R-Squared (R²) Calculator for JMP Users
Calculate the coefficient of determination (R²) from your model’s sum of squares values.
What is R-squared (R²) in JMP?
R-squared (R²), also known as the coefficient of determination, is a statistical measure that represents the proportion of the variance for a dependent variable that’s explained by an independent variable or variables in a regression model. When you perform a regression analysis in JMP using platforms like “Fit Y by X” or “Fit Model”, the output includes a “Summary of Fit” or “Analysis of Variance” table. These tables provide the necessary components for calculating R² using JMP info.
In essence, R-squared provides a measure of how well your model’s predictions approximate the real data points. A value of 1 indicates that the regression predictions perfectly fit the data, while a value of 0 indicates that the model does not explain any of the variability of the response data around its mean. This calculator simplifies the process of manually calculating R² if you have the summary statistics from JMP.
R² Formula and Explanation
The most common formula for R-squared uses the Total Sum of Squares (SST) and the Sum of Squared Errors (SSE). Both of these values are readily available in JMP’s output tables.
The formula is:
R² = 1 – (SSE / SST)
Understanding the components is key to calculating R² using JMP info. See this link for more about the JMP linear regression.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| SST (Total Sum of Squares) | Measures the total variance in the dependent variable. It’s the sum of the squared differences between each observed value and the mean of all observed values. | Unitless (Squared units of the response variable) | Greater than or equal to SSE. |
| SSE (Sum of Squared Errors) | Also called Residual Sum of Squares (RSS). It measures the variance not explained by the model. It’s the sum of the squared differences between the observed value and the predicted value. | Unitless (Squared units of the response variable) | Greater than or equal to 0. |
| R² (R-Squared) | The proportion of variance in the dependent variable that is predictable from the independent variable(s). | Unitless Ratio | Typically 0 to 1, but can be negative. |
Practical Examples
Example 1: Strong Model Fit
Suppose you are modeling house prices and your JMP analysis provides the following from the “Analysis of Variance” table:
- Inputs:
- Total Sum of Squares (SST): 25,000,000
- Sum of Squared Errors (SSE): 3,750,000
- Calculation:
- R² = 1 – (3,750,000 / 25,000,000) = 1 – 0.15 = 0.85
- Result:
- An R-squared of 0.85, or 85%, indicates that your model explains 85% of the variability in house prices. This is generally considered a strong fit.
Example 2: Weaker Model Fit
Imagine you are analyzing student test scores based on hours studied. The ANOVA table JMP output shows:
- Inputs:
- Total Sum of Squares (SST): 1,200
- Sum of Squared Errors (SSE): 810
- Calculation:
- R² = 1 – (810 / 1200) = 1 – 0.675 = 0.325
- Result:
- An R-squared of 0.325, or 32.5%, means that the number of hours studied explains 32.5% of the variability in test scores. While statistically significant, it suggests other factors also play a large role. For more information, read about what is statistical significance.
How to Use This R-Squared Calculator
Follow these steps to easily find your R-squared value:
- Run Your Model in JMP: Perform your regression analysis (e.g., using “Fit Model”).
- Locate the Sum of Squares: In the JMP output report, find the “Analysis of Variance” table.
- Find SST: Identify the “C. Total” (Corrected Total) value for the Sum of Squares. This is your SST.
- Find SSE: Identify the “Error” row’s value for the Sum of Squares. This is your SSE.
- Enter Values: Input the SST and SSE values into the corresponding fields in the calculator above.
- Interpret Results: The calculator will instantly display the R-squared value, the percentage of variance explained, and a visual chart. The closer the R-squared is to 1, the more variance your model explains.
Key Factors That Affect R-squared
Several factors can influence your R-squared value. Understanding them is crucial when you interpret R-squared value.
- Number of Predictors: Adding more variables to a model, even if they are not truly significant, will almost always increase the R-squared value. This can be misleading, which is why adjusted R-squared vs R-squared is an important distinction.
- Model Overfitting: A model that is too complex might fit the sample data perfectly (high R²) but fail to predict new data.
- Linearity: R-squared measures the strength of a *linear* relationship. If the true relationship is non-linear, a low R² doesn’t necessarily mean the variables are unrelated.
- Outliers: Extreme values in your dataset can have a significant impact on the regression line and, consequently, the R-squared value.
- Sample Size: With a very small sample, you might get a high R-squared by chance. A larger sample size provides a more reliable estimate.
- Variance of Predictors: A wider range of values for your independent variables can often lead to a higher R-squared.
Frequently Asked Questions (FAQ)
What is a good R-squared value?
There’s no single answer. In some fields like social sciences, an R² of 0.3 (30%) might be considered good, while in physics or chemistry, a good fit might require an R² above 0.95. The context of your data and research question is critical.
Can R-squared be negative?
Yes. While uncommon in standard linear regression, R-squared can be negative if the chosen model fits the data worse than a simple horizontal line (the mean of the data). This often indicates a fundamentally flawed model.
Where do I find SST and SSE in JMP?
In the “Fit Model” or “Fit Y by X” platforms, look for the “Analysis of Variance” table. SSE is the “Sum of Squares” for the “Error” source. SST is the “Sum of Squares” for the “C. Total” (Corrected Total) source.
Does a high R-squared mean my model is good?
Not necessarily. A high R-squared indicates that the model explains a lot of the variance in your sample data, but it doesn’t prove that the model is correct, unbiased, or will predict new data well. Always check residual plots and other diagnostic tools. Consider using a P-value calculator to check for significance.
What’s the difference between R-squared and Adjusted R-squared?
Adjusted R-squared modifies the R-squared value to account for the number of predictors in the model. It increases only if the new variable improves the model more than would be expected by chance, making it a better metric for comparing models with different numbers of variables.
Why did my R-squared value decrease when I removed a variable?
The standard R-squared will always decrease or stay the same when a variable is removed. Only the adjusted R-squared can increase if the removed variable was not contributing meaningfully to the model.
Is this calculator the same as the one in the “Summary of Fit” in JMP?
Yes, this calculator uses the same underlying formula, R² = 1 – (SSE/SST), that JMP uses to calculate the R-squared value displayed in the “Summary of Fit” report. This tool is for manually verifying the calculation or for situations where you only have the summary numbers.
Can I use this for non-linear models?
While the formula is mathematically applicable, the interpretation of R-squared in non-linear models can be more complex and less intuitive. It’s generally most reliable and straightforward for linear regression models.