Recommendation 5

Kelly McConville (chair), Ben Baumer, Nathan Tintle, Mine Dogucu, Patti Frazer Lock

Encourage multivariate thinking.

Introduction

Multivariable thinking (MVT) is critical for students to be able to navigate our complex, data-driven world. Phenomena in areas such as economics, medicine, social sciences, and natural sciences are influenced by a multitude of factors, and many rich data sets with lots of cases and a great number of variables are readily available. MVT is no longer a skill required only of statisticians and data scientists—everyone should be able to think critically about the relationships among many variables and draw conclusions from that investigation.

MVT is not new to statistics, or even new to the introductory statistics course. However, we must do more than sprinkle traces of MVT in our courses. Rather, we must embrace the inherent complexity of MVT and ensure that students get repeated practice disentangling the relationships in data. MVT is needed throughout the data analysis process and is necessary for understanding current research in applied disciplines such as medicine (NJ 2005) and countless others across the natural and social sciences and beyond. Multiple regression is just one technique for engaging with MVT. More broadly, MVT can span multiple units in introductory classes, including exploratory data analysis, statistical modeling, and statistical inference. In this section, we provide concrete suggestions on how to infuse MVT into introductory statistics and data science courses.

First steps towards MVT

Here are some starting points for incorporating MVT into your classes:

• Even if you focus on one or two variables in a study in order to help students master key learning objectives, go beyond this and require students to reflect on the broader context in which the datasets exist. (Assessment examples/resources: Problem 1 of this document)

• Regularly ask your students to create/find, explore and summarize datasets that have multiple variables. (Assessment examples/resources: First day survey example, student generated game data as explained in depth by Kuiper et al. (2025), problem 2 of this document)

• Provide opportunities for students to interpret and create data visualizations that display more than two variables. This, for example, can be achieved simply by coloring points on a scatterplot to represent a third variable. (Assessment examples/resources: Problem 3 of this document, NYTimes “What’s going on in this graph?, Gapminder Tools, this applet)

• Ask students to identify potential confounding variables and brainstorm how the impact of additional variables could possibly affect their conclusions. Exploring an example that exhibits Simpson’s Paradox can provide a compelling case for the importance of MVT. (Assessment examples/resources: How Much Do Minority Lives Matter?)

Intermediate topics related to MVT

Most of our first steps involved considering how additional variables might impact one’s statistical work. For the intermediate steps, we suggest requiring students to more directly incorporate multiple variables into their analysis:

• Infuse MVT into your lessons about data collection methods. Ask students to brainstorm what potential confounding variables they would like to measure, or how they might employ stratification in random sampling. (Assessment examples/resources: Section 7.3.1 of “Using data-centric methods to teach introductory statistics”)

• Ask students to explore how common data wrangling steps, such as removing observations with missing values or transforming variables, impact the relationships between variables in their datasets. (Assessment examples/resources: Section 7.3.2 of “Using data-centric methods to teach introductory statistics”)

• For student projects, get students asking and assessing MVT questions, such as how to control for variables and what potential confounding variables exist. (Assessment examples/resources: )

• Lay the foundation for modeling by helping students see connections between the two-sample t-test and the linear regression model. (Assessment examples/resources: This activity )

• Take a prediction approach to motivate and teach the idea of multiple linear regression. In particular, have students (Assessment examples/resources:)

  • Explore these models graphically by creating scatterplots where a third categorical variable is represented by the color of the points and include the regression curve.

  • Fit these models and use them for prediction.

  • Consider the assumptions of the model and whether or not any violated assumptions could be resolved by incorporating additional predictors.

  • Connect the interpretation of individual model terms with the idea of confounding.

Aspirational topics related to MVT

We understand that many introductory statistics courses are already overloaded with topics to cover. Given the importance of MVT to modern data work, we recommend instructors take a hard look at what topics can be removed to make room for more MVT. (Our suggestions might include looking up p-values in tables, calculating the standard deviation by hand, or deriving closed form expressions for the expected value of a random variable mathematically.) A few aspirational topics to consider including in your courses include:

• Have students brainstorm sources of variation in a response variable before giving them access to the data. Ask students to determine whether these sources of variation could be controlled by design or analytically or if these sources need to remain as unexplained variation? (Assessment examples/resources:this exploration)

• Continue developing the student’s understanding of multiple linear regression. In particular, have students (Assessment examples/resources: this activity, this app)

  • Graphically explore the differences between equal slopes and different slopes models.

  • Practice interpreting interaction terms between categorical and quantitative predictors.

  • Conduct inference for regression terms.

  • Use a model selection criteria, such as adjusted R-squared or BIC, to compare and evaluate different models.

Additional Resources

• Annotated bibliography

• Examples

• Assessments

References

Kuiper, Shonda, Abhishek Chakraborty, Tyler George, Lisa W Kay, Lawrence M Lesser, Ginger Rowell, Dennis Pearl, Scott Crawford, and Anna Olsen. 2025. “The Greenhouse Effect: Using Student-Generated Agricultural Data to Warm up Students for Data-Based Decision Making.” Journal of Statistics and Data Science Education 33 (2): 143–51.

NJ, HORTON. 2005. “Statistical Methods in the Journal.” N Engl J Med. 353 (18): 1977–79.