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# Regression toward the Mean

#### Machine Learning (ML)

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## Introduction

The statistical concept of Regression toward the mean offers a unique explanation of how extreme measurements or values tend to be followed by measurements that are closer to the mean or average value. In simpler terms, if a variable is measured twice and the first measurement is an extreme value, the second measurement is expected to be closer to the average value, even if no changes were made to the underlying process being measured.

First observed by Sir Francis Galton in the late 19th century, this phenomenon has since been widely applied in various fields, including psychology, sports, economics, and machine learning.

The goal of this article at OpenGenus is to provide a comprehensive understanding of Regression toward the mean, its applications, potential confounding factors, and how to account for it in experimental designs and machine learning models. Through examples, we will explore how this phenomenon can occur in different contexts and how it can be utilized to enhance decision-making and analysis in various fields.

By the end of this article at OpenGenus, you will have gained a comprehensive understanding of this topic, its significance in data analysis and decision-making, and the ability to apply this knowledge to your own work while considering this phenomenon.

## Explanation

This phenomenon is commonly observed in statistics when measuring something that varies over time. In simple terms, it refers to a tendency for extremes (either very high or very low) to move towards the average over time.
For instance, let us consider a group of students who take a test. Some students may score very high, while others may score very low. However, when the same test is given again, the students who had scored very high initially would typically score lower on the second test, and the students who had scored very low initially would score higher on the second test. This is an example of Regression toward the mean.

Another instance could be a Soccer player who performs exceptionally well in one game, but his performance declines in subsequent games. This is because his performance in the first game was an extreme measurement that shifted towards the mean over time.

Regression toward the mean is also prevalent in other fields such as medicine, where it is crucial to comprehend this phenomenon to evaluate treatment effectiveness accurately, and in finance, where it can be beneficial for investment decisions.

## Applications

The phenomenon of Regression toward the mean has numerous practical applications in various fields. Some of these are:

1. In Psychology, Regression toward the mean can clarify why treatments that are implemented after extreme behaviour or symptoms may seem effective, even though they may not be so. For example, if a patient with a severe form of depression experiences a reduction in symptoms after receiving treatment, it is possible that the reduction was due to Regression toward the mean, rather than the effectiveness of the treatment itself.
2. In sports, Regression toward the mean can explain why athletes who perform exceedingly well in one season may not perform as well in the following season. For example, a baseball player who hits a high number of home runs in one season may not hit as many home runs in the next season, as their first performance might have been influenced by chance factors.
3. In economics, Regression toward the mean can explain why stock prices may fluctuate significantly in the short term but tend to revert to the mean over the long term. For instance, a company that experiences a large increase in stock price may not continue to experience such substantial gains in the future, as the initial increase may have been influenced by chance factors.
4. In experimental design, researchers may accidentally choose participants or variables based on extreme measurements or values. This phenomenon can be a confounding factor in experimental designs. For example, if a researcher selects participants for a study based on their high or low scores on a test, the results may be biased, as the extreme scores may not be representative of the underlying distribution.
5. In Machine Learning, Regression toward the mean is important, as extreme values in the data may not be representative of the overall pattern, and relying solely on them may lead to less accurate predictions. To account for this, machine learning practitioners use techniques such as cross-validation and regularization, which help to ensure that their models are trained on data that is more representative of the underlying distribution and less biased towards extreme values.

## Role in Experimental Design

Regression toward the mean can exhibit a puzzling factor in experimental design. To mitigate this effect, researchers can employ several strategies to guarantee that their findings are not biased by extreme measurements or values.

Strategy 1
In this, one opts for random sampling in the selection of participants or variables for a study rather than based on extreme measurements or values. This approach will ensure that the sample is an accurate representation of the underlying distribution and that the possibility of extreme values being chosen is reduced.

Strategy 2
This technique is to take multiple measurements over time instead of relying on a single measurement. This method will consider the fact that extreme values tend to regress toward the mean over time, thereby ensuring that chance factors do not influence the outcomes.

Strategy 3
Researchers can use statistical methods such as analysis of variance (ANOVA) and linear regression to accommodate the effects of Regression toward the mean. These techniques will help to identify the extent to which extreme values are affecting the results and make the necessary adjustments during analysis.

So, researchers need to consider the specific variables and context of their study and acknowledge the potential influence of Regression toward the mean on their results. By adhering to these measures, researchers can ensure the dependability and precision of their findings while decreasing the impact of confounding factors like it.

## Role in Machine Learning

Machine learning relies on Regression toward the mean, which can impact the accuracy of the model's predictions. To mitigate this issue, machine learning practitioners can use several techniques to ensure that their models are trained on representative data.

One approach is to use cross-validation, which evaluates the model's performance on multiple data subsets to avoid bias towards extreme values. By using this approach, the model's ability to generalize to new data that is more representative of the underlying distribution can be improved and the bias towards extreme values can be avoided.

The second approach is Regularization techniques such as L1 and L2 can also penalize the model for relying too heavily on any one feature or predictor variable. So, it is possible to prevent extreme values from having a disproportionate impact on the model's predictions and increase its ability to generalize to new data.

Next is the Outlier detection technique, which is used to identify and treat extreme values separately in the analysis, ensuring that the model's predictions are accurate and preventing the model from being excessively influenced by outliers and enhancing its ability to make precise predictions on new data.

So, practitioners should carefully consider the features and predictors being used in the model and the potential for Regression toward the mean to influence the results. By following these above steps, machine learning practitioners can guarantee accurate and reliable predictions

## Limitations

The concept of Regression toward the mean has some limitations and areas of caution that we should be aware of when applying it in practice:

1. Causality: Just because extreme values regress toward the mean doesn't mean there is a causal relationship. It could just be due to chance, so we should be cautious in interpreting results and look for other evidence of causality before making conclusions.
2. Sample size: Regression toward the mean is more likely to occur in small samples than in large samples, so we should use a large enough sample size to reduce the impact of chance factors.
3. Measurement error: Measurement error can also contribute to Regression toward the mean, so we should use reliable and accurate measurement tools and account for measurement error in the analysis.
4. Selection bias: Regression toward the mean can be confounded by selection bias, so we should use random sampling and ensure that the sample is representative of the underlying population.
5. Contextual factors: Regression toward the mean can be influenced by contextual factors, such as the timing of the measurement or the presence of other variables that are correlated with the outcome, so we should carefully consider the context in which the measurements are being taken and account for these factors in the analysis.
6. Statistical assumptions: Regression toward the mean relies on certain statistical assumptions, such as the assumption that the underlying distribution is normal, so we should check these assumptions carefully and make sure they are met before applying them in practice.

## Conclusion

Regression toward the mean is an important consideration when conducting experiments and analyzing data, as failing to account for it can lead to incorrect results and poor decision-making. Understanding this topic is important because it helps us to better interpret data and make more accurate predictions about future outcomes. By recognizing that extreme measurements are likely to regress towards the mean over time, we can avoid making erroneous conclusions and make better decisions.

#### Anmol Taneja

Student at Panipat Institute of Engineering and Technology