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How do you explain the concept of adjusted r squared to a non-technical audience?

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If you have ever used regression analysis to model the relationship between a dependent variable and one or more independent variables, you have probably encountered the terms r squared and adjusted r squared. These are two common measures of how well your model fits the data, but what do they actually mean and how do they differ? In this article, we will explain the concept of adjusted r squared in a simple and intuitive way, and compare it to r squared and other model selection criteria.

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Samarth Khurana, FRM?, CSPO?

Consultant at Deloitte USI | Certified FRM | CSPO? | Product Management | Model Risk Management
Natalie Rodrigue

Skilled Biostatistician & Epidemiologist - Mentor, Experienced, Knowledgeable & Passionate Instructor - DoE - Expert…
Olayemi O.

1 What is r squared?

R squared, also known as the coefficient of determination, is a statistic that ranges from 0 to 1 and indicates how much of the variation in the dependent variable can be explained by the variation in the independent variables. A higher r squared means a better fit, while a lower r squared means a weaker fit. You can calculate r squared by dividing the sum of squares of the regression (SSR) by the total sum of squares (SST), or by squaring the correlation coefficient between the observed and predicted values.

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Samarth Khurana, FRM?, CSPO?

Consultant at Deloitte USI | Certified FRM | CSPO? | Product Management | Model Risk Management
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Imagine R-squared as a report card grade that measures how well a student understands a subject. Now, think of adjusted R-squared as a modified grade that factors in the difficulty level of the coursework or any extra credit earned. In this context, while R-squared serves as a basic indicator of how much our statistical model comprehends the data, adjusted R-squared goes a step further by considering the challenges and complexities the model encounters. It's like giving the model a more thoughtful and fair evaluation, accounting for various factors to provide a more accurate measure of its performance in real-world scenarios.

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Olayemi O.
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There needs to be an explanation of what sum of squares and total sum of squares. Maybe an analogy can be used here? Or perhaps a real life example used throughout the article to bring readers with no background in statistics/econometrics along.

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Natalie Rodrigue

Skilled Biostatistician & Epidemiologist - Mentor, Experienced, Knowledgeable & Passionate Instructor - DoE - Expert in the Analysis of Genomic Data & Statistical Modelling. + 30 yrs experience.
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The R-squared or coefficient of determination is the percentage of variation explained by the model (explanatory variables included in the model). For instance, if the model r-squared is 82%, this means that 82% of the variation in the response (dependent variable) is explained (or captured) by the explanatory variable(s). It also means that the model fails to explain 18% of the variation in the response. For a simple linear regression (a single explanatory variable), the r-squared is the square of the correlation coefficient between Y (dependent) and X (explanatory variable.

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Gerard Kinneen

CoFounder and CXO of iSort AB | Blending Innovation in Robotics with Practical Solutions for Heavy Load Management | Passionate about Autonomous Vehicles for Construction Sites.
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Adjusted R-squared is like a score that tells us how good our model is at predicting things. It considers both how well our model fits the data and how many things we're trying to predict with. So, a higher adjusted R-squared means our model is doing a good job, but we don't want it to be too high, or we might be making things overly complicated.

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2 What is adjusted r squared?

Adjusted r squared is a modified version of r squared that takes into account the number of independent variables in the model. It adjusts the r squared value downward to penalize models that have too many variables, which can lead to overfitting and multicollinearity. You can calculate adjusted r squared by multiplying r squared by a factor that depends on the sample size (n) and the number of independent variables (k):

adjusted r squared = 1 - [(1 - r squared) * (n - 1) / (n - k - 1)]

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3 Why use adjusted r squared?

Adjusted r squared is useful because it gives you a more realistic estimate of how well your model generalizes to new data. Unlike r squared, adjusted r squared can decrease when you add irrelevant variables to the model, which can help you avoid adding unnecessary complexity. Adjusted r squared also accounts for the degrees of freedom of the model, which reflects the amount of information available to estimate the parameters.

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Natalie Rodrigue

Skilled Biostatistician & Epidemiologist - Mentor, Experienced, Knowledgeable & Passionate Instructor - DoE - Expert in the Analysis of Genomic Data & Statistical Modelling. + 30 yrs experience.
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The more explanatory variables are added into a regression model, the larger the r-square becomes. The adjusted r-squared, as the name suggests, penalises explanatory variables that do not contribute to the explanatory power of the model. It is advisable to rely on the r-squared for multiple linear regression models (several explanatory variables). Discrepancy between the r-squared and the adjusted r-squared indicates that some explanatory variables part of the model do not represent variables that contribute to the explanatory performance of the model. A more parcimonious model is needed.

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4 How to compare models using adjusted r squared?

One way to compare different models using adjusted r squared is to use the principle of parsimony, which states that simpler models are preferable to more complex ones, as long as they have similar explanatory power. In other words, you should choose the model that has the highest adjusted r squared with the fewest independent variables. However, adjusted r squared is not the only criterion that you can use to compare models. There are other metrics, such as the Akaike information criterion (AIC) or the Bayesian information criterion (BIC), that also balance fit and complexity, but in different ways.

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5 How to interpret adjusted r squared?

Interpreting adjusted r squared is similar to interpreting r squared, but with some caveats. Adjusted r squared tells you how much of the variation in the dependent variable can be explained by the variation in the independent variables, after adjusting for the number of variables and the sample size. However, adjusted r squared does not imply causation, nor does it measure the accuracy or precision of the predictions. Adjusted r squared also depends on the choice of variables and the functional form of the model, so it is not a universal measure of goodness of fit.

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6 How to communicate adjusted r squared to a non-technical audience?

Communicating adjusted r squared to a non-technical audience can be difficult, but not impossible. To make it easier, analogies or examples can be used to illustrate the concept of fit and complexity. For instance, you can compare your model to a recipe that uses different ingredients to create a dish. Adjusted r squared tells you how well your recipe matches the taste of the dish, as well as how simple or complicated your recipe is. You want to find the recipe that uses the least amount of ingredients, yet still produces a delicious dish. Additionally, jargon or formulas should be avoided and focus should be placed on the main idea and implications. For example, you can say that adjusted r squared is a number that measures how well your model explains the data, but also how simple your model is. Moreover, visuals or graphs can demonstrate the difference between r squared and adjusted r squared. For example, you can plot the r squared and adjusted r squared values for different models on a chart and highlight the model that has the highest adjusted r squared and the lowest number of variables. You can also show how adding or removing variables affects the adjusted r squared value.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you explain the concept of adjusted r squared to a non-technical audience?

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1 What is r squared?

2 What is adjusted r squared?

3 Why use adjusted r squared?

4 How to compare models using adjusted r squared?

5 How to interpret adjusted r squared?

6 How to communicate adjusted r squared to a non-technical audience?

7 Here’s what else to consider

Regression Analysis

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