How to Interpret R Squared in Regression Analysis?

Regression Analysis is a set of statistical processes that are at the core of data science. It helps in interpreting machine learning algorithms. Linear regression models across diverse fields such as ads, medical research, farming, and sports.  

R-squared tells us how well the model and the thing we're studying are connected. It's on a scale from 0 to 100%, making it easy to figure out how good the model is. In linear regression models, R-squared is a goodness-fit-measure. It considers the relationship strength between the model and the dependent variable. 

In this article, we will learn about R-squared (R2 ), r-squared value interpretation, limitations, how to interpret R squared in regression analysis and a few miscellaneous insights about it. We will also cover machine learning with python fundamentals and more. 

What is Regression Analysis?

Regression Analysis is a statistical technique that examines the relationship between independent (explanatory) and dependent (response) variables. It formulates a mathematical model to estimate values close to the actual ones.

Key terms in Regression Analysis:

• Dependent variables: The factors you want to predict.
• Independent variables: The factors that influence the dependent variable.

For example, using student data on study hours, attendance, and exam scores, regression analysis identifies which factors significantly impact exam scores. It highlights influential aspects like study hours or attendance and disregards less important ones, focusing the analysis by interpreting the r-squared value in regression analysis to determine how well the independent variables explain the variability in exam scores.

Thus, regression analysis reveals connections between study hours, attendance, and exam scores, providing a clear understanding of student performance influences.

The two main objectives of Regression Analysis are:

• Explanatory analysis: Understanding and identifying the influence of the explanatory variable on the response variable.
• Predictive analysis - This analysis is used to predict the value assumed by the dependent variable.

Why Use Regression Analysis?

The technique generates a regression equation where the relationship between the explanatory variable and the response variable is represented by the parameters of the technique. 

You can use the Regression Analysis to perform the following: 

• To model different independent variables. 
• To add continuous and categorical variables having numerous distinct groups based on a characteristic. 
• To model the curvature using polynomial terms. 
• To determine the effect of a certain independent variable on another variable by assessing the interaction terms.  

What is R-squared? 

R squared (R2  ) value in machine learning is referred to as the coefficient of determination or the coefficient of multiple determination in case of multiple regression.  

R-squared value interpretation in regression acts as an evaluation metric to evaluate the scatter of the data points around the fitted regression line. It recognizes the percentage of variation of the dependent variable.

R-squared and the Goodness-of-fit 

R-squared is the proportion of variance in the dependent variable that can be explained by the independent variable.

The value of R-squared stays between 0 and 100%: 

• 0% corresponds to a model that does not explain the variability of the response data around its mean. The mean of the dependent variable helps to predict the dependent variable and also the regression model. 
• On the other hand, 100% corresponds to a model that explains the variability of the response variable around its mean. 

If your value of R2  is large, you have a better chance of your regression model fitting the observations. 

Although you can get essential insights about the regression model in this statistical measure, you should not depend on it for the complete assessment of the model. It does not give information about the relationship between the dependent and the independent variables.  

It also does not inform about the quality of the regression model. Hence, as a user, you should always analyze R2  along with other variables and then derive conclusions about the regression model. 

What is Goodness-of-Fit?  

The Regression Analysis is a part of the linear regression technique. It examines an equation that reduces the distance between the fitted line and all of the data points. Determining how well the model fits the data is crucial in a linear model. 

A general idea is that if the deviations between the observed values and the predicted values of the linear model are small and unbiased, the model has a well-fit data. To evaluate this, it is important to interpret r squared value in Regression Analysis as it provides a measure of how well the observed outcomes are replicated by the model.

In technical terms, “Goodness-of-fit” is a mathematical model that describes the differences between the observed values and the expected values or how well the model fits a set of observations. This measure can be used in statistical hypothesis testing. 

How to Interpret R squared in Regression Analysis?

The simplest r squared interpretation in regression analysis is how well the regression model fits the observed data values. Let us take an example to understand this.

Consider a model where the  R2   value is 70%. Here r squared meaning would be that the model explains 70% of the fitted data in the regression model. Usually, when the R2   value is high, it suggests a better fit for the model.  

The correctness of the statistical measure does not only depend on R2   but can depend on other several factors like the nature of the variables, the units on which the variables are measured, etc. So, a high R-squared value is not always likely for the regression model and can indicate problems too. 

A low R-squared value is a negative indicator for a model in general. However, if we consider the other factors, a low R2  value can also end up in a good predictive model. 

Calculation of R-squared 

R- squared can be evaluated using the following formula: 

Where: 

• SSregression – Explained sum of squares due to the regression model. 
• SStotal – The total sum of squares. 

The sum of squares due to regression assesses how well the model represents the fitted data and the total sum of squares measures the variability in the data used in the regression model. 

Now let me come back to the earlier situation where we have two factors: number of hours of study per day and the score in a particular exam to understand the calculation of R-squared more effectively. Here, the target variable is represented by the score and the independent variable by the number of hours of study per day.

In this case, we will need a simple linear regression model and the equation of the model will be as follows:  

ŷ = w1x1 + b

The parameters w1  and b can be calculated by reducing the squared error over all the data points. The following equation is called the least square function:

minimize ∑(yi –  w1x1i – b)2

Now, to calculate the goodness-of-fit, we need to calculate the variance:

var(u) = 1/n∑(ui – ū)2

where, n represents the number of data points. 

Now, R-squared calculates the amount of variance of the target variable explained by the model, i.e. function of the independent variable. 

However, in order to achieve that, we need to calculate two things: 

• Variance of the target variable: 

var(avg) = ∑(yi – Ӯ)2

• Variance of the target variable around the best-fit line:
   

var(model) = ∑(yi – ŷ)2

Finally, we can calculate the equation of R-squared as follows:

R2  = 1 – [var(model)/var(avg)] = 1 -[∑(yi – ŷ)2/∑(yi – Ӯ)2] 

Limitations of R-squared in Regression Analysis

Some of the limitations of R-squared are: 

• R-squared cannot be used to check if the coefficient estimates and predictions are biased or not. 
• R-squared does not inform if the regression model has an adequate fit or not. 

To determine the biasedness of the model, you need to assess the residuals plots. A good model can have a low R-squared value whereas you can have a high R-squared value for a model that does not have proper goodness-of-fit.

How to Assess Goodness-of-fit in a Regression Model? 

As a statistician, I believe that if the differences between predicted values and actual observations are consistently small and unbiased, it indicates a well-fitted model. Unbiasedness means predicted values don’t veer too high or too low compared to actual observations.

A linear regression model minimizes the differences between observed and predicted values, seeking the smallest sum of squared residuals. Assessing a regression model requires examining residual plots before numerical measures like R-squared. These plots help identify potential biases by revealing any problematic patterns. Evidence of a biased model in the residual plots is a red flag, making the model results questionable. Conversely, if residual plots don’t show issues, it's appropriate to evaluate numerical metrics like r squared value interpretation and other outputs. Interpret R Squared in Regression Analysis to understand the proportion of variance in the dependent variable that is predictable from the independent variables.

What are Residuals? 

Residuals identify the deviation of observed values from the expected values. They are also referred to as error or noise terms. A residual gives an insight into how good our model is against the actual value but there are no real-life representations of residual values. 

Source:  hatarilabs.com

Regression Line and Residual Plots

The calculation of the real values of intercept, slope, and residual terms can be a complicated task. However, the Ordinary Least Square (OLS) regression technique can help us to speculate on an efficient model.  The technique minimizes the sum of the squared residuals. With the help of the residual plots, you can check whether the observed error is consistent with the stochastic error (differences between the expected and observed values must be random and unpredictable).  

Visual Representation of R-squared 

You can have a visual demonstration of the plots of fitted values by observed values in a graphical manner. It illustrates how R-squared values represent the scatter around the regression line. 

As observed in the pictures above, the value of R-squared for the regression model on the left side is 17%, and for the model on the right is 83%. In a regression model, when the variance accounts to be high, the data points tend to fall closer to the fitted regression line.  

However, a regression model with an R2  of 100% is an ideal scenario which is actually not possible. In such a case, the predicted values equal the observed values and it causes all the data points to fall exactly on the regression line. 

Low R-squared and High R-squared values 

Regression models with low R² do not always pose a problem. This is common in areas like studying human behavior, which often results in R² values less than 50% due to the complexity of predicting people compared to physical processes. Essential conclusions can still be drawn if the independent variables in the model have statistical significance, indicating the mean change in the dependent variable when the independent variable shifts by one unit. How to Interpret R Squared in Regression Analysis to understand the proportion of variance in the dependent variable that is predictable from the independent variables.

However, low R² values can be problematic for precise predictions. Conversely, a high R² can lead to specification bias if the model is missing key variables, polynomial terms, or interactions. This can be mitigated by adding appropriate terms or fitting a non-linear model.

Model overfitting and data mining can also inflate R², resulting in deceptively excellent fits. These models might appear to fit the data well but may not perform accurately on new, unseen data.

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Conclusion

We covered Regression Analysis, its importance, Residuals, Goodness-of-fit, and R-squared, including its representation, r-squared value interpretation We discussed low and high R-squared values. While R-squared is intuitive for determining model fit, it doesn't tell the whole story. Full understanding requires in-depth knowledge of R-squared and other statistical measures and residual plots. 

For more on R-squared limitations, learn about how to interpret R squared in regression analysis and Predicted R-squared, which offer different insights into model fit. Also, consider the Standard Error of the Regression. Explore more with upGrad's machine learning with python and related courses.

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