3.1: Explanatory and Response Variables

While it is fundamentally important to know how to describe the distribution of a single variable, most studies pose research questions that involve exploring the relationship between two variables using the collected data.

Here are a few examples of such research questions with the two variables highlighted:

Examples

  1. Is there a relationship between gender and test scores on a particular standardized test?

    Other ways of phrasing the same research question include the following:

    • Is performance on the test related to gender?
    • Is there a gender effect on test scores?
    • Are there differences in test scores between males and females?
  2. How is the number of calories in a hot dog related to (or affected by) the type of hot dog (beef, meat or poultry)? In other words, are there differences in the number of calories among the three types of hot dogs?
  3. Is there a relationship between the type of light a baby sleeps with (no light, night-light, lamp) and whether or not the child develops nearsightedness?
  4. Are the smoking habits of a person (yes, no) related to the person’s gender?
  5. How well can we predict a student’s freshman year GPA from his/her SAT score?
  6. What is the relationship between driver’s age and sign legibility distance (the maximum distance at which the driver can read a sign)?
  7. Is there a relationship between the time a person has practiced driving while having a learner’s permit and whether or not this person passed the driving test?
  8. Can you predict a person’s favorite type of music (classical, rock, jazz) on the basis of his/her IQ level?

In most studies involving two variables, each of the variables has a role. We distinguish between:

  • The explanatory variable (also commonly referred to as the independent variable)—the variable that claims to explain, predict, or affect the response; and
  • The response variable (also commonly referred to as the dependent variable)—the outcome of the study.

Typically, the explanatory (or independent) variable is denoted by X, while the response (or dependent) variable is denoted by Y.

Explanatory and Response Variables

In this course, we use the terms explanatory and response variables instead of independent and dependent variables.

Now, let’s go back to some of the examples and classify the two relevant variables according to their roles in the study.

Example 1

We want to explore whether the outcome of the study—the score on a test—is affected by the test-taker’s gender.

Therefore:
Gender is the explanatory variable
Test score is the response variable

Example 3

In this study, we explore whether the nearsightedness of a person can be explained by the type of light that person slept with as a baby.
Therefore:
Light type is the explanatory variable
Nearsightedness is the response variable

Example 5

Here we are examining whether a student’s SAT score is a good predictor for the student’s GPA freshman year.

Therefore:
SAT score is the explanatory variable
GPA of freshman year is the response variable

Example 7

Here we are examining whether a person’s outcome on the driving test (pass/fail) can be explained by the length of time this person has practiced driving prior to the test.

Therefore:
Time is the explanatory variable
Driving test outcome is the response variable

Learn by Doing

Now, using the same reasoning, the following exercise will help you to classify the two variables in the other examples

(click here to open a separate window with the eight examples above).

Many Students Wonder…

Question: Is the role classification of variables always clear? In other words, is it always clear which of the variables is the explanatory and which is the response?
Answer: No. There are studies in which the role classification is not really clear. This mainly happens in cases when both variables are categorical or both are quantitative. An example is a study that explores the relationship between students’ SAT Math and SAT Verbal scores. In cases like this, any classification choice would be fine (as long as it is consistent throughout the analysis).

If we further classify each of the two relevant variables according to type (categorical or quantitative), we get the following four possibilities for role-type classification:

  1. Categorical explanatory and quantitative response
  2. Categorical explanatory and categorical response
  3. Quantitative explanatory and quantitative response
  4. Quantitative explanatory and categorical response

This role-type classification can be summarized and easily visualized in the following table (note that the explanatory variable is always listed first):

It is possible for any type of explanatory variable to be paired with any type of response variable. The possible pairings are: Categorical Explanatory → Categorical Response (C→C), Categorical Explanatory → Quantitative Response (C→Q), Quantitative Explanatory → Categorical Response (Q→C), and Quantitative Explanatory → Quantitative Response (Q→Q)

This role-type classification serves as the infrastructure for this entire section. In each of the four cases, different statistical tools (displays and numerical measures) should be used to explore the relationship between the two variables. This suggests the following important principle:

Principle

When confronted with a research question that involves exploring the relationship between two variables, the first and most crucial step is to determine which of the four cases represents the data structure of the problem. In other words, the first step should be classifying the two relevant variables according to their role and type, and only then can we determine what statistical tools should be used to analyze them.

Now let’s go back to our eight examples and determine which of the four cases represents the data structure of each:

Example 1

Gender is the explanatory variable, and it is categorical.
Test score is the response variable, and it is quantitative.
Therefore, this is an example of case C→Q.

Example 3

Light Type is the explanatory variable, and it is categorical.
Nearsightedness is the response variable, and it is categorical.
Therefore, this is an example of case C→C.

Example 5

SAT Score is the explanatory variable, and it is quantitative.
GPA of Freshman Year is the response variable, and it is quantitative.
Therefore, this is an example of case Q→Q.

Example 7

Time is the explanatory variable, and it is quantitative.
Driving Test Outcome is the response variable, and it is categorical.
Therefore, this is an example of case Q→C.

Learn by Doing

Now you complete the rest (click here to open a separate window with the eight examples). For cases such as Q→C, type them as Case Q-C.

Example 2:

Example 4:

Example 6:

Example 8:

Did I get this?

In each of the following three problems, you are presented with a brief description of a study involving two variables. Based on the role-type classification of the two variables, you are asked to determine which of the four cases represents the data structure of the problem.

For your convenience, here again is the role-type classification table:

Categorical Explanatory → Categorical Response (C→C), Categorical Explanatory → Quantitative Response (C→Q), Quantitative Explanatory → Categorical Response (Q→C), Quantitative Explanatory → Quantitative Response (Q→Q)

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Intro to Statistics MAT1260 by SBCCOE is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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