--- title: "Data collection and data wrangling" author: "The ASTA team" output: slidy_presentation: fig_caption: no highlight: tango theme: cerulean pdf_document: fig_caption: no highlight: tango number_section: yes toc: yes --- ```{r, include = FALSE} options(digits = 2) ## Remember to add all packages used in the code below! missing_pkgs <- setdiff(c("mosaic", "palmerpenguins"), rownames(installed.packages())) if(length(missing_pkgs)>0) install.packages(missing_pkgs) knitr::opts_chunk$set(warning = FALSE) library(mosaic) ``` # Data ## Data example We use data about pengiuns from the R package [palmerpenguins](https://github.com/allisonhorst/palmerpenguins) ```{r} pingviner <- palmerpenguins::penguins pingviner ``` ```{r include=FALSE} # Ignore this code. It's just used to write correct numbers later in the text. y <- pingviner %>% filter(species == "Gentoo") %>% pull(bill_length_mm) %>% na.omit() nn <- length(y) xx <- sort(y) ``` # Summaries and plots of qualitative variables ## Tables of qualitative variables - The main function to make tables from a data frame of observations is `tally()` which tallies (counts up) the number of observations within a given category. E.g: ```{r} tally(~species, data = pingviner) ``` ```{r} tally(species ~ island, data = pingviner) ``` ## Plots of qualitative variables - The main plotting functions for qualitative variables are `gf_percents()` and `gf_bar()`. E.g: ```{r} gf_percents(~species, data = pingviner) ``` ---- ```{r} gf_percents(~species, fill = ~sex, data = pingviner) ``` ---- ```{r} gf_percents(~species, fill = ~sex, data = pingviner, position = position_dodge()) ``` # Target population and random sampling ## Population parameters * When the sample size grows, then e.g. the mean of the sample, $\overline{y}$, will stabilize around a fixed value, $\mu$, which is usually unknown. The value $\mu$ is called the **population mean**. * Correspondingly, the standard deviation of the sample, $s$, will stabilize around a fixed value, $\sigma$, which is usually unknown. The value $\sigma$ is called the **population standard deviation**. * Notation: + $\mu$ (mu) denotes the population mean. + $\sigma$ (sigma) denotes the population standard deviation. | Population | Sample | |:----------:|:------------:| |$\mu$ |$\overline{y}$| |$\sigma$ |$s$ | ### A word about terminology * **Standard deviation**:\ a measure of variability of a population or a sample. * **Standard error**:\ a measure of variability of an estimate. For example, a measure of variability of the sample mean. ## Aim of statistics * Statistics is all about "saying something" about a population. * Typically, this is done by taking a random sample from the population. * The sample is then analysed and a statement about the population can be made. * The process of making conclusions about a population from analysing a sample is called **statistical inference**. ## Random sampling schemes Possible strategies for obtaining a random sample from the target population are explained in Agresti section 2.4: + **Simple sampling: each possible sample of equal size equally probable** + Systematic sampling + Stratified sampling + Cluster sampling + Multistage sampling + ... # Biases ## Types of biases Agresti section 2.3: * Sampling/selection bias + Probability sampling: each sample of size $n$ has same probability of being sampled + Still problems: undercoverage, groups not represented (inmates, homeless, hospitalized, ...) + Non-probability sampling: probability of sample not possible to determine + E.g. volunteer sampling * Response bias + E.g. poorly worded, confusing or even order of questions + Lying if think socially unacceptable * Non-response bias + Non-response rate high; systematic in non-responses (age, health, believes) ## Example of sample bias: United States presidential election, 1936 (Based on Agresti, [this](https://en.wikipedia.org/wiki/United_States_presidential_election,_1936) and [this](https://www.math.upenn.edu/~deturck/m170/wk4/lecture/case1.html).) * Current president: Franklin D. Roosevelt * Election: Franklin D. Roosevelt vs Alfred Landon (Republican governor of Kansas) * Literary Digest: magazine with history of accurately predicting winner of past 5 presidential elections ---- ### Results * Literary Digest poll: Landon: 57%; Roosevelt: 43% * Actual results: Landon: 38%; Roosevelt: 62% * Sampling error: 57%-38% = 19% + Practically all of the sampling error was the result of **sample bias** + Poll size of > 2 mio. individuals participated -- extremely large poll ---- ### Problems (biases) * Mailing list of about 10 mio. names was created + Based on every telephone directory, lists of magasine subscribers, rosters of clubs and associations, and other sources + Each one of 10 mio. received a mock ballot and asked to return the marked ballot to the magazine * "respondents who returned their questionnaires represented only that subset of the population with a relatively intense interest in the subject at hand, and as such constitute in no sense a random sample ... it seems clear that the minority of anti-Roosevelt voters felt more strongly about the election than did the pro-Roosevelt majority" (*The American Statistician*, 1976) * Biases: + Sample bias - List generated towards middle- and upper-class voters (e.g. 1936 and telephones) - Many unemployed (club memberships and magazine subscribers) + Non-response bias - Only responses from 2.3/2.4 mio out of 10 million people ## Example of response bias: Wording matters New York Times/CBS News poll on attitude to increased fuel taxes * "Are you in favour of a new gasoline tax?" - 12% said yes. * "Are you in favour of a new gasoline tax to decrease US dependency on foreign oil?" - 55% said yes. * "Do you think a new gas tax would help to reduce global warming?" - 59% said yes. ## Example of response bias: Order of questions matter US study during cold war asked two questions: 1 "Do you think that US should let Russian newspaper reporters come here and sent back whatever they want?" 2 "Do you think that Russia should let American newspaper reporters come in and sent back whatever they want?" The percentage of yes to question 1 was 36%, if it was asked first and 73%, when it was asked last. ## Example of survivior bias: Bullet holes of honor (Based on [this](https://www.motherjones.com/kevin-drum/2010/09/counterintuitive-world/).) * World War II * Royal Air Force (RAF), UK + Lost many planes to German anti-aircraft fire * Armor up! + Where? + Count up all the bullet holes in planes that returned from missions - Put extra armor in the areas that attracted the most fire ---- * Hungarian-born mathematician Abraham Wald: + If a plane makes it back safely with a bunch of bullet holes in its wings: holes in the wings aren't very dangerous - **Survivorship bias** + Armor up the areas that (on average) don't have any bullet holes - They never make it back, apparently dangerous |Section of plane$\quad$| Bullet holes per square foot | |:----------------------|:----------------------------:| |Engine | $1.11$| |Fuselage | $1.73$| |Fuel system | $1.55$| |Rest of the plane | $1.80$| ```{r, fig.width=10, echo=FALSE, fig.align='center', eval = FALSE} # was ![](https://asta.math.aau.dk/static-files/asta/img/bulletHoles.jpeg) url <- "https://asta.math.aau.dk/static-files/asta/img/bulletHoles.jpeg" z <- tempfile() download.file(url, z, mode = "wb") grid::grid.raster(jpeg::readJPEG(z)) unlink(z) ``` (See also [this xkcd](https://xkcd.com/1827/))