Question

Suppose we want to estimate the effects of alcohol consumption (alcohol) on college grade point average (colGPA). In addition to collecting information on grade point averages and alcohol usage. we also obtain attendance information (say, percentage of lectures attended, called attend). A standardized test score (say, \(S A T\) ) and high school GPA (hsGPA) are also available. i. Should we include attend along with alcohol as explanatory variables in a multiple regression model? (Think about how you would interpret \(\beta_{\text {alcohbol. }}\) ) ii. Should \(S A T\) and \(h s G P A\) be included as explanatory variables? Explain.

Short answer

Yes, include 'attend', 'SAT', and 'hsGPA' to control for confounding factors.

Step-by-step solution

Understanding the Variables

Before deciding which variables to include in the regression model, it's important to understand what each variable represents. - **colGPA**: College grade point average, the dependent variable we are trying to explain. - **alcohol**: The amount of alcohol consumed, the primary independent variable of interest. - **attend**: Percentage of lectures attended, a potential explanatory variable. - **SAT**: Standardized test score, potentially indicative of academic ability. - **hsGPA**: High school GPA, potentially indicative of prior academic performance.

Assessing the Role of 'attend' in the Model

Including **attend** in the multiple regression model is important for determining if there is a relationship between lecture attendance and college GPA. - If attendance influences colGPA, omitting it from the analysis could bias the estimation of the effect of alcohol consumption on colGPA. - Including "attend" allows for a clearer interpretation of the coefficient \( \beta_{\text{alcohol}} \), which would then reflect the effect of alcohol consumption on colGPA, holding lecture attendance constant. Thus, "attend" should be included.

Evaluating the Inclusion of 'SAT' and 'hsGPA'

The **SAT** score and **hsGPA** are indicators of a student's prior academic prowess, which could influence their college GPA. - Including these variables helps control for differences in academic preparedness among students, which could otherwise be confounded with the effects of alcohol consumption. - Failing to include them might overstate the effect of alcohol consumption on college GPA if more prepared students drink less or perform better regardless of drinking. Therefore, both "SAT" and "hsGPA" should be included to accurately estimate the effect of alcohol consumption.

Key concepts

Explanatory Variables

Explanatory variables, also known as independent variables, are the factors that help explain or predict changes in a dependent variable. In a multiple regression analysis, these variables are crucial for modeling and understanding the relationship between factors and outcomes.
To determine the impact of alcohol consumption on college GPA, we incorporate several explanatory variables beyond just the amount of alcohol consumed:

• Alcohol: The primary variable of interest, representing the level of alcohol consumption among college students. We are interested in understanding how changes in this variable influence college GPA.
• Attend: The percentage of lectures attended. This variable is crucial because lecture attendance can have a significant impact on academic performance.
• SAT and hsGPA: These variables represent prior academic performance and standardized test outcomes, which can affect current academic success.

By including these variables, the model becomes more robust and capable of providing a reliable estimation of how alcohol consumption affects college GPA, while controlling for other influences.

College GPA

College Grade Point Average (GPA) is the primary dependent variable in this analysis. As a measure of a student's academic performance, it serves as the central outcome that the regression model aims to predict and explain.
Understanding the factors that influence college GPA is important for both educators and students. In this context, we're interested in seeing whether and how alcohol consumption impacts academic performance, as measured by college GPA. To effectively assess this relationship, we incorporate various explanatory variables like attendance, high school GPA, and standardized test scores.
College GPA can be influenced by a multitude of factors such as study habits, social activities, financial stress, and health. By analyzing it in the context of a multiple regression analysis, we can tease out the specific impact of individual factors, while controlling for others.

Control Variables

Control variables play a crucial role in multiple regression analyses. They are included to account for additional factors that may influence the dependent variable, thus allowing the model to isolate and estimate the effect of the primary variables of interest.
In our analysis, SAT and hsGPA are considered control variables. They are important because they help control for prior academic ability and standardized performance, which can vary significantly among students. Including them in the model helps ensure that the estimation of alcohol's impact on college GPA is not confounded by these factors.
Moreover, controlling for variables like attendance ensures the measured effect of alcohol on GPA reflects alcohol's true impact, independent of how much students attend classes, as attendance itself can greatly influence academic success.

Regression Coefficients

Regression coefficients are essential in interpreting the results of a multiple regression analysis. They quantify the relationship between each explanatory variable and the dependent variable.
In our context, the coefficient for alcohol consumption (denoted as \( \beta_{\text{alcohol}} \)) tells us how much college GPA is expected to change with each unit increase in alcohol consumption, holding all other factors constant. This specific interpretation provides insights into the isolated effect of alcohol on academic performance.
Similarly, coefficients for other explanatory variables, like attendance or SAT scores, help us understand their individual contributions to changes in college GPA. These coefficients allow researchers and educators to assess the relative importance of each factor in influencing academic success, providing a clearer picture of what interventions or supports might be beneficial for students.