## Regression Analysis and Residuals: Understanding Standardized Residuals in Python

Regression analysis is a statistical tool used to understand the relationship between two or more variables. It is a commonly used method in data analysis, helping researchers to identify patterns and make predictions.

One essential aspect of regression analysis is the residuals. In this article, we’ll be exploring what residuals are and how to calculate standardized residuals in Python.

## Definition of Residuals

In regression analysis, a residual is the difference between the observed value and the predicted value of the dependent variable. Simply put, it is the vertical distance between the observed value and the regression line.

Residuals can have either a positive or negative value, depending on whether the observed value is greater or less than the predicted value. Residuals are an important tool in regression analysis because they help to assess the accuracy of the model.

A good regression model will have residuals that are small and random. If the residuals are large or have a pattern, it could indicate that there is a problem with the model, such as an incorrect specification of the functional form.

## Standardized Residuals

Standardized residuals are a type of residual that has been adjusted for the influence of the predictor variables. They are useful in identifying outliers, which are observations that are far away from the regression line.

Outliers can have a significant impact on the analysis, so it’s essential to identify them. The formula for calculating standardized residuals is as follows:

standardized residual = residual / standard error of the estimate

The standard error of the estimate is a measure of the variability in the errors.

### It is calculated as follows:

standard error of the estimate = square root of (sum of squared residuals / degrees of freedom)

Degrees of freedom refer to the number of observations minus the number of parameters estimated in the model.

## Example of Calculating Standardized Residuals in Python

To demonstrate how to calculate standardized residuals in Python, we’ll use a simple dataset consisting of the height and weight of ten individuals.

We’ll fit a linear regression model to predict weight based on height and then calculate the standardized residuals.

### Creating a Dataset

First, we’ll create a dataset using the Pandas library:

```
import pandas as pd
data = {'Height': [63, 65, 68, 70, 71, 72, 73, 75, 76, 78],
'Weight': [127, 140, 172, 186, 190, 203, 208, 230, 253, 280]}
df = pd.DataFrame(data)
```

The dataset contains ten observations, with height and weight as the variables.

### Fitting a Regression Model

Next, we’ll fit a linear regression model to predict weight based on height:

```
from sklearn.linear_model import LinearRegression
X = df[['Height']]
y = df['Weight']
model = LinearRegression().fit(X, y)
```

### Calculating Standardized Residuals

To calculate the standardized residuals, we’ll use the “resid_studentized_internal” function:

```
from statsmodels.stats.outliers_influence import OLSInfluence
resid = OLSInfluence(model).resid_studentized_internal
```

The function calculates the standardized residuals for each observation in the dataset.

### Visualizing Standardized Residuals

Finally, we’ll visualize the standardized residuals using a scatterplot:

```
import matplotlib.pyplot as plt
plt.scatter(X, resid)
plt.axhline(y=0, color='r', linestyle='-')
plt.show()
```

The scatterplot shows the standardized residuals for each observation in the dataset, with the predictor variable (height) on the x-axis and the residuals on the y-axis.

The red line at y=0 represents the regression line.

## Conclusion

In conclusion, the residuals are a useful tool in regression analysis as they help to evaluate the accuracy of the model. Standardized residuals, in particular, are helpful in identifying outliers and assessing the impact they may have on the analysis.

With the use of Python libraries, we can easily calculate and visualize standardized residuals to better understand our datasets and improve our regression models. To summarize, the article explained what residuals are and their importance in regression analysis.

Residuals help to assess the accuracy of regression models and identify the presence of outliers. Standardized residuals, specifically, are adjusted for predictor variables and are useful in identifying and assessing the impact of outliers.

With the use of Python libraries, we can easily calculate and visualize standardized residuals. An understanding of residuals is essential in data analysis and can lead to improved predictive modeling.

Overall, residuals provide insight into the accuracy and reliability of regression models, and by calculating standardized residuals, we can identify outliers and improve our understanding of the data.