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What is R Squared ?
How to calculate R Squared ?
How to Calculate Adjusted R Squared ?
Which Method is Better ?
What is R Squared ?

R-squared is a goodness-of-fit measure for linear regression models. This statistic indicates the percentage of the variance in the dependent variable that the independent variables explain collectively. R-squared measures the strength of the relationship between your model and the dependent variable on a convenient 0 – 100% scale.
Image of a large R-squared.After fitting a linear regression model, you need to determine how well the model fits the data. Does it do a good job of explaining changes in the dependent variable? There are a several key goodness-of-fit statistics for regression analysis. In this post, we’ll examine R-squared (R2 ), highlight some of its limitations, and discover some surprises. For instance, small R-squared values are not always a problem, and high R-squared values are not necessarily good! Linear regression identifies the equation that produces the smallest difference between all of the observed values and their fitted values. To be precise, linear regression finds the smallest sum of squared residuals that is possible for the dataset.
Statisticians say that a regression model fits the data well if the differences between the observations and the predicted values are small and unbiased. Unbiased in this context means that the fitted values are not systematically too high or too low anywhere in the observation space.
However, before assessing numeric measures of goodness-of-fit, like R-squared, you should evaluate the residual plots. Residual plots can expose a biased model far more effectively than the numeric output by displaying problematic patterns in the residuals. If your model is biased, you cannot trust the results.

R-squared and the Goodness-of-Fit

R-squared evaluates the scatter of the data points around the fitted regression line. It is also called the coefficient of determination, or the coefficient of multiple determination for multiple regression. For the same data set, higher R-squared values represent smaller differences between the observed data and the fitted values.
R-squared is the percentage of the dependent variable variation that a linear model explains.

Formulae:

1- (Sum of Errors / Total Sum of errors)

R-squared is always between 0 and 100%:
0% represents a model that does not explain any of the variation in the response variable around its mean. The mean of the dependent variable predicts the dependent variable as well as the regression model. 100% represents a model that explains all of the variation in the response variable around its mean. Usually, the larger the R2, the better the regression model fits your observations. However, this guideline has important caveats that I’ll discuss in both this post and the next post. What are the limitations of R Squared ?

R2 increases with every predictor added to a model. As R2 always increases and never decreases, it can appear to be a better fit with the more terms you add to the model. This can be completely misleading.
Similarly, if your model has too many terms and too many high-order polynomials you can run into the problem of over-fitting the data. When you over-fit data, a misleadingly high R2 value can lead to misleading projections.

What is Adjusted R Squared ?

R2 shows how well terms (data points) fit a curve or line. Adjusted R2 also indicates how well terms fit a curve or line, but adjusts for the number of terms in a model. If you add more and more useless variables to a model, adjusted r-squared will decrease. If you add more useful variables, adjusted r-squared will increase. Adjusted R2 will always be less than or equal to R2.

Formulae:

Adjusted R Squared = 1 – [((1 – R2) * (n – 1)) / (n – k – 1)]

Where:
n – Number of points in your data set.
k – Number of independent variables in the model, excluding the constant

Fortunately, if you have a low R-squared value but the independent variables are statistically significant, you can still draw important conclusions about the relationships between the variables. Statistically significant coefficients continue to represent the mean change in the dependent variable given a one-unit shift in the independent variable. Clearly, being able to draw conclusions like this is vital.

Which Method is Better ?

Both R2 and the adjusted R2 give you an idea of how many data points fall within the line of the regression equation. However, there is one main difference between R2 and the adjusted R2: R2 assumes that every single variable explains the variation in the dependent variable. The adjusted R2 tells you the percentage of variation explained by only the independent variables that actually affect the dependent variable.
The adjusted R2 will penalize you for adding independent variables (K in the equation) that do not fit the model. Why? In regression analysis, it can be tempting to add more variables to the data as you think of them. Some of those variables will be significant, but you can’t be sure that significance is just by chance. The adjusted R2 will compensate for this by that penalizing you for those extra variables.
While values are usually positive, they can be negative as well. This could happen if your R2 is zero; After the adjustment, the value can dip below zero. This usually indicates that your model is a poor fit for your data. Other problems with your model can also cause sub-zero values, such as not putting a constant term in your model.

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