library(magrittr)

# Remove Cibola since cases have been counted towards Valencia. Ideally, this
# should be accounted for when creating the zones.
zones <- NM_geo %>%
  filter(county != "cibola") %>%
  select(seat_long, seat_lat) %>%
  as.matrix %>%
  spDists(x = ., y = ., longlat = TRUE) %>%
  dist_to_knn(k = 15) %>%
  knn_zones

Baselines

The advantage of expectation-based scan statistics is that parameters such as the expected value can be modelled and estimated from past data e.g. by some form of regression. For the expectation-based Poisson scan statistic, we can use a (very simple) Poisson GLM to estimate the expected value of the count in each county and year, accounting for the different populations in each region. Similar to the counts argument, the expected values should be passed as a matrix to the scan_eb_poisson function:

mod <- glm(count ~ offset(log(population)) + 1 + I(year - 1985),
           family = poisson(link = "log"),
           data = NM_popcas %>% filter(year < 1986))

ebp_baselines <- NM_popcas %>% 
  filter(year >= 1986 & year < 1990) %>%
  mutate(mu = predict(mod, newdata = ., type = "response")) %>%
  df_to_matrix(value_col = "mu")

Note that the population numbers are (perhaps poorly) interpolated from the censuses conducted in 1973, 1982, and 1991.

Calculation

We can now calculate the Poisson scan statistic. To give us more confidence in our detection results, we will perform 999 Monte Carlo replications, by which data is generated using the parameters from the null hypothesis and a new scan statistic calculated. This is then summarized in a \(P\)-value, calculated as the proportion of times the replicated scan statistics exceeded the observed one. The output of scan_poisson is an object of class “scanstatistic”, which comes with the print method seen below.

set.seed(1)
poisson_result <- scan_eb_poisson(counts = counts, 
                                  zones = zones, 
                                  baselines = ebp_baselines,
                                  n_mcsim = 999)
print(poisson_result)

## Data distribution:                Poisson
## Type of scan statistic:           expectation-based
## Setting:                          univariate
## Number of locations considered:   32
## Maximum duration considered:      4
## Number of spatial zones:          415
## Number of Monte Carlo replicates: 999
## Monte Carlo P-value:              0.005
## Gumbel P-value:                   NULL
## Most likely event duration:       4
## ID of locations in MLC:           15, 26

As we can see, the most likely cluster for an anomaly stretches from 1986-1989 and involves the locations numbered 15 and 26, which correspond to the counties

counties <- as.character(NM_geo$county)
counties[c(15, 26)]

[1] "losalamos" "santafe"  

These are the same counties detected by Kulldorff et al. (1998), though their analysis was retrospective rather than prospective as ours was. Ours was also data dredging as we used the same study period with hopes of detecting the same cluster.

A heuristic score for locations

We can score each county according to how likely it is to be part of a cluster in a heuristic fashion using the function score_locations, and visualize the results on a heatmap as follows:

# Calculate scores and add column with county names
county_scores <- score_locations(poisson_result, zones)
county_scores %<>% mutate(county = factor(counties[-length(counties)], 
                                          levels = levels(NM_geo$county)))

# Create a table for plotting
score_map_df <- merge(NM_map, county_scores, by = "county", all.x = TRUE) %>%
  arrange(group, order)

# As noted before, Cibola county counts have been attributed to Valencia county
score_map_df[score_map_df$subregion == "cibola", ] %<>%
  mutate(relative_score = score_map_df %>% 
                          filter(subregion == "valencia") %>% 
                          select(relative_score) %>% 
                          .[[1]] %>% .[1])

ggplot() + 
  geom_polygon(data = score_map_df,
               mapping = aes(x = long, y = lat, group = group, 
                             fill = relative_score),
               color = "grey") +
  scale_fill_gradient(low = "#e5f5f9", high = "darkgreen",
                      guide = guide_colorbar(title = "Relative\nScore")) +
  geom_text(data = NM_geo, 
            mapping = aes(x = center_long, y = center_lat, label = county),
            alpha = 0.5) +
  ggtitle("County scores")

A warning though: the score_locations function can be quite slow for large data sets. This might change in future versions of the package.

Finding the top-scoring clusters

Finally, if we want to know not just the most likely cluster, but say the five top-scoring space-time clusters, we can use the function top_clusters. The clusters returned can either be overlapping or non-overlapping in the spatial dimension, according to our liking.

top5 <- top_clusters(poisson_result, zones, k = 5, overlapping = FALSE)

# Find the counties corresponding to the spatial zones of the 5 clusters.
top5_counties <- top5$zone %>%
  purrr::map(get_zone, zones = zones) %>%
  purrr::map(function(x) counties[x])

# Add the counties corresponding to the zones as a column
top5 %<>% mutate(counties = top5_counties)

The top_clusters function includes Monte Carlo and Gumbel \(P\)-values for each cluster. These \(P\)-values are conservative, since secondary clusters from the original data are compared to the most likely clusters from the replicate data sets.