Studentized residual

In statistics, a studentized residual is the dimensionless ratio resulting from the division of a residual by an estimate of its standard deviation, both expressed in the same units. It is a form of a Student's t-statistic, with the estimate of error varying between points.

This is an important technique in the detection of outliers. It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym "Student" (e.g., Student's distribution). Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.

Motivation

The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions.

Consider the simple linear regression (SLR) model

Y

=

α

0

+

α

1

X

+

ε

.

{\displaystyle Y=\alpha _{0}+\alpha _{1}X+\varepsilon .\,}

Given a random sample (Xi, Yi), i = 1, ..., n, each pair (Xi, Yi) satisfies

Y

i

=

α

0

+

α

1

X

i

+

ε

i

,

{\displaystyle Y_{i}=\alpha _{0}+\alpha _{1}X_{i}+\varepsilon _{i},\,}

where the errors

ε

i

{\displaystyle \varepsilon _{i}}

, are independent and all have the same variance

σ

2

{\displaystyle \sigma ^{2}}

. The residuals are not the true errors, but estimates, based on the observable data.