Residual values in linear regression analysis need to be calculated for several purposes. In linear regression using the ordinary least squares method, one of the assumptions that must be met is that residuals must be normally distributed, hence the necessity to first calculate residual values. However, before calculating the residual values, we need to first calculate the predicted Y values. Therefore, on this occasion, we will discuss how to calculate predicted Y values and residual values.
First, let’s understand how to obtain the predicted Y value in simple linear regression. This model consists of only one dependent variable and one independent variable. The intercept (b0) and the coefficient b1 will be used to calculate the predicted Y value.
As we know, the general equation for simple linear regression is:
Y = b0 + b1X
Where: Y = dependent variable; X = independent variable; b0 = intercept; and b1 = regression coefficient
To calculate the predicted Y value, we need to plug the intercept value (b0) and the regression coefficient (b1) into the regression equation. Then, we input the appropriate value of the independent variable (X) to obtain the predicted Y value.
Next, after calculating the predicted Y value, we can proceed to calculate the residual value. The residual value is the difference between the actual Y value and the predicted Y value. This residual indicates how far the regression model’s predictions are from the actual observed values. This step is crucial because residuals must meet several assumptions for the validity of the regression model, including normal distribution.
It is important to note that calculating and analyzing residuals helps us understand how well the regression model predicts the dependent variable. If the residuals are randomly dispersed and do not show a specific pattern, then the regression model can be considered quite good. Conversely, if there is a clear pattern in the residual plot, it might indicate that the regression model is not suitable or that there are important variables not included in the model.
Calculating Predicted Y Values
The predicted Y value can be calculated for each observation based on the regression equation. The way to calculate it is by summing the product of each coefficient from the estimation result with the initial observation value of the independent variable.
For example, if the intercept value is 218.38 and the estimated coefficient value for the X variable is -0.0014, and the first observation value for the X variable is 6000, then the way to calculate the predicted Y for this first observation is as follows:
Y = b0 + b1X
Y = 218,38 + (– 0,0014 x 6000)
Y = 210,003
You can use Microsoft Excel to simplify and speed up the process of calculating the predicted Y values. In the same way, you can calculate the predicted Y values for all existing observation data or sample data. Congratulations, you have successfully calculated the predicted Y values correctly.
Calculating Residual Values
Referring to the beginning of the paragraph, the predicted Y value is used to calculate the residual value in simple linear regression analysis. Before calculating the residual value, it is important to understand the definition of the residual value in regression analysis. The residual value is the difference between the actual observed value of the dependent variable (Y) and the predicted Y value. The formula to calculate it is shown in the following equation:
Residual = Y actual – Y predicted
Residual = 213 – 210,003
Residual = 2,997
You have successfully calculated the residual value for the first observation or sample from this calculation. Next, you need to calculate the residual values for all observations or samples in your study. With this understanding, you can now comprehensively calculate both the predicted Y values and residual values. This is crucial in regression data analysis as it helps evaluate how well your regression model predicts the dependent variable. If you have a lot of data, using tools like Microsoft Excel is highly recommended for efficiency.
Additionally, it is important to remember that residual analysis can also provide insights into the fit of the regression model to the data. Randomly dispersed residuals indicate a good model, while specific patterns in the residuals can suggest problems such as a poor-fitting model or missing important variables.
This article aims to provide a thorough understanding of how to calculate and interpret predicted Y values and residuals in simple linear regression. Thank you for visiting this blog, and don’t forget to follow future updates from Kanda Data. Stay motivated and always stay healthy! 😊
