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How to Interpret R-squared in Regression Analysis?

Introduction

Regression analysis helps determine the relationship among the variables and how one variable affects the other. In regression analysis, R-squared is a model that evaluates how the data is scattered around a fitted regression line. The variation of the dependent data across the regression line is calculated in terms of percentage. 

You can pursue an Executive PG Programme in Machine Learning & AI from IIITB to understand the relevance of R-squared regression analysis in data science. This blog discusses in detail the different aspects of R-squared in regression analysis.

The Meaning and Components of R-squared

We must first understand what dependent and independent variables mean in this context. Dependent variables are the factors to be predicted or understood, and independent variables are the factors on which the dependent variables rely or are based. 

Correlation helps determine the strength of the relation between the dependent and independent variables. On the other hand, R-squared meaning determines the extent to which the variables are related. A higher value of the R-squared determines more dependence of the performance of the dependent variable on the independent variable. In other words, a higher R-squared value indicates more variability explained by the regression model.

The following components are required to calculate the value of R-squared:

  • Total values of the dependent and the independent variables.
  • The sum of the product of the dependent and independent variables. 
  • The sum of the squares of the dependent and independent variable values. 

When put in a mathematical formula, these values help determine the extent of the relation between the dependent and the independent variables. 

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The Usefulness of R-squared in Regression Analysis

One of the main reasons why R-squared is used in regression analysis is because of its capability to determine the probability of the occurrence of future events within the predicted outcomes. 

R-squared is also known as the coefficient of determination. The value of the coefficient of determination ranges between 0 and 1 and helps measure the perfection of a statistical model with which it can predict an outcome. If the coefficient of determination is 0, it does not predict an outcome. If the result is 1, then the model predicts the outcome accurately. 

R-squared is used for tasks like identifying the risks in hedge funds, tracking the performance of mutual funds, etc. 

Interpreting R-squared Values: What They Mean

Knowing how to interpret R-squared values is imperative to arrive at conclusions. As mentioned above, R-squared values lie between 0 (0%) and 1 (100%).

R-squared regression helps determine how well the model explains the data around the mean. Obtaining 0% does not explain any variation in the response variable around the mean and vice versa. 

Let us understand this with an R-squared interpretation example. In the above picture, the value of R-squared for the first picture is 14.3%, and that in the second picture is 86.5%. When the value of R-squared is high, the data points will be closer to the regression line, and vice versa. If the value of R-squared is 100%, all the data points will lie on the regression line. However, it is an impossible occurrence in practical application. 

It is also important to note that the R-squared interpretation depends mainly on factors such as the variables’ nature, the unit of measure, etc. Therefore, one must acknowledge that a high or low value of R-squared cannot be considered an indicator of a good regression model. 

This is explained in the next section of the blog. 

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High R-squared vs Low R-squared: Which is Better?

Ideally, a high value of R-squared is believed to be better than a low value of R-squared. However, that is not always the case. A high value of R-squared can often indicate problems with the model. 

R-squared is a biased estimator. A biased estimator is either lower or higher than the population value in statistics. In case of a bias, the regression line continuously overpredicts or underpredicts the data across the curve.

Overfitting is another reason why the R-squared value may be high. Overfitting occurs when one includes more terms in a model than the number of observations. This causes the regression model to tailor itself so that the noise in the sample can fit easily, and it does not reflect the overall population. 

Other conditions that can cause the R-squared value to appear high include data mining. Therefore, the preference for a high or low value of R-squared varies depending on the condition. Pursuing an MS in Full Stack AI and ML can help you understand the relevance of these subjects in AI and ML. 

The Limitations of R-squared

There are some limitations to using the R-squared model for regression analysis. 

  • Although R-squared explains the relationship between the dependent and independent variables, it does not offer adequate information on whether the predictions and the data are biased. 
  • The results obtained from R-squared may be misleading if one is not careful. The model also needs to be revised in the case of categorical values. 

The Importance of Considering Other Regression Statistics

The R-squared method often fails to offer sufficient information regarding the causation relationship existing between the dependent and independent variables. It also does not reveal whether a regression model is correct. Therefore, it is important to always cross-check with other regression models before arriving at conclusions. 

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An Example To Show How To Calculate R-squared in Regression Analysis

Let us understand how to calculate R-squared with an example. Take the following data set: 

X Y
35 39
43 48
24 29
66 69

First, we will have to find out the XY, X^2, Y^2 and their sums:

X Y XY X^2 Y^2
35 39 1365 1225 1521
43 48 2064 1849 2304
24 29 696 576 841
66 69 4554 4356 4761
Sum: 168 185 8679 8006 9427

Here, the number of observations (n) is 4.

Next, we will have to find out the correlation coefficient (R), which is:

 R= (4*8679)-(185*168)/Sq. rt [(4*8006)-(168^2)]*[(4*9427)-(185^2)]

X Y XY X^2 Y^2
35 39 1365 1225 1521
43 48 2064 1849 2304
24 29 696 576 841
66 69 4554 4356 4761
Sum: 168 185 8679 8006 9427
Correlation Coefficient 0.999437

Finally, squaring this value will give us the R-squared value, 0.998874. 

Troubleshooting Common Issues in R-squared Analysis

Several reasons can cause the value of R-squared linear regression to appear inflated. Being a biased indicator, R-squared may deliver over or under-fitted values. Some other reasons include overfitting and data mining. 

We use a concept known as adjusted R-squared to eliminate the chances of erroneous analysis. Adjusted R-squared helps reduce the R-squared’s value, resulting in an unbiased estimate. This method is also termed R-squared shrinkage. 

Adjusted R-squared interpretation has the potential to deliver more accurate results and also troubleshoot the problems arising out of R-squared analysis. 

Comparing R-squared from Multiple Regression Models

There are different regression models, the most common being linear, logistic, and nonlinear. 

The above discussion on the value of R-squared holds true for a linear regression model. However, the value of R-squared stands invalid for non-linear regression. Similarly, R-squared does not apply when one analyses data with logistic regression.

However, R-squared can be calculated in ridge regression, a method used when multicollinearity of data is present. It also holds valid for the lasso or quantile regression method. 

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Interpreting R-squared in Nonlinear Regression Analysis

The R-squared value is not valid for nonlinear regression and is statistically incorrect. The mathematical formula of R-squared is designed not to deliver accurate results if not used on a linear model. 

R-squared in Time Series Forecasting

Forecasting helps to predict the future based on past occurrences. Determining the accuracy of the time series forecasting models is extremely important because all future decisions rely on the insights the forecast generates. 

The  R2 regression method in time series forecasting compares a model’s stationary part to a simple mean model. However, it is important to note that R-squared regression analysis does not accurately determine a model’s capability to predict the future. It determines whether a model is a good fit for the values considered. 

Conclusion

Interpreting R-squared in regression analysis involves understanding the proportion of variability in the dependent variable that can be explained by the model’s independent variable(s). It provides insights into the goodness of fit and the strength of the relationship between variables. However, it is crucial to consider the limitations of R-squared and supplement its interpretation with other statistical measures when concluding the regression analysis.  

If you look forward to learning its implication in data science, you can pursue an Executive PG Program in Data Science & Machine Learning from the University of Maryland from upGrad. Enrol now to give your career in data science a boost. 

Frequently Asked Questions

Is a high R2 regression value a good model?

Although a high R-squared value is considered good, it is not always true. A high R-squared value can overpredict or underpredict the data along the curve, resulting in a bias.

Is R-squared valid for non-linear models?

Although many statisticians use R-squared for non-linear regression, it is not statistically correct.

What does an adjusted R-square tell you?

Adjusted R-squared is a modified version of R-squared. When the value of the R-squared is inflated unusually, the adjusted R-squared helps adjust the number of terms in the model, thereby shrinking the R-squared value.

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