epiR_descriptive.knit
Author: Mark Stevenson
Date: 2024-04-27

#

# Descriptive epidemiology using

#### Mark Stevenson

#### 2024-04-27

Epidemiology is the study of the frequency, distribution and determinants of health-related states in populations and the application of such knowledge to control health problems (Disease Control and Prevention 2006).

This vignette provides instruction on the use of R and `epiR` for descriptive epidemiological analyses, that is, to describe how the frequency of disease varies by individual, place and time.

The EpiTools app for iPhone and Android devices provides access to many of the descriptive epidemiology functions in `epiR` using a smart phone.

## Individual

Disease frequency 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% 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.

Lets say we 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 `n`th 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("2022-07-26"), to = as.Date("2022-12-13"), 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 2022.

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 2022. 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 2022, conditioned by sex.

Let’s say an event occurred on 31 October 2022. 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("31/10/2022", 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 2022, conditioned by sex. An event that occurred on 31 October 2022 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("31/10/2022", 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 2022, 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("31/10/2022", 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 2022, 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("2020-02-24"), to = as.Date("2020-07-20"), 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 2020-02-24    1
    #> 2 2020-02-25    0
    #> 3 2020-02-26    0
    #> 4 2020-02-27    1
    #> 5 2020-02-28    0
    #> 6 2020-02-29    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 2020 to 20 July 2020.

This plot has features of a common point source epidemic for the period April 2020 to May 2020. After May 2020 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 2020 to 20 July 2020. 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 (`geom_line()`).

Frequency histogram showing counts of incident cases of disease as a function of calendar date, 24 February 2020 to 20 July 2020. 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 sudden infant death syndrome (SIDS) babies in North Carolina counties for 1974 using the `nc.sids` data provided with the `spData` package. In the code that follows `nc` refers to North Carolina, `sids` refers to sudden infant death syndrome and `ll` refers to the projection of the `sf` object (latitude, longitude). 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)

    ncsidsll.sf <- st_read(dsn = system.file("shapes/sids.shp", package = "spData")[1])
    #> Reading layer `sids' from data source
    #>   `C:\Program Files\R\R-4.3.2\library\spData\shapes\sids.shp'
    #>   using driver `ESRI Shapefile'
    #> Simple feature collection with 100 features and 22 fields
    #> Geometry type: MULTIPOLYGON
    #> Dimension:     XY
    #> Bounding box:  xmin: -84.32385 ymin: 33.88199 xmax: -75.45698 ymax: 36.58965
    #> CRS:           NA
    ncsidsll.sf <- ncsidsll.sf[,c("BIR74","SID74")]
    head(ncsidsll.sf)
    #> Simple feature collection with 6 features and 2 fields
    #> Geometry type: MULTIPOLYGON
    #> Dimension:     XY
    #> Bounding box:  xmin: -81.74107 ymin: 36.07282 xmax: -75.77316 ymax: 36.58965
    #> CRS:           NA
    #>   BIR74 SID74                       geometry
    #> 1  1091     1 MULTIPOLYGON (((-81.47276 3...
    #> 2   487     0 MULTIPOLYGON (((-81.23989 3...
    #> 3  3188     5 MULTIPOLYGON (((-80.45634 3...
    #> 4   508     1 MULTIPOLYGON (((-76.00897 3...
    #> 5  1421     9 MULTIPOLYGON (((-77.21767 3...
    #> 6  1452     7 MULTIPOLYGON (((-76.74506 3...

The `ncsidsll.sf` simple features object lists for each county in the North Carolina USA the number SIDS deaths for 1974. Plot a choropleth map of the counties of the North Carolina showing SIDS counts for 1974:

    ggplot() +
       theme_bw() +
       geom_sf(data = ncsidsll.sf, aes(fill = SID74), colour = "dark grey") +
       scale_fill_gradientn(limits = c(0,60), colours = brewer.pal(n = 5, "Reds"), guide = "colourbar") +
       scale_x_continuous(name = "Longitude") +
       scale_y_continuous(name = "Latitude") +
       labs(fill = "SIDS 1974")

Map of North Carolina, USA showing the number of sudden infant death syndrome cases, by county for 1974.

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.

## References

Diggle, PJ. 1990. “A Point Process Modeling Approach to Raised Incidence of a Rare Phenomenon in the Vicinity of a Prespecified Point.” Journal of the Royal Statistical Society Series A 153: 349–62.
Disease Control, Centers for, and Prevention. 2006. Principles of Epidemiology in Public Health Practice: An Introduction to Applied Epidemiology and Biostatistics. Atlanta, Georgia: Centers for Disease Control; Prevention.
Feychting, M, B Osterlund, and A Ahlbom. 1998. “Reduced Cancer Incidence Among the Blind.” Epidemiology 9: 490–94.