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. When the method of least squares is used to estimate α 0 {\displaystyle \alpha _{0}} and α 1 {\displaystyle \alpha _{1}} , then the residuals ε ^ {\displaystyle {\widehat {\varepsilon \,}}} , unlike the errors ε {\displaystyle \varepsilon } , cannot be independent since they satisfy the two constraints
∑ i = 1 n ε ^ i = 0 {\displaystyle \sum _{i=1}^{n}{\widehat {\varepsilon \,}}_{i}=0}
and
∑ i = 1 n ε ^ i x i = 0. {\displaystyle \sum _{i=1}^{n}{\widehat {\varepsilon \,}}_{i}x_{i}=0.}
(Here εi is the ith error, and ε ^ i {\displaystyle {\widehat {\varepsilon \,}}_{i}} is the ith residual.) The residuals, unlike the errors, do not all have the same variance: the variance decreases as the corresponding x-value gets farther from the average x-value. This is not a feature of the data itself, but of the regression better fitting values at the ends of the domain. It is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence. This can also be seen because the residuals at endpoints depend greatly on the slope of a fitted line, while the residuals at the middle are relatively insensitive to the slope. The fact that the variances of the residuals differ, even though the variances of the true errors are all equal to each other, is the principal reason for the need for studentization. It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.
Background For this simple model, the design matrix is
X = [ 1 x 1 ⋮ ⋮ 1 x n ] {\displaystyle X=\left[{\begin{matrix}1&x_{1}\\\vdots &\vdots \\1&x_{n}\end{matrix}}\right]}
and the hat matrix H is the matrix of the orthogonal projection onto the column space of the design matrix:
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