---
title: "<h1 style='text-align:center;'>Descriptive epidemiology using <img src='../man/figures/logo.svg' style='vertical-align:middle; width:10%; margin-right:10px;'/></h1>"
author: "Mark Stevenson"
date: "`r Sys.Date()`"
bibliography: epiR_descriptive.bib
link-citations: yes
output:
  knitr:::html_vignette:
  toc: true
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Descriptive epidemiology}
  %\usepackage[utf8]{inputenc}
---

\setmainfont{Calibri Light}

```{r, echo = FALSE, message = FALSE}

# If you want to create a PDF document paste the following after line 9 above:
#   pdf_document:
#     toc: true
#     highlight: tango
#     number_sections: no
#     latex_engine: xelatex    
# header-includes: 
#    - \usepackage{fontspec}

knitr::opts_chunk$set(collapse = T, comment = "#>")
options(tibble.print_min = 4L, tibble.print_max = 4L)
```

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 [@cdc: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](https://apps.apple.com/vn/app/epi-tools/id1611139482) and [Android](https://play.google.com/store/apps/details?id=au.melbourne.uni.epitools) 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?

```{r message = FALSE}
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
```

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_et_al: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?

```{r}
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
```

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:

```{r}
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:

```{r}
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)
```

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`:

```{r}
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.

```{r dfreq02-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:dfreq02}Ranked error bar plot showing the prevalence of disease (and its 95% confidence interval) for 100 population units."}
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))
```
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.

```{r}
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:

```{r dfreq03-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:dfreq03}Ranked error bar plot showing the prevalence of disease (and its 95% confidence interval) for 100 population units."}
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))
```

## 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:

```{r}
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:

```{r epicurve01-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve01}Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2022."}
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))
```

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

```{r epicurve02-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve02}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."}

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))

```

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

```{r epicurve03-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve03}Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2022, conditioned by sex."}
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)
```

Let's say an event occurred on 31 October 2022. Mark this date on your epidemic curve using `geom_vline`:

```{r epicurve04-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve04}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."}
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")
```

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

```{r epicurve05-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve05}Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2022, grouped by 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))
```

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:

```{r epicurve06-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve06}Frequency histogram showing counts of incident cases of disease as a function of calendar date, 26 July to 13 December 2022, grouped by 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, 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))
```

### 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`.

```{r}
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)
```

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

```{r epicurve07-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve07}Frequency histogram showing counts of incident cases of disease as a function of calendar date, 24 February 2020 to 20 July 2020."}
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))
```

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:

```{r}
max(cumsum(dat.df$ncas))
```

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:

```{r epicurve08-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve08}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."}

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))
```

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:

```{r epicurve09-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:epicurve09}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."}

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))
```

## 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_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.

```{r message = FALSE, warning = FALSE}
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])
head(nyage65utm.sf)
```

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

```{r spatial01-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:spatial01}Map of an area of New York, USA showing 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")
```

**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:

```{r message = FALSE}
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:

```{r}
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:

```{r}
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):

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

```{r spatial02-fig, warnings = FALSE, echo = TRUE, fig.cap="\\label{fig:spatial02}Point map showing the place of residence of individuals diagnosed with laryngeal cancer (Pos) and lung cancer (Neg), Copull Lancashire, UK, 1972 to 1980."}
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))
```

## References
