index.Rmd

---
title: "A Gentle but Critical Introduction to Statistical Inference, Moderation, and Mediation"
author: "Wouter de Nooy "
date: "2016-09-20 - `r Sys.Date()`"
site: bookdown::bookdown_site
output: 
  bookdown::gitbook:
    css: styleX.css
    number_sections : TRUE
    split_by : "section+number"
    split_bib: FALSE
    download: ["GentleIntro.pdf"]
    highlight: monochrome
    config:
      toc:
        collapse: section
      sharing: no
    includes:
      in_header: hypothesis.html
  bookdown::pdf_book:
    keep_tex: yes
    toc : TRUE
    toc_depth : 3
    toc_unnumbered : yes
    number_sections : TRUE
    highlight: monochrome
    includes: 
      in_header: preambleX.tex
      before_body: lists.tex
documentclass: book
geometry: left=40mm, right=30mm, top=30mm, bottom = 25mm
bibliography: ["RefList.bib"]
biblio-style: "apa"
csl: apa.csl
link-citations: TRUE
---

```{r setup, include=FALSE, echo=FALSE}
# All code chunks that should never be shown must have the echo=FALSE option.
# Chunks with answers to exercises have the echo option not set, so their visibility depends on the the echo option set for all chunks here. Set it to FALSE if the answers are always visible.
knitr::opts_chunk$set(echo=TRUE, warning=FALSE, message=FALSE, eval=TRUE, tidy=TRUE)
# Set output format for kableExtra tables: "html" or "latex"
options(knitr.table.format = "latex")
# Don't show NAs in kable tables.
options(knitr.kable.NA = "")
# Libraries used.
library(tidyverse)
library(knitr)
library(kableExtra)
library(visNetwork)
library(bookdown)
library(haven)
# Basic colors and layout.
source("../apps/plottheme/styling.R")
# Chunks with answers to Test Your Understanding questions are set to a boolean variable for each chapter, which can be set to TRUE for showing the answers.
# Show TYU answers per chapter. Change between FALSE and TRUE.
ch1 <- FALSE
ch2 <- FALSE
ch3 <- FALSE
ch4 <- FALSE
ch5 <- FALSE
ch6 <- FALSE
ch7 <- FALSE
ch8 <- FALSE
ch9 <- FALSE
ch10 <- FALSE
ch11 <- FALSE
```

```{r include=FALSE, echo=FALSE, eval=FALSE}
# Notes

# PDF output:
#* Embedded shiny apps: Lots of whitespace within the screenshot. Helps to define out.height in chunk - no. cliprect = 'viewport' does not work. selector = "#sampling-distribution-summary1" does not work. vwidth=420, vheight=620 does not work. Setting dev="png" works even though some whitespace remains under the app. Specify out.width (to get a caption) but don't specify out.height (increases the app size).
#* Interactive network visualizations (flow chart in 4.2 Formulating Statistical Hypotheses, mediation models in 9.6 Path Model with Regression Analysis) badly captured (due to fig.asp="50%"?) with lots of whitespace (and of course no link to interactive version). Adding out.width and/or out.height seems to create new problems (mean-centering of consecutive text, caption rendered as LaTex...). Manual screenshots shown with screenshot.alt = "".
#* No screen shots for videos embedded with iframe (4.2.9 Comparing Means in SPSS). Also, a separate figure caption is not numbered. Solved by embedding them with knitr::include_url and dev = "png" but this puts videos in a floating context, so they are no longer in the Instructions section in the PDF. 
#* Manually adjust .tex file: replace \begin{figure} by \begin{figure}[H]. Also apply to tables to stop them from floating. Manual screenshots of videos (with correct file names) in figures\videoshots. Copy all to _book\GentleIntro_files\figure-latex and recompile .tex file. Note: Also copy all files in figure-latex directory from _bookdown_files\GentleIntro_files  to the _book\GentleIntro_files.
#* Including a manual screen shot with screenshot.alt = "" solves the white-space problem but it does not add a link to the interactive content (shiny app). Only applicable to dynamic networks that have no link anyhow.

#* Only caption to figure if out.width is set.
#* kable tables are not adjusted to the page margins (see 2.5.2 Conditions for the use of theoretical probability distributions, 9.6.3 Partial and full mediation). out.width="100%" does not help.

#* Equations: use aligned instead of align. But aligned does not work in HTML - need different package to render equations in HTML? 
#* Equations are numbered with the section number. The number is not displayed with the equation itself.

#* HTML table to arrange text and pictures does not work: 2.1 (baron), 3.5.1 (Neyman), 5.2 (metal detector), 5.3.1 (Pearson), 6.3.2 (Gauss, LaPlace). Size of picture must be made explicit. Embed with knitr::include_graphics and set output.width? Note that include_graphics makes the picture float, which is not what I want with portraits. Also, the dpi seems to matter to eventual out.width of r chunck with include_graphics.

# Links to external resources such as (SPSS) data must be absolute. They cannot be stored in the R Markdown doc root directory, so I store them in a data subdirectory of the shiny server http://82.196.4.233:3838/data. Embed links as HTML with target="_blank" to open the resource in a new window otherwise the Shiny server displays the R Markdown document in grey (must be reloaded).

# Include SPSS tabular output:
#- Right-click table, select Style Output, Select as Group (Continue), Table Look, Compact.
#- Select tales to be exported. Don't select separate analysis titles, they are badly formatted.
#- Right-click selected table and export as HTML, Change Options: Export visible layer only, Export without styling, Include footnotes and captions, Export visible view only.
#- Export HTML file to R project directory.
#- Include HTML file in R Markdown within <iframe>. Set width="100%" and experiment to find the right height.
# Note that html content embedded in this way yields problems with webshot capturing if the document is knitted into PDF. 
```

```{r example-data, include=FALSE, echo=FALSE, eval=FALSE}
# Do not overwrite the data files (unless it is really necessary),
# because data sets may be slightly different from the ones used in the book
# since R version 4.

source("norm10.R") #function to normalize scores to [1-10] or [0-10]

# SPSS data set candy colour (uniform) and weight (normal, M = 2.8, SD = 0.2), N= 50.
set.seed(seed = 12345)
# Candy bags example.
candies <- data.frame(
  colour = c(1, 2, 4, 3, 5), 
  weight = rnorm(50, mean = 2.8, sd = 0.2), 
  sticky = c(rep(c(2, 2, 1, 2, 1), 6), rep(c(2, 1), 9), c(1, 2)), 
  colour_pre = rnorm(50, mean = 7.3, sd = 0.5)
  )
set.seed(seed = 8734)
candies$colour_post <- candies$colour_pre + ifelse(candies$colour %in% c(1, 2, 3), rnorm(50, mean = -1.1, sd = 0.1), ifelse(candies$colour == 5, rnorm(50, mean = -2.4, sd = 0.2), rnorm(50, mean = -3.7, sd = 0.4)))
set.seed(seed = 3295)
candies$sweetness <- 12 - candies$colour_post + rnorm(50, mean = 0, sd = 1.7)
candies$sweetness <- norm10(candies$sweetness) #normalize to 1-10
set.seed(seed = 9501)
candies$help <- runif(50, 0, 1)
candies$spotted <- ifelse((candies$colour %in% c(1, 2, 4) & candies$help > 0.3) | (candies$colour %in% c(3, 5) & candies$help > 0.7), 2, 1) 
candies$help <- NULL
# add labels
candies <- candies %>%
  mutate(
    colour = labelled_spss(colour,
      labels = c(`blue` = 1,`green` = 2,`yellow` = 5,`orange`= 3,`red`= 4),
      label = "Candy colour"
    ),
    weight = labelled_spss(weight,
      label = "Candy weight (grams)"
    ),
    sticky = labelled_spss(sticky,
      labels = c(`not sticky`= 1, `sticky` = 2),
      label = "Candy is sticky"
    ),
    colour_pre = labelled_spss(colour_pre,
      labels = c(`Totally faded` = 1, `Maximum colourfulness` = 10),
      label = "Candy colourfulness before exposure to sun light"
    ),
    colour_post = labelled_spss(colour_post,
      labels = c(`Totally faded` = 1, `Maximum colourfulness` = 10),
      label = "Candy colourfulness after exposure to sun light"
    ),
    sweetness = labelled_spss(sweetness,
      label = "Candy sweetness",
      labels = c(`Not a bit sweet` = 1, `Super sweet` = 10),
    ),
    spotted = labelled_spss(spotted,
      labels = c(`Not spotted` = 1, `Spotted` = 2),
      label = "Candy is spotted"
    )
  )
# save as SPSS file
write_sav(candies, "data/candies.sav")

# Households: tv station reach, region, household income.
set.seed(2583)
random <- runif(120)
households <- data.frame(region = ifelse(random <= 0.2, "North", ifelse(random <= 0.45, "East", ifelse(random <= 0.75, "South", "West"))))
set.seed(6111)
random <- runif(120)
households$tv_reach <- "no"
households$tv_reach[(households$region == "North" & random < 0.2) | (households$region == "East" & random < 0.7) | (households$region == "South" & random < 0.4) | (households$region == "West" & random < 0.9)] <- "yes"
households$tv_reach <- factor(households$tv_reach)
households$income <- rnorm(120, mean = 40000, sd = 12000) + ifelse(households$region == "West" | households$region == "East", runif(120) * 10000, 0)
attributes(households$region)$label <- "Region of residence"
attributes(households$tv_reach)$label <- "Can the household receive the tv station?"
attributes(households$income)$label <- "Household income"
write_sav(households, "data/households.sav")

# Children: media literacy, sex, age, supervision.
set.seed(2583)
n <- 472 # 87
children <- data.frame(
  medliter = rnorm(n, mean = 4.3, sd = 1.8), 
  sex = c(rep(3, round(n * 46 / 87)), rep(2, round(n * 40 / 87)), rep(1, round(n / 87))) # One/some impossible value(s).
  )
children$medliter <- children$medliter - min(children$medliter) + 1
children <- children %>%
  mutate(
    medliter = case_when(
      medliter < 0 ~ 0,
      medliter > 10 ~ 10,
      TRUE ~ medliter
    )
  )
set.seed(1174)
children$age <- round((children$medliter + 22 + rnorm(n, mean = 0, sd = 5.1))/3)
set.seed(3228)
children$supervision <- (children$medliter + rnorm(n, mean = 0, sd = 3.6))
children$supervision <- norm10(children$supervision) #normalize to 1-10
children$supervision[30] <- 25 # Impossible value.
# Set labels.
children <- children %>%
  mutate(
    medliter = labelled_spss(medliter,
      labels = c(`No media literacy at all` = 1, `Maximum media literacy` = 10),
      label = "Media literacy"
    ),
    sex = labelled_spss(sex,
      labels = c(` ` = 1, `boy` = 2, `girl` = 3),
      label = "Sex"
    ),
    age  = labelled_spss(age,
      label = "Age (in years)"
    ),
    supervision = labelled_spss(supervision,
      labels = c(`Never supervised` = 1, `Supervised all the time` = 10),
      label = "Parental supervision during media use"
    )
  )
# attributes(children$sex)$label <- "Sex"
# attributes(children$age)$label <- "Age (in years)"
# attributes(children$medliter)$label <- "Media literacy"
# children$medliter <- labelled::labelled(children$medliter, labels = c(`No media literacy at all` = 1, `Maximum media literacy` = 10))
# attributes(children$supervision)$label <- "Parental supervision during media use"
# children$supervision <- labelled::labelled(children$supervision, labels = c(`Never supervised` = 1, `Supervised all the time` = 10))
haven::write_sav(children, "data/allchildren.sav") # data/children.sav

# Voters: immigrant opinion, age (class: young vs. old).
set.seed(1324)
voters <- data.frame(age = round(runif(66) * 70 + 17))
voters$age_group <- ifelse(voters$age < 30, 2, 1)
set.seed(2431)
voters$immigrant <- ifelse(voters$age_group == 2, round(rnorm(66, mean = 5, sd = 1.4)), round(rnorm(66, mean = 5.1, sd = 1.8)))
voters$immigrant[voters$immigrant < 1] <- 1
voters$immigrant[voters$immigrant > 10] <- 10
# add labels
voters <- voters %>%
  mutate(
    age = labelled_spss(age,
      label = "Age"
    ),
    age_group = labelled_spss(age_group,
      labels = c(`young (<30)` = 2, `old (30+)` = 1),
      label = "Age group"
    ),
    immigrant = labelled_spss(immigrant,
      labels = c(`Very hostile towards immigrants` = 1, `Very welcoming to immigrants` = 10),
      label = "Attitude towards immigrants"
  )
    
  )
# save as SPSS file
haven::write_sav(voters, "data/voters.sav")

# Donors: endorser of the campaign, remember campaign, exposure to campaign, willingness to donate to the campaign (pre and post), sex.
# Create predictors.
set.seed(1245)
remember <- sample(c(0,1), size = 143, replace = TRUE, prob = c(0.5, 0.5))
set.seed(320)
exposure <- norm10(rnorm(143, 0, 1))
sex <- c(rep(0, 71), rep(1, 72))
endorser <- c(rep(0,22), rep(1,24), rep(2, 25), rep(0,23), rep(1,25), rep(2, 24))
# Create outcome.
Clooney <- ifelse(endorser==1, 1, 0)
Jolie <- ifelse(endorser==2, 1, 0)
set.seed(1245)
willing_post <- 3 + 0.84*remember + 0.51*sex + 0.1*exposure + 0.03*Clooney + 1.13*Jolie + 1.23*Clooney*sex - 0.77*Jolie*sex + rnorm(143, 0, 1.48)
# Create pre-measurement of outcome.
set.seed(4944)
willing_pre <- willing_post + runif(143, -0.6, 0.3)
# Join in dataframe.
donors <- data.frame(
  willing_post = labelled_spss(willing_post,
                               labels = c(`I am very sure not to donate` = 1, `I am very sure to donate` = 10),
                               label = "Respondent's willingness to donate at the campaign end"), 
  willing_pre = labelled_spss(willing_pre,
                              labels = c(`I am very sure not to donate` = 1, `I am very sure to donate` = 10),
                              label = "Respondent's willingness to donate at the campaign start"), 
  exposure = labelled_spss(
    exposure, labels = c(`No exposure` = 1, `Maximum exposure` = 10),
    label = "Respondent's exposure to the campaign"), 
  remember = labelled_spss(remember, 
                           labels = c(`Does not recognize the campaign` = 0, `Recognizes the campaign` = 1),
                           label = "Respondent remembers the campaign?"),
  endorser = labelled_spss(endorser, 
                           labels = c(`Nobody` = 0, `George Clooney` = 1, `Angelina Jolie` = 2),
                           label = "Celebrity endorsing the campaign"), 
  sex = labelled_spss(sex, 
                      labels = c(`Male` = 0, `Female` = 1),
                      label = "Respondent's sex")
  )
# Cleanup.
rm(Clooney, Jolie, willing_post, willing_pre, exposure, remember, endorser, sex)
write_sav(donors, "data/donors.sav")
# Note: Set factors to nominal scale in SPSS.

#Consumers: brand awareness, ad exposure, word of mouth, gender (and intention to buy).
set.seed(834)
n <- 520 #62
consumers <- data.frame(
  ad_expo = norm10(sqrt(abs(rnorm(n, 10, 4.8)))), 
  wom = c(rep(2, round(n * 18 / 62)), rep(1, round(n * 44 / 62))), 
  gender = c(rep(1, round(n * 12 / 62)), rep(2, round(n * 6 / 62)), rep(1, round(n * 19 / 62)), rep(2, round(n * 25 / 62))))
set.seed(825)
consumers$brand_aw <- norm10(consumers$ad_expo - as.integer(consumers$gender) * 1.2 - as.integer(consumers$wom) * 1.7 + rnorm(n, 0, 3.3))
set.seed(426)
consumers$intention <- norm10(8.4 * consumers$ad_expo + 7.7 * consumers$brand_aw - 1 * consumers$ad_expo * consumers$brand_aw + as.integer(consumers$gender) * 1 - as.integer(consumers$wom) * 0.3 + rnorm(n, 0, 14.3))
#summary(lm(intention ~ ad_expo*brand_aw + gender, data = consumers))
#summary(lm(intention ~ scale(ad_expo, scale = F)*scale(brand_aw, scale = F) + gender, data = consumers))
# add labels
consumers <- consumers %>%
  mutate(
    ad_expo = labelled_spss(ad_expo,
      labels = c(`No exposure to ads` = 1, `Maximum exposure to ads` = 10),
      label = "Exposure to brand advertisements"
    ),
    wom = labelled_spss(wom,
      labels = c(`no` = 1, `yes` = 2),
      label = "Heard about the brand by word of mouth?"
    ),
    gender = labelled_spss(gender,
      labels = c(`female` = 1, `male` = 2),
      label = "Consumer's sex"
    ),
    brand_aw = labelled_spss(brand_aw,
      labels = c(`Does not recognize the brand at all` = 1, `Knows the brand very well` = 10),
      label = "Brand awareness"
    ),
    intention = labelled_spss(intention,
      labels = c(`No intention to buy at all` = 1, `Determined to buy` = 10),
      label = "Intention to buy"
    )
  )
# save SPSS data file
haven::write_sav(consumers, "data/allconsumers.sav") # data/consumers.sav

#Smokers: attitude [-5, 5], exposure [0, 10], (smoking) status [2 or 3 cat.], contact (with smokers) [0, 10].
set.seed(4932)
n <- 312 #85
smokers <- data.frame(exposure = runif(n)*10, status2 = rbinom(n, 1, 0.2))
smokers$status3[smokers$status2 == 1] <- 2
set.seed(8220)
smokers$status3[smokers$status2 != 1] <- rbinom(sum(smokers$status2 != 1), 1, 0.3)
set.seed(4321)
smokers$contact <- 0.26*(10 - smokers$exposure) + rnorm(n, mean = 0, sd = 2)
smokers$contact <- norm10(smokers$contact, lbound1 = FALSE)
set.seed(390)
smokers$attitude <- 1.5*(-0.26*smokers$exposure + 
  0.03*smokers$exposure*smokers$contact +
  -0.8*(smokers$status3 == 1) +
  -0.24*smokers$exposure*(smokers$status3 == 1) +
  0.7*(smokers$status3 == 2) +
  -0.20*smokers$exposure*(smokers$status3 == 2) +
  rnorm(n, mean = 1, sd = 0.6))
# recode values that are too high or too low
smokers <- smokers %>%
  mutate(attitude = case_when(
    attitude < -5 ~ -5,
    attitude > 5 ~ 5,
    TRUE ~ attitude
  ))
# add labels
smokers <- smokers %>%
  mutate(
    exposure = labelled_spss(exposure,
      labels = c(`No exposure` = 0, `Maximum exposure` = 10),
      label = "Exposure to anti-smoking campaign"
    ),
    status2 = labelled_spss(status2,
      labels = c(`Non-smoker` = 0, `Smoker` = 1),
      label = "Smoking status"
    ),
    status3 = labelled_spss(status3,
      labels = c(`Non-smoker` = 0, `Former smoker` = 1, `Smoker` = 2),
      label = "Smoking status"
    ),
    contact = labelled_spss(contact,
      labels = c(`No contact` = 0, `Maximum contact` = 10),
      label = "Contact with smokers"
    ),
    attitude = labelled_spss(attitude,
      labels = c(`Very negative` = -5, `Neutral`= 0, `Very positive` = 5),
      label = "Attitude towards smoking"
    )
  )
# save SPSS data file
haven::write_sav(smokers, "data/allsmokers.sav") # data/smokers.sav

# Readers: Predicting newspaper reading time from news site use, political scepticism, political interest, education level, and age.
# Age (in decades).
set.seed(5554)
readers <- data.frame(age = round(runif(n = 312, min = 1.8, max = 8.2), digits = 1))
# Education level (in years).
set.seed(4466)
readers$education <- round(-readers$age + rnorm(n = 312, mean = 0, sd = 15), digits = 0)
readers$education <- 8 + round(norm10(readers$education, lbound1 = FALSE), digits = 0)
# Political interest (10-point scale).
set.seed(7373)
readers$polinterest <- round(norm10(0.3*readers$age + 1.2*readers$education + rnorm(n = 312, mean = 0, sd = 6)), digits = 1)
# Political cynicism (10-point scale).
set.seed(3737)
readers$polcynic <- round(norm10(-0.8*readers$age - 1.4*readers$education + rnorm(n = 312, mean = 0, sd = 6)), digits = 1)
# News site use (7-point scale).
set.seed(2176)
readers$newssite <- round(norm10(-8.6*readers$age + 1.4*readers$polinterest + rnorm(n = 312, mean = 0, sd = 6)), digits = 0)
readers$newssite[readers$newssite == 1] <- 2
readers$newssite[readers$newssite > 8] <- 8
readers$newssite <- readers$newssite - 2
# Newspaper reading time (minutes per day on average).
set.seed(3310)
readers$readingtime <- round(4.6*readers$age + 0.3*readers$education + 0.5*readers$polinterest - 0.1*readers$polcynic - 1.3*readers$newssite + rnorm(n = 312, mean = 0, sd = 6), digits = 0) 
readers$readingtime <- readers$readingtime - min(readers$readingtime)
# Add labels
readers <- readers %>%
  mutate(
    age = labelled_spss(age,
      labels = ,
      label = "Age (in decades)"
    ),
    education = labelled_spss(education,
      labels = ,
      label = "Education level (in years)"
    ),
    polinterest = labelled_spss(polinterest,
      labels = c(`No interest` = 1, `Very much interested` = 10),
      label = "Interest in politics (scale)"
    ),
    polcynic = labelled_spss(polcynic,
      labels = c(`Not cynical at all` = 1, `Very cynical` = 10),
      label = "Political cynicism (scale)"
    ),
    newssite = labelled_spss(newssite,
      labels = c(`Never` = 0, `At most once a week` = 1, `Twice a week` = 2, `Once a day` = 3, `Twice a day` = 4, `Three times a day` = 5, `At least four times a day` = 6),
      label = "News sites use"
    ),
    readingtime = labelled_spss(readingtime,
      labels = ,
      label = "Average newspaper reading time per day (minutes)"
    )
  )
# save SPSS data file
haven::write_sav(readers, "data/readers.sav")
```

# Introduction and Reader's Guide {-}

In the years that I have been teaching inferential statistics to bachelor students in Communication Science, I have learned two things. First, it is paramount that students thoroughly understand the principles of statistical inference before they can apply statistical inference correctly themselves. Second, formal notation, manual calculation, and estimation details distract rather than help students understand what they are doing. This book offers a non-technical but thorough introduction to statistical inference. It discusses a minimal set of concepts needed to understand both the possibilities and pitfalls of estimation, null hypothesis testing, moderation, and mediation analysis. It uses a minimum of formal notation.

### Intended Audience and Setting {-}

This book is written as reading material for a follow-up course in statistics, in the bachelor of Communication Science at the University of Amsterdam. Students enrolled in this course have passe an introductory course in statistics that explained how to change research questions into variables and associations between variables, how to select and execute the correct analysis or test (in SPSS) to answer their research question, and how to interpret the results in a language that is both comprehensible for the average reader and complying with professional standards (APA standard for reporting test results). In addition, they have learned the very basics of inferential statistics: How to decide which null hypothesis to reject based on reported _p_ values, and how to interpret confidence intervals.  

This book is meant for use in a flipped-classroom setting. Students should read the text, watch embedded videos, and play with the interactive content before they meet in class. Class meetings are used to answer questions raised by the students, do group work to exercise with the concepts and techniques presented in the text, and do short tests to check understanding.  

### Interactive Content {-}

The interactive content in this book replaces simulations that used to be demonstrated during lectures. I expect that doing simulations yourself rather than watching them being done by someone else enhances understanding. I have tried to break down the simulations into smaller steps, confronting the student several times with essentially the same simulation, but with added complexity. I hope that this approach enhances understanding and remembrance and, at the same time, avoids frustration caused by complex dashboards offering all options at once. 

### Disclaimer {-}

The example data sets have been generated for the purpose of demonstrating statistical techniques. These are not real data and no conclusions should be drawn from the results obtained from the data.

### Acknowledgements {-}

Adam Sasiadek developed the more complicated Shiny apps in this book. The College of Communication at the University of Amsterdam generously supported the creation of the apps whereas this university's Grassroots Project for ICT in education refused to support it. Renske van Bronswijk corrected my English. Any remaining errors result from changes and additions that I applied afterwards. My colleague Peter Neijens commented on a draft of this text. Among the first tutors using this book, Chei Billedo, Marcel van Egmond, Matthijs Elenbaas, Andreas Goldberg, Bregje van Groningen, Luzia Helfer, Rhianne Hoek, Laura Jacobs, Jeroen Jonkman, Fam te Poel, Sanne Schinkel, Christin Scholz, Ragnheiður Torfadóttir, Philipp Mendoza, and she whose name I am not allowed to mention offered many suggestions that improved the text substantially. Among the students who helped to improve the book, Alissa Hilbertz stands out for her careful language corrections and suggestions.

<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png" /></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a>.