Individual

The frequency of disease can be reported in terms of either prevalence or incidence.

Some definitions. Strictly speaking, ‘prevalence’ equals the number of cases of a given disease or attribute that exists in a population at a specified point in time. Prevalence risk is the proportion of a population that has a given disease or attribute at a specified point in time. Many authors use the term ‘prevalence’ when they really mean prevalence risk, and this vignette will follow this convention.

Two types of prevalence are reported in the literature: (1) point prevalence equals the proportion of a population in a diseased state at a single point in time; (2) period prevalence equals the proportion of a population with a given disease or condition over a specific period of time (i.e., the number of existing cases at the start of a follow-up period plus the number of incident cases that occur during the follow-up period).

Incidence provides a measure of how frequently susceptible individuals become disease cases as they are observed over time. An incident case occurs when an individual changes from being susceptible to being diseased. The count of incident cases is the number of such events that occur in a population over a defined follow-up period. There are two ways to express incidence:

Incidence risk (also known as cumulative incidence) is the proportion of initially susceptible individuals in a population that become new cases over a defined follow-up period.

Incidence rate (also known as incidence density) is the number of new cases of disease that occur per unit of individual time at risk over a defined follow-up period.

In addition to reporting the point estimate of disease frequency, it is important to provide an indication of the uncertainty around that point estimate. The epi.conf function in the epiR package allows you to calculate confidence intervals for prevalence, incidence risks and incidence rates.

Let’s say we’re interested in the prevalence of disease X in a population comprised of 1000 individuals. Two hundred are tested and four returned a positive result. Assuming 100% diagnostic test sensitivity and specificity, what is the estimated prevalence of disease X in this population?

library(epiR); library(ggplot2); library(scales); library(zoo)

ncas <- 4; npop <- 200
tmp <- as.matrix(cbind(ncas, npop))
epi.conf(tmp, ctype = "prevalence", method = "exact", N = 1000, design = 1, 
   conf.level = 0.95) * 100
#>   est     lower    upper
#> 1   2 0.5475566 5.041361

The estimated prevalence of disease X in this population is 2.0 (95% confidence interval [CI] 0.55 to 5.0) cases per 100 individuals at risk.

Another example. A study was conducted by Feychting, Osterlund, and Ahlbom (1998) to report the frequency of cancer among the blind. A total of 136 diagnoses of cancer were made from 22,050 person-years at risk. What was the incidence rate of cancer in this population?

ncas <- 136; ntar <- 22050
tmp <- as.matrix(cbind(ncas, ntar))
epi.conf(tmp, ctype = "inc.rate", method = "exact", N = 1000, design = 1, 
   conf.level = 0.95) * 1000
#>         est    lower    upper
#> ncas 6.1678 5.174806 7.295817

The incidence rate of cancer in this population was 6.2 (95% CI 5.2 to 7.3) cases per 1000 person-years at risk.

We now want to compare the frequency of disease across several populations. An effective way to do this is to use a ranked error bar plot. With a ranked error bar plot the points represent the point estimate of the measure of disease frequency and the error bars indicate the 95% confidence interval around each estimate. With a ranked error bar plot the disease frequency estimates are plotted from lowest to highest. Generate some data:

ncas <- c(347,444,145,156,56,618,203,113,10,30,663,447,213,52,256,216,745,97,31,250,430,494,96,544,352)
npop <- c(477,515,1114,625,69,1301,309,840,68,100,1375,1290,1289,95,307,354,1393,307,35,364,494,1097,261,615,508)
rname <- paste("Region ", 1:length(npop), sep = "")
dat.df <- data.frame(rname,ncas,npop)

Calculate the prevalence of disease in each region and its 95% confidence interval. The function epi.conf provides several options for confidence interval calculation methods for prevalence. For this example we’ll use the exact method:

tmp <- as.matrix(cbind(dat.df$ncas, dat.df$npop))
tmp <- epi.conf(tmp, ctype = "prevalence", method = "exact", N = 1000, design = 1, 
   conf.level = 0.95) * 100
dat.df <- cbind(dat.df, tmp)
head(dat.df)
#>      rname ncas npop      est    lower    upper
#> 1 Region 1  347  477 72.74633 68.51271 76.69532
#> 2 Region 2  444  515 86.21359 82.93082 89.07325
#> 3 Region 3  145 1114 13.01616 11.09506 15.13489
#> 4 Region 4  156  625 24.96000 21.61207 28.54645
#> 5 Region 5   56   69 81.15942 69.93958 89.56878
#> 6 Region 6  618 1301 47.50192 44.75821 50.25695

Sort the data in order of variable est, assign a 1 to n identifier as variable rank and make a copy of dat.df$rname:

dat.df <- dat.df[sort.list(dat.df$est),]
dat.df$rank <- 1:nrow(dat.df)
dat.df$labels <- dat.df$rname

Create a ranked error bar plot. Because its useful to provide the region-area names on the horizontal axis rotate the horizontal axis labels by 90 degrees.

ggplot(data = dat.df, aes(x = rank, y = est)) +
  theme_bw() +
  geom_errorbar(aes(ymin = lower, ymax = upper), width = 0.1) +
  geom_point() +
  scale_x_continuous(limits = c(0,25), breaks = dat.df$rank, labels = dat.df$labels, name = "Region") +
  scale_y_continuous(limits = c(0,100), name = "Cases per 100 individuals at risk") + 
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

Ranked error bar plot showing the prevalence of disease (and its 95% confidence interval) for 100 population units.

If you have a large number of regions the horizontal axis labels can become crowded and difficult to read. Use the ndelete function to drop every nth region name.

ndelete <- function(x, n){
  id <- seq(from = 1, to = length(x), by = n)
  rval <- rep("", times = length(x))
  rval[id] <- x[id]
  rval
}

dat.df$labels <- ndelete(x = dat.df$rname, n = 2)

Re-draw the ranked error bar plot:

ggplot(data = dat.df, aes(x = rank, y = est)) +
  theme_bw() +
  geom_errorbar(aes(ymin = lower, ymax = upper), width = 0.1) +
  geom_point() +
  scale_x_continuous(limits = c(0,25), breaks = dat.df$rank, labels = dat.df$labels, name = "Region") +
  scale_y_continuous(limits = c(0,100), name = "Cases per 100 individuals at risk") + 
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

Ranked error bar plot showing the prevalence of disease (and its 95% confidence interval) for 100 population units.

Time

Epidemic curves are used to describe patterns of disease over time. Epidemic curve data are often presented in one of two formats:

1. One row for each individual identified as a case with an event date assigned to each.

2. One row for every event date with an integer representing the number of cases identified on that date.

In the notes that follow we provide details on how to produce an epidemic curve when you’re data are in these formats.

One row of data for each case

Generate some data, with one row for every individual identified as a case:

n.males <- 100; n.females <- 50
odate <- seq(from = as.Date("2024-07-26"), to = as.Date("2024-12-13"), by = 1)
prob <- c(1:100, 41:1); prob <- prob / sum(prob)
modate <- sample(x = odate, size = n.males, replace = TRUE, p = prob)
fodate <- sample(x = odate, size = n.females, replace = TRUE)

dat.df <- data.frame(sex = c(rep("Male", n.males), rep("Female", n.females)), 
   odate = c(modate, fodate))

# Sort the data in order of odate:
dat.df <- dat.df[sort.list(dat.df$odate),] 

Plot the epidemic curve using the ggplot2 and scales packages:

ggplot(data = dat.df, aes(x = as.Date(odate))) +
  theme_bw() +
  geom_histogram(binwidth = 7, colour = "gray", fill = "dark blue", linewidth = 0.1) +
  scale_x_date(breaks = date_breaks("7 days"), labels = date_format("%d %b"), 
     name = "Date") +
  scale_y_continuous(breaks = seq(from = 0, to = 30, by = 5), limits = c(0,30), name = "Number of cases") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2024.

You may want to superimpose a smoothed line to better appreciate trend. Do this using the geom_density function in ggplot2:

ggplot(data = dat.df, aes(x = odate)) +
  theme_bw() +
  geom_histogram(binwidth = 7, colour = "gray", fill = "dark blue", linewidth = 0.1) +
  geom_density(aes(y = after_stat(density) * (nrow(dat.df) * 7)), colour = "red") +
  scale_x_date(breaks = date_breaks("7 days"), labels = date_format("%d %b"), 
     name = "Date") +
  scale_y_continuous(breaks = seq(from = 0, to = 30, by = 5), limits = c(0,30), name = "Number of cases") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2024. Superimposed on this plot is a smoothed estimate of case density.

Produce a separate epidemic curve for males and females using the facet_grid option in ggplot2:

ggplot(data = dat.df, aes(x = as.Date(odate))) +
  theme_bw() +
  geom_histogram(binwidth = 7, colour = "gray", fill = "dark blue", linewidth = 0.1) +
  scale_x_date(breaks = date_breaks("1 week"), labels = date_format("%d %b"), 
     name = "Date") +
  scale_y_continuous(breaks = seq(from = 0, to = 30, by = 5), limits = c(0,30), name = "Number of cases") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1)) + 
  facet_grid( ~ sex)

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2024, conditioned by sex.

Let’s say an event occurred on 31 October 2024. Mark this date on your epidemic curve using geom_vline:

ggplot(data = dat.df, aes(x = as.Date(odate))) +
  theme_bw() +
  geom_histogram(binwidth = 7, colour = "gray", fill = "dark blue", linewidth = 0.1) +
  scale_x_date(breaks = date_breaks("1 week"), labels = date_format("%d %b"), 
     name = "Date") +
  scale_y_continuous(breaks = seq(from = 0, to = 30, by = 5), limits = c(0,30), name = "Number of cases") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1)) + 
  facet_grid( ~ sex) +
  geom_vline(aes(xintercept = as.numeric(as.Date("31/10/2024", format = "%d/%m/%Y"))), 
   linetype = "dashed")

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2024, conditioned by sex. An event that occurred on 31 October 2024 is indicated by the vertical dashed line.

Plot the total number of disease events by day, coloured according to sex:

ggplot(data = dat.df, aes(x = as.Date(odate), group = sex, fill = sex)) +
  theme_bw() +
  geom_histogram(binwidth = 7, colour = "gray", linewidth = 0.1) +
  scale_x_date(breaks = date_breaks("1 week"), labels = date_format("%d %b"), 
     name = "Date") +
  scale_y_continuous(breaks = seq(from = 0, to = 30, by = 5), limits = c(0,30), name = "Number of cases") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1)) + 
  geom_vline(aes(xintercept = as.numeric(as.Date("31/10/2024", format = "%d/%m/%Y"))), 
   linetype = "dashed") + 
  scale_fill_manual(values = c("#d46a6a", "#738ca6"), name = "Sex") +
  theme(legend.position = c(0.90, 0.80))
#> Warning: A numeric `legend.position` argument in `theme()` was deprecated in ggplot2
#> 3.5.0.
#> ℹ Please use the `legend.position.inside` argument of `theme()` instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2024, grouped by sex.

It can be difficult to appreciate differences in male and female disease counts as a function of date with the above plot format so dodge the data instead:

ggplot(data = dat.df, aes(x = as.Date(odate), group = sex, fill = sex)) +
  theme_bw() +
  geom_histogram(binwidth = 7, colour = "gray", linewidth = 0.1, position = "dodge") +
  scale_x_date(breaks = date_breaks("1 week"), labels = date_format("%d %b"), 
     name = "Date") +
  scale_y_continuous(breaks = seq(from = 0, to = 30, by = 5), limits = c(0,30), name = "Number of cases") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1)) + 
  geom_vline(aes(xintercept = as.numeric(as.Date("31/10/2024", format = "%d/%m/%Y"))), 
   linetype = "dashed") + 
  scale_fill_manual(values = c("#d46a6a", "#738ca6"), name = "Sex") + 
  theme(legend.position = c(0.90, 0.80))

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2024, grouped by sex.

Integers representing counts of cases on each date

We now provide code to deal with the situation where the data are presented with one row for every date during an outbreak and an integer representing the number of cases identified on each date.

Actual outbreak data will be used for this example. In the code below edate represents the event date (i.e., the date of case detection) and ncas represents the number of cases identified on each edate.

edate <- seq(from = as.Date("2024-02-24"), to = as.Date("2024-07-20"), by = 1)
ncas <- c(1,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,2,
   0,0,1,0,1,1,2,3,2,5,10,15,5,7,17,37,31,34,42,46,73,58,67,57,54,104,77,52,
   90,59,64,61,21,26,25,32,24,14,11,23,6,8,9,4,5,7,14,14,1,5,1,1,5,3,3,1,3,3,
   7,5,10,11,21,14,16,15,13,13,8,5,16,7,9,19,13,5,6,6,5,5,10,9,2,2,5,8,10,6,
   8,8,4,9,7,8,3,1,4,2,0,4,8,5,8,10,12,8,20,16,11,25,19)  

dat.df <- data.frame(edate, ncas)
dat.df$edate <- as.Date(dat.df$edate, format = "%Y-%m-%d")
head(dat.df)
#>        edate ncas
#> 1 2024-02-24    1
#> 2 2024-02-25    0
#> 3 2024-02-26    0
#> 4 2024-02-27    1
#> 5 2024-02-28    0
#> 6 2024-02-29    1

Generate an epidemic curve. Note weight = ncas in the aesthetics argument for ggplot2:

ggplot() +
  theme_bw() +
  geom_histogram(dat.df, mapping = aes(x = edate, weight = ncas), binwidth = 1, fill = "#738ca6", colour = "grey", linewidth = 0.1) +
  scale_x_date(breaks = date_breaks("2 weeks"), labels = date_format("%b %Y"), 
     name = "Date") +
  scale_y_continuous(limits = c(0,125), name = "Number of cases") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 24 February 2024 to 20 July 2024.

This plot has features of a common point source epidemic for the period April 2024 to May 2024. After May 2024 the plot shows feature of a propagated epidemic pattern.

Add a line to the plot to show the cumulative number of cases detected as a function of calendar date. The coding here requires some thought. First question: What was the cumulative number of cases at the end of the follow-up period? Use the cumsum (cumulative sum) function in base R:

max(cumsum(dat.df$ncas))
#> [1] 1834

At the end of the follow-up period the cumulative number of cases was 1834. What we need to do is to get our 0 to 1834 cumulative case numbers to ‘fit’ into the 0 to 125 vertical axis limits of the epidemic curve. A reasonable approach would be to: (1) divide cumulative case numbers by a number so that the maximum cumulative case number divided by our selected number roughly equals the maximum number of cases identified per day; for this example, 15 would be a good choice (1834 / 15 = 122); and (2) set sec.axis = sec_axis(~ . * 15) to multiply the values that appear on the primary vertical axis by 15 for the labels that appear on the secondary vertical axis:

ggplot() +
  theme_bw() +
  geom_histogram(data = dat.df, mapping = aes(x = edate, weight = ncas), binwidth = 1, fill = "#738ca6", colour = "grey", linewidth = 0.1) +
  geom_line(data = dat.df, mapping = aes(x = edate, y = cumsum(ncas) / 15)) + 
  scale_x_date(breaks = date_breaks("2 weeks"), labels = date_format("%b %Y"), 
     name = "Date") +
  scale_y_continuous(limits = c(0,125), name = "Number of cases", 
      sec.axis = sec_axis(~ . * 15, name = "Cumulative number of cases")) +
  guides(fill = "none") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 24 February 2024 to 20 July 2024. Superimposed on this plot is a line showing cumulative case numbers.

Finally, we might want to superimpose a line representing the rolling average of case numbers. Calculate the 5-day rolling mean use the rollmean function in the contributed zoo package:

dat.df$rncas <- rollmean(x = dat.df$ncas, k = 5, fill = NA)

ggplot() +
  theme_bw() +
  geom_histogram(data = dat.df, mapping = aes(x = edate, weight = ncas), binwidth = 1, fill = "#738ca6", colour = "grey", linewidth = 0.1) +
  geom_line(data = dat.df, mapping = aes(x = edate, y = rncas), colour = "red") + 
  scale_x_date(breaks = date_breaks("2 weeks"), labels = date_format("%b %Y"), 
     name = "Date") +
  scale_y_continuous(limits = c(0,125), name = "Number of cases") +
  guides(fill = "none") +
  theme(axis.text.x = element_text(angle = 90, hjust = 1))
#> Warning: Removed 4 rows containing missing values or values outside the scale range
#> (`geom_line()`).

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 24 February 2024 to 20 July 2024. Superimposed on this plot is the 5-day rolling mean number of cases per day.

Place

Two types of maps are often used when describing patterns of disease by place:

1. Choropleth maps. Choropleth mapping involves producing a summary statistic of the outcome of interest (e.g. count of disease events, prevalence, incidence) for each component area within a study region. A map is created by ‘filling’ (i.e. colouring) each component area with colour, providing an indication of the magnitude of the variable of interest and how it varies geographically.

2. Point maps.

Choropleth maps

For illustration we make a choropleth map of the percentage of individuals aged greater than 65 years in an area of New York, USA. These data are taken from the data sets supporting Waller and Gotway (2004). In the code that follows ny refers to New York, age65 refers to individuals aged greater than 65 years and utm refers to the spatial projection of the sf object (Universal Transverse Mercator). The object name suffix .sf tells you that this is a spatial features object.

library(sf); library(spData); library(plyr); library(RColorBrewer); library(sp); library(spatstat)

nyage65utm.sf <- st_read(dsn = system.file("shapes/NY8_bna_utm18.gpkg", package = "spData")[1])
#> Reading layer `sf_bna2_utm18' from data source 
#>   `C:\Program Files\R\R-4.4.1\library\spData\shapes\NY8_bna_utm18.gpkg' 
#>   using driver `GPKG'
#> Simple feature collection with 281 features and 12 fields
#> Geometry type: MULTIPOLYGON
#> Dimension:     XY
#> Bounding box:  xmin: 357628 ymin: 4649538 xmax: 480360.3 ymax: 4808317
#> Projected CRS: UTM Zone 18, Northern Hemisphere
head(nyage65utm.sf)
#> Simple feature collection with 6 features and 12 fields
#> Geometry type: MULTIPOLYGON
#> Dimension:     XY
#> Bounding box:  xmin: 421391.6 ymin: 4661130 xmax: 427059.6 ymax: 4664600
#> Projected CRS: UTM Zone 18, Northern Hemisphere
#>       AREAKEY        AREANAME        X        Y POP8 TRACTCAS  PROPCAS
#> 1 36007000100 Binghamton city 4.069397 -67.3533 3540     3.08 0.000870
#> 2 36007000200 Binghamton city 4.639371 -66.8619 3560     4.08 0.001146
#> 3 36007000300 Binghamton city 5.709063 -66.9775 3739     1.09 0.000292
#> 4 36007000400 Binghamton city 7.613831 -65.9958 2784     1.07 0.000384
#> 5 36007000500 Binghamton city 7.315968 -67.3183 2571     3.06 0.001190
#> 6 36007000600 Binghamton city 8.558753 -66.9344 2729     1.06 0.000388
#>   PCTOWNHOME PCTAGE65P        Z  AVGIDIST PEXPOSURE
#> 1  0.3277311 0.1466102  0.14197 0.2373852  3.167099
#> 2  0.4268293 0.2351124  0.35555 0.2087413  3.038511
#> 3  0.3377396 0.1380048 -0.58165 0.1708548  2.838229
#> 4  0.4616048 0.1188937 -0.29634 0.1406045  2.643366
#> 5  0.1924370 0.1415791  0.45689 0.1577753  2.758587
#> 6  0.3651786 0.1410773 -0.28123 0.1726033  2.848411
#>                             geom
#> 1 MULTIPOLYGON (((421808.5 46...
#> 2 MULTIPOLYGON (((421794.6 46...
#> 3 MULTIPOLYGON (((423127.5 46...
#> 4 MULTIPOLYGON (((425229.9 46...
#> 5 MULTIPOLYGON (((425001.4 46...
#> 6 MULTIPOLYGON (((426143.9 46...

The nyage65utm.sf simple features object lists for each census tract the percentage of individuals aged greater than 65 years:

ggplot() + 
   theme_bw() +
   geom_sf(data = nyage65utm.sf, aes(fill = PCTAGE65P), colour = "dark grey") + 
   scale_fill_gradientn(limits = c(0,0.5), colours = brewer.pal(n = 5, "Reds"), guide = "colourbar") +
   scale_x_continuous(name = "Longitude") +
   scale_y_continuous(name = "Latitude") +
   labs(fill = "SIDS 1974")

Map of an area of New York, USA showing for each census tract the percentage of individuals aged greater than 65 years.

Point maps

Between 1972 and 1980 an industrial waste incinerator operated at a site about 2 kilometres southwest of the town of Coppull in Lancashire, England. Addressing community concerns that there were greater than expected numbers of laryngeal cancer cases in close proximity to the incinerator Diggle (1990) conducted a study investigating risks for laryngeal cancer, using recorded cases of lung cancer as controls. The study area is 20 km x 20 km in size and includes location of residence of patients diagnosed with each cancer type from 1974 to 1983.

Load the chorley data set from the spatstat package. The point locations in this data are projected using the British National Grid coordinate reference system (EPSG code 27700). Create an observation window for the data as coppull.ow and a ppp object for plotting:

data(chorley)
chorley.df <- data.frame(xcoord = chorley$x * 1000, ycoord = chorley$y * 1000, status = chorley$marks)
chorley.df$status <- factor(chorley.df$status, levels = c("lung","larynx"), labels = c("Lung","Larynx"))

chlarynxbng.sf <- st_as_sf(chorley.df, coords = c("xcoord","ycoord"), remove = FALSE)
st_crs(chlarynxbng.sf) <- 27700

chlarynxbng.ow <- chorley$window

Create a simple features polygon object from coppull.ow. First we convert chlarynxbng.ow to a SpatialPolygonsDataFrame object:

coords <- matrix(c(chlarynxbng.ow$bdry[[1]]$x * 1000, chlarynxbng.ow$bdry[[1]]$y * 1000), ncol = 2, byrow = FALSE)
pol <- Polygon(coords, hole = FALSE)
pol <- Polygons(list(pol),1)
pol <- SpatialPolygons(list(pol))
chpolbng.spdf <- SpatialPolygonsDataFrame(Sr = pol, data = data.frame(id = 1), match.ID = TRUE)

Convert the SpatialPolygonsDataFrame to an sf object and set the coordinate reference system:

chpolbng.sf <- as(chpolbng.spdf, "sf")
st_crs(chpolbng.sf) <- 27700

The mformat function is used to plot the axis labels in kilometres (instead of metres):

mformat <- function(){
   function(x) format(x / 1000, digits = 2)
}

ggplot() +
  theme_bw() +
  geom_sf(data = chlarynxbng.sf, aes(colour = status, shape = status)) +
  geom_sf(data = chpolbng.sf, fill = "transparent", colour = "black") +
  coord_sf(datum = st_crs(chpolbng.sf)) +
  scale_colour_manual(name = "Type", values = c("grey","red")) +
  scale_shape_manual(name = "Type", values = c(1,16)) +
  scale_x_continuous(name = "Easting (km)", labels = mformat()) +
  scale_y_continuous(name = "Northing (km)", labels = mformat()) +
  theme(legend.position = c(0.10, 0.12))

Point map showing the place of residence of individuals diagnosed with laryngeal cancer (Pos) and lung cancer (Neg), Copull Lancashire, UK, 1972 to 1980.