## How to Interpret R Squared and Goodness of Fit in Regression Analysis

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# How to Interpret R Squared and Goodness of Fit in Regression Analysis

10K Regression Analysis is a set of statistical processes that are at the core of data science. In the field of numerical simulation, it represents the most well-understood models and helps in interpreting machine learning algorithms. Their real-life applications can be seen in a wide range of domains, ranging from advertising and medical research to agricultural science and even different sports.

In linear regression models, R-squared is a goodness-fit-measure. It takes into account the strength of the relationship between the model and the dependent variable. Its convenience is measured on a scale of 0 – 100%.

Once you have a fit linear regression model, there are few considerations that you need to address:

• How well does the model fit the data?
• How well does it explain the changes in the dependent variable?

Let us first understand the fundamentals of Regression Analysis and its necessity.

## What is Regression Analysis?

Regression Analysis is a well-known statistical learning technique that allows you to examine the relationship between the independent variables (or explanatory variables) and the dependent variables (or response variables). It requires you to formulate a mathematical model that can be used to determine an estimated value which is nearly close to the actual value.

The two terms essential to understanding Regression Analysis:

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

Consider a situation where you are given data about a group of students on certain factors: number of hours of study per day, attendance, and scores in a particular exam. The Regression technique allows you to identify the most essential factors, the factors that can be ignored and the dependence of one factor on others.

There are mainly two objectives of a Regression Analysis technique:

• Explanatory analysis - This analysis understands and identifies the influence of the explanatory variable on the response variable concerning a certain model.
• 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 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).

## 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.

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 assess Goodness-of-fit in a regression model?

According to statisticians, if the differences between the observations and the predicted values tend to be small and unbiased, we can say that the model fits the data well. The meaning of unbiasedness in this context is that the fitted values do not reach the extremes, i.e. too high or too low during observations.

As we have seen earlier, a linear regression model gives you the outlook of the equation which represents the minimal difference between the observed values and the predicted values. In simpler terms, we can say that linear regression identifies the smallest sum of squared residuals probable for the dataset.

Determining the residual plots represents a crucial part of a regression model and it should be performed before evaluating the numerical measures of goodness-of-fit, like R-squared. They help to recognize a biased model by identifying problematic patterns in the residual plots.

However, if you have a biased model, you cannot depend on the results. If the residual plots look good, you can assess the value of R-squared and other numerical outputs.

## What is R-squared?

In data science, R-squared (R2) is referred to as the coefficient of determination or the coefficient of multiple determination in case of multiple regression.

In the linear regression model, R-squared 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 R 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.

### 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.

### Interpretation of R-squared

The simplest interpretation of R-squared 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%. This would mean that the model explains 70% of the fitted data in the regression model. Usually, when the R 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 Rvalue 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 us 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

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.

## Low R-squared and High R-squared values

Regression models with low R2 do not always pose a problem. There are some areas where you are bound to have low Rvalues. One such case is when you study human behavior. They tend to have R values less than 50%. The reason behind this is that predicting people is a more difficult task than predicting a physical process.

You can draw essential conclusions about your model having a low Rvalue when the independent variables of the model have some statistical significance. They represent the mean change in the dependent variable when the independent variable shifts by one unit.

However, if you are working on a model to generate precise predictions, low R-squared values can cause problems.

Now, let us look at the other side of the coin. A regression model with high R2  value can lead to – as the statisticians call it – specification bias. This type of situation arises when the linear model is underspecified due to missing important independent variables, polynomial terms, and interaction terms.

To overcome this situation, you can produce random residuals by adding the appropriate terms or by fitting a non-linear model.

Model overfitting and data mining techniques can also inflate the value of R2. The model they generate might provide an excellent fit to the data but actually the results tend to be completely deceptive.

## Conclusion

Let us summarize what we have covered in this article so far:

• Regression Analysis and its importance
• Residuals and Goodness-of-fit
• R-squared: Representation, Interpretation, Calculation, Limitations
• Low and High Rvalues

Although R-squared is a very intuitive measure to determine how well a regression model fits a dataset, it does not narrate the complete story. If you want to get the full picture, you need to have an in-depth knowledge of R2  along with other statistical measures and residual plots.

For gaining more information on the limitations of the R-squared, you can learn about Adjusted R-squared and Predicted R-squared which provide different insights to assess a model’s goodness-of-fit. You can also take a look at a different type of goodness-of-fit measure, i.e. Standard Error of the Regression.

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## Data Science: Correlation vs Regression in Statistics

In this article, we will understand the key differences between correlation and regression, and their significance. Correlation and regression are two different types of analyses that are performed on multi-variate distributions of data. They are mathematical concepts that help in understanding the extent of the relation between two variables: and the nature of the relationship between the two variables respectively. Correlation Correlation, as the name suggests is a word formed by combining ‘co’ and ‘relation’. It refers to the analysis of the relationship that is established between two variables in a given dataset. It helps in understanding (or measuring) the linear relationship between two variables.  Two variables are said to be correlated when a change in the value of one variable results in a corresponding change in the value of the other variable. This could be a direct or an indirect change in the value of variables. This indicates a relationship between both the variables.  Correlation is a statistical measure that deals with the strength of the relation between the two variables in question.  Correlation can be a positive or negative value. Positive Correlation Two variables are considered to be positively correlated when the value of one variable increases or decreases following an increase or decrease in the value of the other variable respectively.  Let us understand this better with the help of an example: Suppose you start saving your money in a bank, and they offer some amount of interest on the amount you save in the bank. The more the amount you store in the bank, the more interest you get on your money. This way, the money stored in a bank and the interest obtained on it are positively correlated. Let us take another example: While investing in stocks, it is usually said that higher the risk while investing in a stock, higher is the rate of returns on such stocks.  This shows a direct inverse relationship between the two variables since both of them increase/decrease when the other variable increases/decreases respectively. Negative Correlation Two variables are considered to be negatively correlated when the value of one variable increases following a decrease in the value of the other variable. Let us understand this with an example: Suppose a person is looking to lose weight. The one basic idea behind weight loss is reducing the number of calorie intake. When fewer calories are consumed and a significant number of calories are burnt, the rate of weight loss is quicker. This means when the amount of junk food eaten is decreased, weight loss increases. Let us take another example: Suppose a popular non-essential product that is being sold faces an increase in the price. When this happens, the number of people who purchase it will reduce and the demand would also reduce. This means, when the popularity and price of the product increases, the demand for the product reduces. An inverse proportion relationship is observed between the two variables since one value increases and the other value decreases or one value decreases and the other value increases.  Zero Correlation This indicates that there is no relationship between two variables. It is also known as a zero correlation. This is when a change in one variable doesn't affect the other variable in any way. Let us understand this with the help of an example: When the increase in height of our friend/neighbour doesn’t affect our height, since our height is independent of our friend’s height.  Correlation is used when there is a requirement to see if the two variables that are being worked upon are related to each other, and if they are, what the extent of this relationship is, and whether the values are positively or negatively correlated.  Pearson’s correlation coefficient is a popular measure to understand the correlation between two values.  Regression Regression is the type of analysis that helps in the prediction of a dependant value when the value of the independent variable is given. For example, given a dataset that contains two variables (or columns, if visualized as a table), a few rows of values for both the variables would be given. One or more of one of the variables (or column) would be missing, that needs to be found out. One of the variables would depend on the other, thereby forming an equation that relevantly represents the relationship between the two variables. Regression helps in predicting the missing value. Note: The idea behind any regression technique is to ensure that the difference between the predicted and the actual value is minimal, thereby reducing the error that occurs during the prediction of the dependent variable with the help of the independent variable. There are different types of regression and some of them have been listed below: Linear Regression This is one of the basic kinds of regression, which usually involves two variables, where one variable is known as the ‘dependent’ variable and the other one is known as an ‘independent’ variable. Given a dataset, a pattern has to be formed (linear equation) with the help of these two variables and this equation has to be used to fit the given data to a straight line. This straight-line needs to be used to predict the value for a given variable. The predicted values are usually continuous. Logistic Regression There are different types of logistic regression:  Binary logistic regression is a regression technique wherein there are only two types or categories of input that are possible, i.e 0 or 1, yes or no, true or false and so on. Multinomial logistic regression helps predict output wherein the outcome would belong to one of the more than two classes or categories. In other words, this algorithm is used to predict a nominal dependent variable. Ordinal logistic regression deals with dependant variables that need to be ranked while predicting it with the help of independent variables.  Ridge Regression It is also known as L2 regularization. It is a regression technique that helps in finding the best coefficients for a linear regression model with the help of an estimator that is known as ridge estimator. It is used in contrast to the popular ordinary least square method since the former has low variance and hence it calculates better coefficients. It doesn’t eliminate coefficients thereby not producing sparse, simple models.  Lasso Regression LASSO is an acronym that stands for ‘Least Absolute Shrinkage and Selection Operator’. It is a type of linear regression that uses the concept of ‘shrinkage’. Shrinkage is a process with the help of which values in a data set are reduced/shrunk to a certain base point (this could be mean, median, etc). It helps in creating simple, easy to understand, sparse models, i.e the models that have fewer parameters to deal with, thereby being simple.  Lasso regression is highly suited for models that have high collinearity levels, i.e a model where certain processes (such as model selection or parameter selection or variable selection) is automated.  It is used to perform L1 and L2 regularization. L1 regularization is a technique that adds a penalty to the given values of coefficients in the equation. This also results in simple, easy to use, sparse models that would contain lesser coefficients. Some of these coefficients can also be estimated off to 0 and hence eliminated from the model altogether. This way, the model becomes simple.  It is said that Lasso regression is easier to work with and understand in comparison to ridge regression.  There are significant differences between both these statistical concepts.  Difference between Correlation and Regression Let us summarize the difference between correlation and regression with the help of a table: CorrelationRegressionThere are two variables, and their relationship is understood and measured.Two variables are represented as 'dependent' and 'independent' variables, and the dependent variable is predicted.The relationship between the two variables is analysed.This concept tells about how one variable affects the other and tries to predict the dependant variable.The relationship between two variables (say ‘x’ and ‘y’) is the same if it is expressed as ‘x is related to y’ or ‘y is related to x’.There is a significant difference when we say ‘x depends on y’ and ‘y depends on x’. This is because the independent and dependent variables change.Correlation between two variables can be expressed through a single point on a graph, visually.A line or a curve is fitted to the given data, and the line or the curve is extrapolated to predict the data and make sure the line or the curve fits the data on the graph.It is a numerical value that tells about the strength of the relation between two variables.It predicts one variable based on the independent variables. (this predicted value can be continuous or discrete, depending on the type of regression) by fitting a straight line to the data.Conclusion In this article, we understood the significant differences between two statistical techniques, namely- correlation and regression with the help of examples. Correlation establishes a relationship between two variables whereas regression deals with the prediction of values and curve fitting.
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