Adjusted R Squared Formula (Table of Contents)
Adjusted R Squared Formula
Before jumping to the adjusted r squared formula, we need to understand what is R2. In statistics, R2 also known as the coefficient of determination is a tool to which determines and assesses the variation in the dependent variable which is explained by an independent variable in a statistical model. So if R2 is said 0.6, it means that 60% of the variation in the dependent variable is explained by the independent variable. But the problem with R2 is that its value increase with the addition of more variables irrespective of the significance of that variable. So to overcome that, the concept of adjusted r square has been introduced. The idea behind R2 and adjusted R Squared is the same but the difference is that adjusted r squared adjusts the r square value for the number of terms in the model.
Formula For Adjusted R Squared:
Before we calculate adjusted r squared, we need r square first. There are different ways to calculate r square:
- Using Correlation Coefficient :
Correlation Coefficient = Σ [(X – Xm) * (Y – Ym)] / √ [Σ (X – Xm)2 * Σ (Y – Ym)2]
Where:
- X – Data points in data set X
- Y – Data points in data set Y
- Xm– Mean of data set X
- Ym– Mean of data set Y
So
R2 = (Correlation Coefficient)2
Where:
- n – Number of points in your data set.
- k – Number of independent variables in the model, excluding the constant
- Using Regression outputs
R2 = Explained Variation / Total Variation
R2 = MSS / TSS
R2= (TSS – RSS) / TSS
Where:
- TSS – Total Sum of Squares = Σ (Yi – Ym)2
- MSS – Model Sum of Squares = Σ (Y^ – Ym)2
- RSS – Residual Sum of Squares =Σ (Yi – Y^)2
Y^ is the predicted value of the model, Yi is the ith value and Ym is the mean value
Examples of Adjusted R Squared Formula (With Excel Template)
Let’s take an example to understand the calculation of the Adjusted R Squared in a better manner.
Adjusted R Squared Formula – Example #1
Let’s say we have two data sets X & Y and each contains 20 random data points. Calculate the Adjusted R Squared for the data set X & Y.
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Mean is calculated as:
- Mean of Data Set X = 49.2
- Mean of Data Set Y = 53.8
Now, we need to calculate the difference between the data points and the mean value.
Similarly, calculate for all the data set of X.
Similarly, calculate it for data set Y also.
Calculate the square of the difference for both the data sets X and Y.
Multiply the difference in X with Y.
Correlation Coefficient is calculated using the formula given below
Correlation Coefficient = Σ [(X – Xm) * (Y – Ym)] / √ [Σ (X – Xm)2 * Σ (Y – Ym)2]
Correlation Coefficient = 0.325784
R2 is calculated using the formula given below
R2 = (Correlation Coefficient)2
R2 = 10.61%
Adjusted R Squared is calculated using the formula given below
Adjusted R Squared = 1 – [((1 – R2) * (n – 1)) / (n – k – 1)]
- Adjusted R Squared = 1 – ((1 – 10.61%) * (20 – 1)/(20 – 1 – 1))
- Adjusted R Squared = 5.65%
Adjusted R Squared Formula – Example #2
Let’s use another method to calculate the r square and then adjusted r squared. Let’s say you have actual and predicted dependent variable values with you ( Y and Y^):
Mean is calculated as
Now, we need to calculate the difference between actual and predicted dependent variable values.
Calculate the difference between the data points and the mean value.
Calculate the square of the differences.
R2 is calculated using the formula given below
R2 = (TSS – RSS) / TSS
- TSS = Σ (Y – Ym)2
- RSS = Σ (Y – Y^)2
R2 = 64.11%
Now let’s say we have 3 independent variables: i.e. k=3.
Adjusted R Squared is calculated using the formula given below
Adjusted R Squared = 1 – [((1 – R2) * (n – 1)) / (n – k – 1)]
- Adjusted R Squared = 1 – (((1 – 64.11%) * (10-1)) / (10 – 3 – 1))
- Adjusted R Squared = 46.16%
Explanation
R2 or Coefficient of determination, as explained above is the square of the correlation between 2 data sets. If R2 is 0, it means that there is no correlation and independent variable cannot predict the value of the dependent variable. Similarly, if its value is 1, it means that independent variable will always be successful in predicting the dependent variable. But there are some limitations also. As the number of independent variable increase in the statistical model, the R2 also increases whether that new variables make sense or not. That is the reason that adjusted r squared is calculated since it adjusts the R2 value for that increase in a number of variables. Adjusted r squared value decrease if that independent variable is not significant and increases if that has significance.
Relevance and Uses of Adjusted R Squared Formula
Adjusted r squared is more useful when we have more than 1 independent variables since it adjusts the r square and takes only into consideration the relevant independent variable, which actually explains the variation in the dependent variable. Its value is always less than the R2 value. In general, there are many practical applications this tool like a comparison of portfolio performance with the market and future prediction, risk modeling in Hedge Funds, etc.
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