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Concept
Studentized residual
Summary
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. 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. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.
Errors and residuals in statistics
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 model
Given a random sample (Xi, Yi), i = 1, ..., n, each pair (Xi, Yi) satisfies
where the errors , are independent and all have the same variance . The residuals are not the true errors, but estimates, based on the observable data. When the method of least squares is used to estimate and , then the residuals , unlike the errors , cannot be independent since they satisfy the two constraints
and
(Here εi is the ith error, and 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.
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