# Residual Analysis

## Introduction to Residual Analysis

If you have studied the regression model, you must have come across the term ‘residual analysis.’ In general, the model is deemed valid if the error term associated with the regression model is in accordance with the four assumptions commonly considered in the model. However, if the assumptions are not satisfied, the conclusions from significance tests associated with it are also considered.

## Residuals in Regression Analysis

The estimated regression equation is used to calculate the residual value. For any dependent variable yi, the ith residual value is the difference between its estimated value and the observed value. The residual values thus calculated are considered as estimates arising from model error, and statisticians use these values to place their assumptions. Therefore, you can understand that experience and good judging skills play an important role in placing the estimates, thus generating residuals’ values.

### Residual Plots

Residual plots are often considered for graphical representation of the residual values. In such graphs, the residual values are plotted on the y-axis (vertical axis), while the independent variables are plotted on the x-axis (horizontal axis). There can be two types of residual plots- linear and nonlinear.

If the residual values are dispersed around the horizontal axis, the linear residual plots are preferred. For example, out of five values of residuals, if two are negative, statisticians will prefer a linear graph.

If the residual values show a pattern change, for example, forming a U or an inverted U on the graph, a non-linear graph can be preferred. Some examples of residual plots are given below.

Random Pattern

Non-Random: U-shaped

Non-Random: Inverted U

A lot of information can be obtained while interpreting residual plots. If the assumptions related to the error term are satisfied by the residual plot, you will obtain a horizontal line of points. However, if the assumptions are not satisfied, the analysis suggests better modifications of the model to obtain better results. Most statisticians consider residual plot analysis to be important in considering the assumptions made about the error term.

### ANOVA Residuals

Residuals are an important concept in ANOVA statistical analysis. ANOVA residuals are important in the interpretation of several biological calculations. Previously, you have learned that residuals are the difference between the predicted and the observed value of the dependent variable. In ANOVA, it is also known as the partition of sums of squares.

SST = SSR + SSE

Where,

SST stands for total variability of the data observed

SSR stands for a fraction of variability explained by the linear regression model. It is considered to be better if the SSR value is high.

SSE stands for a fraction of variability not explained by the linear regression model. It is considered to be better if the SSE value is low.

In this regard, the residual formula is represented as

SSE =  $\sum_{N}^{i=1}$ (yi - yi)2

### Important Software That Can be Used To Calculate Residual Analysis

Different software is routinely used by statisticians to calculate residual analysis. This software is fed in with all the required algorithms to identify the problems based on a number of formulas provided by the user. Most statistical analysis formulas are included in this software. Let us look at some of them.

• SPSS Software

SPSS software is quite famous amongst most statisticians. They have also been given profound importance in biological systems as well. They have a separate section for linear regression plots SPSS that also has the option of including residual analysis in linear regression plots. The statistical analysts can use the feature of SPSS residual plots. They can also perform such residual analysis SPSS and make their assumptions from such models.

• MATLAB Software

MATLAB is another software that most statisticians commonly used for their research. It also has all the necessary formulas to carry out important statistical experiments. For example, you can go for residual plot MATLAB. You can also make assumptions from error models in this software.

These are some of the common formulas, concepts, and software associated with residual analysis. You need to learn these techniques properly if you wish to plot residual plots. You can also use software like SPSS and MATLAB to prepare such plots. You can also analyze them to calculate the error models in this software.

1. What is Residual Analysis in terms of Regression?

Ans: According to the residual analysis definition, it is the difference between the estimated value and the observed value of any dependent variable. For any dependent variable yi, the ith residual value is the difference between its observed and expected value. This term is synonymous with any regression model. Most statisticians make any of the four assumptions generally associated with any regression model.

The model is considered valid if the error term is in sync with these four assumptions, i.e., the assumptions are satisfied. If they are not satisfied, more significant tests are to be performed, and the conclusions from such tests are needed to be considered.

Since the value of residual analysis depends on the expected values, the experience and skills of the statisticians analyzing the data are also important.  Statisticians who have worked with such problems can use their experience to put forward their expected values for the dependent variables.

2. How to Prepare a Residual Plot?

Ans: There are different software that you can use to prepare residual plots. Some of the commonly used ones are SPSS and MATLAB. This software has the right algorithms to analyze the input data and prepare the residual plot with all the related pieces of analysis information.

In any residual plot, there are two types of variables- dependent and independent. The dependent variables are plotted on the y-axis or the vertical axis. The independent variables are plotted on the x-axis or the horizontal axis. There are two types of residual analysis plots- linear and nonlinear. The arrangements of the data points on the graph determine the type of residual plot.

For example, if the residual data points are arranged randomly, it results in a linear graph with the best fit line passing between the data points. However, if the data points fall on the graph in a defined pattern, the graph is non-linear. It can be a U-shaped curve or an inverted U-shaped curve.