BMR 617 - Types of Variable

We make an important distinction between categorical and quantitative variables. Generally speaking, a categorial variable is one which takes on one of a fixed set of values, whereas a quantitative variable is one which is the result of a measurement.

This distinction is extremely important in deciding how to explore, analyze, and make inferences from your data.
Asking the question "Are the explanatory and response variables categorical or quantitative?" goes a long way to determining the correct analysis to perform.

We further classify categorical variables as nominal or ordinal, and quantitative variables as interval or ratio.

Remember the NOIR mnemonic:

Nominal: Categorical, no ordering. Examples: Gender, Race, Genotype.
Ordinal: Categorical with an order. Examples: Socio-economic status, Pain scale
Interval: Numerical data on a scale (i.e. it has units) but no meaningful zero. Examples: temperature in Celcius or Fahrenheit. Time of day or date.
Ratio: Numerical data with scale and zero. Most measurement data is in this category.

To determine the type of variable we are using, we can ask the following questions:

Is there an ordering for values of the variable?
If there is an ordering, is there a scale?
- I.e., does an increase in one unit always mean the same thing?
If there is an ordering and a scale, does the value zero have a specific meaning?

Nominal variables are those whose values have no ordering; they are just qualitative categories. Examples:

Gender. Values might be "Female", "Male", "Other"
Race. Values might be "African American", "Asian", "Caucasian", "Native American", etc.

Ordinal variables are variables with qualitative categories which have an ordering, but no scale.

Example: Economic status.

Values are typically stated as "Low", "Medium", or "High", which are computed using a number of factors (income, education level, occupation, wealth)
These are ordered because there is a natural ordering low → medium → high.
They have no scale because the difference between low and medium is not necessarily the same as the difference between medium and high.

Interval variables are variables with ordering and scale, but with no meaningful zero.

Examples:

Temperature in celsius or fahrenheit
- There is a scale,because a difference of one degree means the same thing, no matter what the starting temperature is
- However, the choice of a zero value (in either unit) is essentially arbitrary
Time of day or date

Computing differences of values of interval variables makes sense.

For example, computing a change in temperature (difference between two temperatures) makes sense, because a change of one unit (one degree) always means the same "amount of temperature".

Computing ratios of values of interval variables does not make sense, because there is no meaningful zero

For example, it doesn't make sense to say that 100°C is double 50°C.
In Farenheit, these values are 212°F and 132°F, respectively.

Ratio variables have both scale and a meaningful zero. Most measurements you work with will be ratio variables.

Examples: Length, mass, count data (e.g. number of cells), etc.

It makes sense to compute differences and ratios of ratio variables.

A blood pressure of 120 is double the blood pressure of 60.

Note that the difference of values of an interval variable is always a ratio variable.

E.g. age (the difference between two dates) is a ratio variable

Variable	Type
Tumor grade	Nominal Ordinal Interval Ratio	Incorrect Tumors are graded Grade I through Grade IV, with Grade I being the most differentiated and Grade IV being the least differentiated. Incorrect Tumors are graded Grade I through Grade IV, with Grade I being the most differentiated and Grade IV being the least differentiated. Even though these are ordered, the "difference between grades" is not really defined and there are no units. So these are not quantitative. Correct
Heart Rate	Nominal Ordinal Interval Ratio	Incorrect Heart rate is a measurement, typically in units of beats/minute. It is quantitative. Incorrect A heart rate of zero has a specific meaning, and it makes sense to talk about doubling a heart rate. Correct
Color	Nominal Ordinal Interval Ratio	Incorrect Colors do not have a specific order associated with them. (Even if you think about color spectra, those are really circular.) Incorrect Colors are categorical,not quantitative. Correct
Weight (mass)	Nominal Ordinal Interval Ratio	Incorrect Mass is a measurement and is quantitative. Incorrect A zero mass has a specific meaning (there is no matter there) and it makes sense to talk about doubling the mass of something. Correct
Disease status	Nominal Ordinal Interval Ratio	Incorrect Disease status is typically "Diseased" or "Not diseased". This do not have an ordering associated with them. Incorrect Disease status is typically "Diseased" or "Not diseased". This is categorical, not quantitative. Correct
Pain scale	Nominal Ordinal Interval Ratio	Incorrect Typically, hospital admissions ask for a pain rating on a scale of 1-10. These have an ordering to them, so they are not nominal. Incorrect Even though pain scales are ordered, there are no real units, and there's no real sense in which the difference between, for example, a pain rating of 2 and of 3 is the "same as" the difference between a pain rating of 9 and of 10. These are really just 10 arbitrary categories in an order. Correct Even though pain scales are ordered, there are no real units, and there's no real sense in which the difference between, for example, a pain rating of 2 and of 3 is the "same as" the difference between a pain rating of 9 and of 10. These are really just 10 arbitrary categories in an order.
Age	Nominal Ordinal Interval Ratio	Incorrect Age is a quantitative measure of time (typically in years, but it could just as well be measured in seconds, or any other time unit). Incorrect Age is quantitative, and zero has a specific meaning. Age is a measure of time. Correct
Genotype	Nominal Ordinal Interval Ratio	Incorrect Genotype is categorical (for example, if the possible alleles are C and T, the values are CC, CT, TT), but there is no sense of ordering of these values. Incorrect Genotype is categorical (for example, if the possible alleles are C and T, the values are CC, CT, TT). Correct
C_T values from RT-PCR	Nominal Ordinal Interval Ratio	Incorrect C_T values are quantitative. They represent the (interpolated) number of cycles on a thermocycler to reach some threshold quantity of DNA. Correct C_T values are definitely quantitative. It is somewhat debatable whether a C_T of zero has a specific meaning.

Nominal and Ordinal variables are categorical. They take on specific values only.

Interval and ratio variables are quantitative. They measure some value.

When we come to visualize and analyze data, the distinction between categorical and quantitative variables is the most important one. Knowing which variables are categorical and which are quantitative goes a long way to determining the correct presentation and analysis of the data.

R supports the notion of types of variable

Open Rstudio and type the following in the console (don't worry about what these functions mean yet):

 	
 	x <- rep(c("a", "b", "c"),  each=2)
 	y <- rnorm(6)

Look in the environment tab? What is the value of x? Can you guess what the function rep means?

What is the value of y?

rnorm gives random values from the normal distribution.

In R, we can ask what type a variable is using the class function.

Try the following:

 	
 	class(x)
 	class(y)

Look in the "Environment" tab. Can you interpret everything that's displayed there?

Thinking in statistical terms, are x and y categorical or quantitative?

So in R, a character variable is , and a numeric variable is .

Imagine a genetic study of obesity, in which we want to determine if the genotype of a particular locus confers obesity.

We could recruit a cohort of patients, measure their BMI, categorize them as obese (yes or no), and determine their genotype at a the locus of interest.

Let’s assume the locus of interest has two possible alleles, C and T

Variable	Categorical or Quantitative
BMI	Categorical Quantitative	Incorrect BMI is a measure; it's a person's weight divided by the square of their height. Correct BMI is a measure; it's a person's weight divided by the square of their height.
Obese	Categorical Quantitative	Incorrect "Obese" takes on the specific values "yes" or "no", so it's categorical. Correct "Obese" takes on the specific values "yes" or "no", so it's categorical.
Genotype	Categorical Quantitative	Incorrect "Genotype" takes on the specific values "CC", "CT", or "TT". Correct "Genotype" takes on the specific values "CC", "CT", or "TT".

Factors in R

In R, we could represent our genotype variable with a character. Try:

 	
    gt <- c("CC", "CC", "CT", "TT", "CT", "CC")

Look in the environment. What is the type of gt? Is this what you expect?

What does the following give?

	
	gt[[3]]

What happens if you do

	
	gt[[2]] <- "CT"

What about

	
	gt[[2]] <- "Meaningless"

When we have a variable that can only take on a fixed set of values, it’s useful to force R to only let it have those values.

This is a common feature of many categorical variables.

In R, a factor gives this functionality.

Try the following:

	
	gt <- factor(c("CC","CC","CT","TT","CT","CC"))

What does the environment tab display for gt now?

What if you display it in the console? (Just type gt in the console.)

Missing values and incorrect values in factors

Many times when working with data, some values are missing.

Particularly true for clinical and patient/subject-based studies.

R reserves the special value NA to represent a missing value.

Now we have gt as a factor, what happens if you do

	
	gt[[2]] <- “nonsense”
	gt

Using a factor in R allows us to force all values to either be meaningful, or missing.

Summary

There are four types of variable:

Nominal
Ordinal
Interval
Ratio

Nominal and ordinal variables are categorical.

Interval and ratio variables are quantitative.

The distinction between categorical and quantitative drives the decision as to how to visualize and analyze data.

Summary of R work

In R, we have learned:

The <- combination of symbols assigns a value to a variable.
- Note you can also use = here: x=c(2,3,5,7,8), but I prefer <-
You can access individual elements of a variable using [[ ]]
- x[[3]] will give the third element of x.
x[[2]] <- 5 will change the second element of x to 5.
You can find the type of a variable x using class(x)
Use character and factor types for categorical variables, numeric for quantitative variables
- We’ll see other types during the course
The special value NA represents a missing value.

BMR 617: Statistical Techniques for the Biomedical Sciences

Types of Variable

Types of Data

Determining the type of the variable

Nominal Variables

Ordinal Variables

Interval Variables

Operations on Interval Variables

Ratio Variables

Examples

Categorical and Quantitative Variables

Types of variable in R

Variable Types

Example

Factors in R

Missing values and incorrect values in factors

Summary

Summary of R work