 # Residuals analysis

Give a method that can analyze residuals. Are residuals informative? These questions were asked during my SPSS research lab defence.

### 1 comment to Residuals analysis

• Peter

Hope you are advance level user of statistics or do SPSS research or SPSS analysis as you want to know about residual analysis. According to the definition, we can say that the residuals from a fitted model are defined as the differences between the response data and the fit to the response data at each predictor value. We can write it in below form:

residual = data – fit

Mathematically, the residual for a specific predictor value is the difference between the response value y and the predicted response value ŷ.

r = y – ŷ

We are assuming here the model are fit to the data is correct and the residuals approximate the random errors. Therefore, if the residuals appear to behave randomly, it suggests that the model fits the data well. However, if the residuals display a systematic pattern, it is a clear sign that the model fits the data poorly.

Let me get the chance to explain it another way. A residual is actually nothing but the error in a result. Precisely, if we want to find x such that

f(x)=b

Given an approximation x0 of x, the residual is

b-f(x0)

whereas the error is

x0-x

If we do not know x, we cannot compute the error but we can compute the residual.

As I say before, residuals are of course informative for searching for error in result. If you do not know the exact solution, you can look for the approximation with small residual. This term or way “residuals” appear in many areas in mathematics, from iterative solvers such as the generalized minimal residual method, which seeks solutions to equations by systematically minimizing the residual.

There are two kinds of residuals.

Raw residuals: The observed values of the raw residuals are given by the fitted residuals

Standardized residuals: The standardized residuals are designed to overcome the problem of different variances of the raw residuals. The problem is solved by dividing each of the raw residuals by an appropriate term.

However, I tried to find something in SPSS help, but failed.