What are residuals?

Residuals are the differences between the observed values of a dependent variable and the values predicted by a statistical model. They are the errors or discrepancies between the actual values and the predicted values of the dependent variable.

Examples of Residuals

Residuals can be used to assess the goodness of fit of a statistical model. If the residuals are small and random, then the model is considered to be a good fit for the data. However, if the residuals are large and systematic, then the model may not be a good fit, and additional analysis or modifications to the model may be necessary.

Residuals can also be used to identify outliers or influential data points that may be affecting the results of a statistical analysis. By examining the residuals, analysts can determine whether certain data points are having a disproportionate impact on the model and may need to be removed or given special consideration in the analysis

The chart below shows the data in a residual plot.

Why are residuals important?

Residuals are important for several reasons in statistical analysis:

1. Assessing Model Fit: Residuals are used to assess how well a statistical model fits the data. If the residuals are small and random, it indicates that the model is a good fit for the data. However, if the residuals are large and systematic, it indicates that the model may not be an accurate representation of the data.

2. Identifying Outliers: Residuals can help identify outliers, which are data points that are significantly different from the rest of the data. Outliers can have a large impact on the results of a statistical analysis, and identifying them can help improve the accuracy of the analysis.

3. Checking Assumptions: Residuals can be used to check whether the assumptions of a statistical model are met. For example, residuals should be normally distributed if the model assumes that the errors are normally distributed.

4. Comparing Models: Residuals can be used to compare different statistical models. The model with smaller residuals is generally considered to be a better fit for the data.

5. Prediction: Residuals can be used to make predictions about future data points. By using the residuals from a statistical model, analysts can estimate how much error is likely in their predictions.

Overall, residuals are an important tool for assessing the accuracy of statistical models and identifying areas for improvement. They can help analysts make more informed decisions and improve the quality of their analyses.

Formula for residuals‍

The formula for residuals is rather straightforward:

Residual = observed y – predicted y

It is important to note that the predicted value comes from our regression line. The observed value comes from our data set.

The different type of residuals

There are several types of residuals that are commonly used in statistical analysis. Here are some of the most important types:

1. Raw Residuals: Raw residuals are simply the differences between the observed values of the dependent variable and the predicted values from a statistical model.

2. Standardized Residuals: Standardized residuals are the raw residuals divided by an estimate of the standard deviation of the errors. These are useful for identifying outliers and checking assumptions of the model.

3. Studentized Residuals: Studentized residuals are similar to standardized residuals, but they are divided by an estimate of the standard error of the residuals. These are used for detecting outliers and influential data points.

4. Deleted Residuals: Deleted residuals are the residuals from a statistical model in which a single observation is removed. These can be used to identify influential data points that are having a disproportionate impact on the model.

5. Press Residuals: Press residuals are the residuals obtained from a statistical model that has been fit to all of the data except for a single observation. These are used to evaluate how well the model predicts new data points.

6. Deviance Residuals: Deviance residuals are used in generalized linear models and are a measure of how well the model fits the data. They are based on the difference between the observed response and the predicted response, after adjusting for the scale parameter of the model.

These are some of the most common types of residuals used in statistical analysis, and they can be useful for a variety of purposes, including model assessment, outlier detection, and prediction.