Introduction to Biostatistics

1 Learning objectives

1.1 Learning objectives

Understand the different types of variables that can be measured
Think about variables in terms of central tendency or location
Think about variables in terms of dispersion or spread
Start to think about variable distributions and plots

2 Types of variables

2.1 Types of variables

There are a lot of ways to talk about variables
- Continuous vs categorical
- Quantitative vs qualitative
- Numeric vs non-numeric
Often loose, ambiguous, and arbitrary

2.2 Levels of measurement

Four ordered levels of measurement based on the mathematical operations that can be performed
- Nominal
- Ordinal
- Interval
- Ratio

Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103(2684), 677-680.

2.3 Nominal variables

Categories with no intrinsic ordering
- Nominal = “name”
Examples
- Department: Psychology, Epidemiology, Statistics, Business
- Religion: Christian, Jewish, Muslim, Atheist
- Ice cream flavor: vanilla, chocolate, strawberry

2.4 Ordinal variables

Categories with some intrinsic ordering
- Ordinal = “ordered”
- Differences between categories are not meaningful/equal
Examples
- Dose of treatment: low, medium, high
- Rankings: 1st, 2nd, 3rd, 4th
- Education: high school, some college, college grad, graduate
- Likert scales: agree, neutral, disagree

2.5 Interval variables

Quantitative variables with no meaningful 0 point
- (“Meaningful 0”: value of 0 = nothing)
- Differences between values are meaningful but ratios are not!
Example: Temperature in Fahrenheit or Celsius
- Difference from 100F to 90F = difference from 90F to 80F
- But 100F is not twice 50F (because 0F is arbitrary)
Most “continuous” variables you deal with are interval
- Most statistical procedures assume interval-level measurement

2.6 Ratio variables

Quantitative variables with meaningful 0 point
- (“Meaningful 0”: value of 0 = nothing)
- Differences between values are meaningful and so are ratios!
Example: Temperature in Kelvin
- Difference from 100K to 90K = difference from 90K to 80K
- 100K is twice as hot as 50K (0K is zero molecular movement)
Height, weight, age, counts

2.7 Stevens (1946)

The levels of measurement determine what mathematical (and statistical) operations you can perform

Mathematical operation	Nominal	Ordinal	Interval	Ratio
equal, not equal	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
greater or less than		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
add, subtract			\(\checkmark\)	\(\checkmark\)
multiply, divide				\(\checkmark\)
central tendency	mode	median	mean	mean

2.8 R “translations” of variable types

Nominal
- char = “character”: Text variable, also called “string”
Nominal or ordinal
- fctr = “factor”: Ordered or unordered categories
- log = “logical” or “boolean”: TRUE or FALSE
Interval or ratio
- int = “integer”: Whole number (no decimals)
- dbl = “double” (num): Number w possible decimal places

3 Central tendency

3.1 Central tendency

Also called “location”
- Where are the observations located?
Several measures of central tendency
- Depend on scale type
- Mode, median, mean

3.2 Mode

The most frequently occurring value
- Value actually exists in the data
Used for nominal, ordinal
- Can also be used for interval, and ratio variables but less useful

3.3 Median

Order observations from smallest to largest
- Select the middle observation
- Value actually exists in the data (if \(n\) is even)
Used for ordinal, interval, and ratio variables

3.4 Mean

Add up all observations and divide by the number of observations
- Value may not actually occur in the data
Mathematically: \(\frac{\sum_{i=1}^n X_i}{n}\)
Used for interval and ratio variables

3.5 gapminder dataset

gap <- gapminder
head(gap)

# A tibble: 6 × 6
  country     continent  year lifeExp      pop gdpPercap
  <fct>       <fct>     <int>   <dbl>    <int>     <dbl>
1 Afghanistan Asia       1952    28.8  8425333      779.
2 Afghanistan Asia       1957    30.3  9240934      821.
3 Afghanistan Asia       1962    32.0 10267083      853.
4 Afghanistan Asia       1967    34.0 11537966      836.
5 Afghanistan Asia       1972    36.1 13079460      740.
6 Afghanistan Asia       1977    38.4 14880372      786.

3.6 gapminder variables

Variable	R variable type	Scale type
`country`	factor	nominal
`continent`	factor	nominal
`year`	integer	ordinal / interval
`lifeExp`	double / numeric	ratio
`pop`	integer	ratio (count)
`gdpPercap`	double / numeric	ratio

3.7 Nominal variables: Mode

Mode() function from DescTools package

Mode(gap$country)

  [1] Afghanistan              Albania                  Algeria                 
  [4] Angola                   Argentina                Australia               
  [7] Austria                  Bahrain                  Bangladesh              
 [10] Belgium                  Benin                    Bolivia                 
 [13] Bosnia and Herzegovina   Botswana                 Brazil                  
 [16] Bulgaria                 Burkina Faso             Burundi                 
 [19] Cambodia                 Cameroon                 Canada                  
 [22] Central African Republic Chad                     Chile                   
 [25] China                    Colombia                 Comoros                 
 [28] Congo, Dem. Rep.         Congo, Rep.              Costa Rica              
 [31] Cote d'Ivoire            Croatia                  Cuba                    
 [34] Czech Republic           Denmark                  Djibouti                
 [37] Dominican Republic       Ecuador                  Egypt                   
 [40] El Salvador              Equatorial Guinea        Eritrea                 
 [43] Ethiopia                 Finland                  France                  
 [46] Gabon                    Gambia                   Germany                 
 [49] Ghana                    Greece                   Guatemala               
 [52] Guinea                   Guinea-Bissau            Haiti                   
 [55] Honduras                 Hong Kong, China         Hungary                 
 [58] Iceland                  India                    Indonesia               
 [61] Iran                     Iraq                     Ireland                 
 [64] Israel                   Italy                    Jamaica                 
 [67] Japan                    Jordan                   Kenya                   
 [70] Korea, Dem. Rep.         Korea, Rep.              Kuwait                  
 [73] Lebanon                  Lesotho                  Liberia                 
 [76] Libya                    Madagascar               Malawi                  
 [79] Malaysia                 Mali                     Mauritania              
 [82] Mauritius                Mexico                   Mongolia                
 [85] Montenegro               Morocco                  Mozambique              
 [88] Myanmar                  Namibia                  Nepal                   
 [91] Netherlands              New Zealand              Nicaragua               
 [94] Niger                    Nigeria                  Norway                  
 [97] Oman                     Pakistan                 Panama                  
[100] Paraguay                 Peru                     Philippines             
[103] Poland                   Portugal                 Puerto Rico             
[106] Reunion                  Romania                  Rwanda                  
[109] Sao Tome and Principe    Saudi Arabia             Senegal                 
[112] Serbia                   Sierra Leone             Singapore               
[115] Slovak Republic          Slovenia                 Somalia                 
[118] South Africa             Spain                    Sri Lanka               
[121] Sudan                    Swaziland                Sweden                  
[124] Switzerland              Syria                    Taiwan                  
[127] Tanzania                 Thailand                 Togo                    
[130] Trinidad and Tobago      Tunisia                  Turkey                  
[133] Uganda                   United Kingdom           United States           
[136] Uruguay                  Venezuela                Vietnam                 
[139] West Bank and Gaza       Yemen, Rep.              Zambia                  
[142] Zimbabwe                
attr(,"freq")
[1] 12
142 Levels: Afghanistan Albania Algeria Angola Argentina Australia ... Zimbabwe

There is no single mode for country

3.8 Nominal variables: Mode

Mode() function from DescTools package

Mode(gap$continent)

[1] Africa
attr(,"freq")
[1] 624
Levels: Africa Americas Asia Europe Oceania

There are 52 countries (52 countries * 12 years = 624) in Africa
- The value of “Africa” is the mode of the continent variable

3.9 Ordinal variables: Median

median(gap$year, na.rm = TRUE)

[1] 1979.5

1952, 1957, 1962, 1967, 1972, 1977, 1982, 1987, 1992, 1997, 2002, 2007
- Median is halfway between 1977 and 1982
na.rm = TRUE option removes missing values
- If you have missing values and don’t include this, you will get NA

3.10 Interval variables: Mean

mean(gap$year, na.rm = TRUE)

[1] 1979.5

\((1952 + 1957 + \dots + 2002 + 2007)/12\)
You can also calculate the median of an interval variable, but we just did that for this variable (treating it as ordinal)
na.rm = TRUE option removes missing values
- If you have missing values and don’t include this, you will get NA

3.11 Use only data from 2007

Including all data gives us a confusing mish-mash of information
- I’m just going to include data from 2007 for these variables
Also going to create the gdp variable at the same time

gap_2007 <- gapminder %>% 
  mutate(gdp = gdpPercap * pop) %>%
  filter(year == 2007)

3.12 Ratio variables: Median

median(gap_2007$lifeExp, na.rm = TRUE)

[1] 71.9355

median(gap_2007$pop, na.rm = TRUE)

[1] 10517531

median(gap_2007$gdpPercap, na.rm = TRUE)

[1] 6124.371

median(gap_2007$gdp, na.rm = TRUE)

[1] 57869055458

50% of countries have a life expectancy less than 71.9355 years
50% of countries have a life expectancy greater than 71.9355 years

3.13 Ratio variables: Mean

mean(gap_2007$lifeExp, na.rm = TRUE)

[1] 67.00742

mean(gap_2007$pop, na.rm = TRUE)

[1] 44021220

mean(gap_2007$gdpPercap, na.rm = TRUE)

[1] 11680.07

mean(gap_2007$gdp, na.rm = TRUE)

[1] 409220666999

The average life expectancy is 67.00742 years

3.14 Compare mean and median

Mean and median are both measures of central tendency for numbers
- What does it mean when they differ?
Median involves only values near it
- Extreme values don’t impact it
Mean involves all values
- Extreme values impact the mean
- Mean is “pulled toward” extreme values

3.15 Mean and median of `lifeExp`

Code

ggplot(data = gap_2007, aes(x = lifeExp)) + 
  geom_histogram(bins = 30, color = "grey80") +
  geom_vline(xintercept = mean(gap_2007$lifeExp), color = "blue", linewidth = 1.5) +
  geom_vline(xintercept = median(gap_2007$lifeExp), color = "red", linewidth = 1.5, linetype = "dashed")

3.16 Mean and median of `pop`

Code

ggplot(data = gap_2007, aes(x = pop)) + 
  geom_histogram(bins = 30, color = "grey80") +
  geom_vline(xintercept = mean(gap_2007$pop), color = "blue", linewidth = 1.5) +
  geom_vline(xintercept = median(gap_2007$pop), color = "red", linewidth = 1.5, linetype = "dashed")

3.17 Mean and median of `gdpPercap`

Code

ggplot(data = gap_2007, aes(x = gdpPercap)) + 
  geom_histogram(bins = 30, color = "grey80") +
  geom_vline(xintercept = mean(gap_2007$gdpPercap), color = "blue", linewidth = 1.5) +
  geom_vline(xintercept = median(gap_2007$gdpPercap), color = "red", linewidth = 1.5, linetype = "dashed")

4 Dispersion

4.1 Dispersion

Also called “spread”
- How spread out or dispersed are the observations?
- Only applies to numeric (interval or ratio) variables
Several measures of dispersion
- Depend on scale type and variable distribution
- Minimum and maximum, percentiles, standard deviation

4.2 Minimum and maximum

Minimum

min(gap_2007$lifeExp, na.rm = TRUE)

[1] 39.613

min(gap_2007$pop, na.rm = TRUE)

[1] 199579

min(gap_2007$gdpPercap, na.rm = TRUE)

[1] 277.5519

Maximum

max(gap_2007$lifeExp, na.rm = TRUE)

[1] 82.603

max(gap_2007$pop, na.rm = TRUE)

[1] 1318683096

max(gap_2007$gdpPercap, na.rm = TRUE)

[1] 49357.19

4.3 Percentiles

Value below which some percentage of observations lie
- 67th percentile: 67% of observations have values below this value
- Median is the 50th percentile: 50% below (and 50% above)
- Quartiles divide into 4 parts: 25th, 50th, 75th percentiles
- Standardized tests: 97th percentile means you scored higher than 97% of people taking the test

4.4 Percentiles

quantile(gap_2007$lifeExp, c(0.25, 0.5, 0.75))

     25%      50%      75% 
57.16025 71.93550 76.41325

quantile(gap_2007$pop, c(0.25, 0.5, 0.75))

     25%      50%      75% 
 4508034 10517531 31210042

quantile(gap_2007$gdpPercap, c(0.1, 0.9))

       10%        90% 
  887.2871 33644.0530

4.5 Standard deviation (& variance)

Spread around the mean
Same units as the variable
- Unlike variance, with is \(SD^2\)
Influenced by extreme values
Mathematically: \(\sqrt{\frac{\sum (X_i - \overline{X})^2}{n-1}}\)

4.6 Presenting location & dispersion

Symmetric distribution
- Mean \(\pm\) standard deviation
- Median (25th to 75th percentile)
- Median (min, max)
Asymmetric distribution
- Median (25th to 75th percentile)
- Median (min, max)

4.7 Presenting

Code

lifeexp_perc <- quantile(gap_2007$lifeExp, c(0, 0.25, 0.5, 0.75, 1))
lifeexp_sd <- sd(gap_2007$lifeExp, na.rm = TRUE)
lifeexp_mean <- mean(gap_2007$lifeExp, na.rm= TRUE)
ggplot(data = gap_2007, aes(x = lifeExp)) + 
  geom_histogram(bins = 30, color = "grey80")

4.8 Presenting: Mean plus/minus 1 SD

Code

ggplot(data = gap_2007, aes(x = lifeExp)) + 
  geom_histogram(bins = 30, color = "grey80") +
  annotate("pointrange", x = lifeexp_mean, y = 5, 
            xmin = mean(gap_2007$lifeExp) - lifeexp_sd,
            xmax = mean(gap_2007$lifeExp) + lifeexp_sd,
            size = 1.5, linewidth = 1.5, color = "red") +
  ylab("count")

4.9 Presenting: Median, 25th, 75th %iles

Code

ggplot(data = gap_2007, aes(x = lifeExp)) + 
  geom_histogram(bins = 30, color = "grey80") +
  annotate("pointrange", x = lifeexp_perc[3], y = 5, 
            xmin = lifeexp_perc[2],
            xmax = lifeexp_perc[4],
            size = 1.5, linewidth = 1.5, color = "red") +
  ylab("count")

5 In-class activities

5.1 In-class activities

Look at some data and variables
- Think about what type(s) of variables they are
Examine central tendency and dispersion for the variables
- Present the best values to describe the data
Start to think about variable distributions and plots

Introduction to Biostatistics

1 Learning objectives

1.1 Learning objectives

2 Types of variables

2.1 Types of variables

2.2 Levels of measurement

2.3 Nominal variables

2.4 Ordinal variables

2.5 Interval variables

2.6 Ratio variables

2.7 Stevens (1946)

2.8 R “translations” of variable types

3 Central tendency

3.1 Central tendency

3.2 Mode

3.3 Median

3.4 Mean

3.5 gapminder dataset

3.6 gapminder variables

3.7 Nominal variables: Mode

3.8 Nominal variables: Mode

3.9 Ordinal variables: Median

3.10 Interval variables: Mean

3.11 Use only data from 2007

3.12 Ratio variables: Median

3.13 Ratio variables: Mean

3.14 Compare mean and median

3.15 Mean and median of lifeExp

3.16 Mean and median of pop

3.17 Mean and median of gdpPercap

4 Dispersion

4.1 Dispersion

4.2 Minimum and maximum

4.3 Percentiles

4.4 Percentiles

4.5 Standard deviation (& variance)

4.6 Presenting location & dispersion

4.7 Presenting

4.8 Presenting: Mean plus/minus 1 SD

4.9 Presenting: Median, 25th, 75th %iles

5 In-class activities

5.1 In-class activities

3.15 Mean and median of `lifeExp`

3.16 Mean and median of `pop`

3.17 Mean and median of `gdpPercap`