Data example

We use data about pengiuns from the R package palmerpenguins

pingviner <- palmerpenguins::penguins
pingviner

## # A tibble: 344 x 8
##    species island    bill_length_mm bill_depth_mm flipp… body… sex    year
##    <fctr>  <fctr>             <dbl>         <dbl>  <int> <int> <fct> <int>
##  1 Adelie  Torgersen           39.1          18.7    181  3750 male   2007
##  2 Adelie  Torgersen           39.5          17.4    186  3800 fema…  2007
##  3 Adelie  Torgersen           40.3          18.0    195  3250 fema…  2007
##  4 Adelie  Torgersen           NA            NA       NA    NA <NA>   2007
##  5 Adelie  Torgersen           36.7          19.3    193  3450 fema…  2007
##  6 Adelie  Torgersen           39.3          20.6    190  3650 male   2007
##  7 Adelie  Torgersen           38.9          17.8    181  3625 fema…  2007
##  8 Adelie  Torgersen           39.2          19.6    195  4675 male   2007
##  9 Adelie  Torgersen           34.1          18.1    193  3475 <NA>   2007
## 10 Adelie  Torgersen           42.0          20.2    190  4250 <NA>   2007
## # ... with 334 more rows

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:

tally(~species, data = pingviner)

## species
##    Adelie Chinstrap    Gentoo 
##       152        68       124

tally(species ~ island, data = pingviner)

##            island
## species     Biscoe Dream Torgersen
##   Adelie        44    56        52
##   Chinstrap      0    68         0
##   Gentoo       124     0         0

Plots of qualitative variables

• The main plotting functions for qualitative variables are gf_percents() and gf_bar(). E.g:

gf_percents(~species, data = pingviner)

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.

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
• …

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 and this.)

• 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.)

• 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	$1.80$

(See also this xkcd)