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Nov 24, 2024

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MATH 1281-01 - AY2024-T2 In the field of statistics and machine learning, particularly in regression analysis, R2 and adjusted R2 are both metrics used to evaluate the goodness of fit of a regression model, but they have some differences. 1. Difference between R2 and adjusted R2: R2 (Coefficient of Determination): R2 measures the proportion of the variance in the dependent variable that is predictable from the independent variables in the model. It ranges between 0 and 1, with 1 indicating a perfect fit where the model explains all the variability of the response data around its mean. However, R2 tends to increase with the addition of more predictors, whether they are relevant or not, which is a limitation. Adjusted R2: Adjusted R2 is a modification of R2 that adjusts for the number of predictors in the model. It penalizes the addition of unnecessary predictors by considering the degrees of freedom. Adjusted R2 increases only if the new term improves the model more than would be expected by chance. It penalizes overly complex models that might overfit the data. 2. Which one is higher and better measure: Generally, R2 will be higher than adjusted R2 because R2 doesn’t account for model complexity. It tends to increase with the addition of more predictors, even if those predictors don’t significantly improve the model. Adjusted R2 will usually be lower than R2 because it adjusts for the number of predictors in the model. It penalizes excessive complexity and will increase only if adding a new variable genuinely improves the model beyond what would be expected by chance. In terms of a better measure, adjusted R2 is often preferred when comparing models with different numbers of predictors. It offers a more conservative estimate of the model's goodness of fit and is more reliable in assessing the true explanatory power of the model. Example1:

Let's say we're analyzing a dataset to predict housing prices based on various factors like square footage, number of bedrooms, bathrooms, and location. If you have a simple model with just square footage as the predictor, R2 might be high because it captures a significant portion of the variance. However, adjusted R2 might be lower when you add more predictors like bedrooms and bathrooms if they don’t significantly contribute to improving the model. In this scenario, the adjusted R2 would be a better measure as it considers the trade-off between model complexity and goodness of fit, giving a more accurate representation of how well the model explains the variation in housing prices while considering the number of predictors used. Example2 : We're working on a project that aims to predict students' exam scores based on various factors such as study hours per week, previous exam scores, and attendance. We've built two regression models: one with just study hours as the predictor (Model A), and another with study hours, previous exam scores, and attendance (Model B). Model A: This simpler model might result in a relatively high R2 because study hours could explain a significant portion of the variance in exam scores. Consequently, R2 might suggest that the model fits the data well. Model B: Adding more predictors in this model might increase the R2 further, as it captures additional variance due to more factors being considered. However, some of these added predictors might not significantly contribute to predicting exam scores. Here’s where adjusted R2 becomes valuable: Model A's R2 vs. adjusted R2: Since Model A has fewer predictors, its adjusted R2 might not differ much from its R2. In Model B, the adjusted R2 might be notably lower than its R2 due to the penalty for the additional predictors that don’t sufficiently improve the model. Interpretation: While R2 might suggest that Model B is better because it has a higher R2 compared to Model A, the adjusted R2 would help in revealing that the improvement gained by including extra predictors may not justify their addition. It provides a more conservative estimate of the model’s goodness of fit, considering the trade-off between model complexity and actual explanatory power. If we were to choose between Model A and Model B, the adjusted R2 would guide us to prefer the simpler Model A unless the additional predictors in Model B significantly contribute to improving the prediction of exam scores. This example emphasizes how adjusted R2 helps select the most appropriate model by penalizing excessive complexity, aiding in the accurate assessment of a model's true explanatory power. Conclusion:

In the realm of regression analysis, selecting an appropriate model requires careful consideration beyond just the R2 value. While R2 measures the goodness of fit, adjusted R2 serves as a crucial metric that accounts for model complexity, penalizing unnecessary predictors. It provides a more conservative estimate of a model's explanatory power, guiding the selection towards simpler yet effective models that maintain strong predictive performance by focusing on the most impactful variables. References: 1. Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to Linear Regression Analysis. John Wiley & Sons., https://ocd.lcwu.edu.pk/cfiles/Statistics/Stat- 503/IntroductiontoLinearRegressionAnalysisbyDouglasC.MontgomeryElizabethA.PeckG .GeoffreyViningz-lib.org.pdf 2. Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2004). Applied Linear Statistical Models. McGraw-Hill/Irwin., https://users.stat.ufl.edu/~winner/sta4211/ALSM_5Ed_Kutner.pdf

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