Studentized residual

In statistics, a studentized residual, named in honor of William Sealey Gosset, who wrote under the pseudonym Student, is a residual adjusted by dividing it by an estimate of its standard deviation. Studentization of residuals is an important technique in the detection of outliers.

Errors versus residuals
It is very important to understand the difference between errors and residuals in statistics. Consider the simple linear regression model


 * $$Y_i=\alpha_0+\alpha_1 x_i+\varepsilon_i,$$

where the errors εi, i = 1, ..., n, are independent and all have the same variance σ2. The residuals are not the true, and unobservable, errors, but rather are estimates, based on the observable data, of the errors. When the method of least squares is used to estimate α0 and α1, then the residuals, unlike the errors, cannot be independent since they satisfy the two constraints


 * $$\sum_{i=1}^n \widehat{\varepsilon}_i=0$$

and


 * $$\sum_{i=1}^n \widehat{\varepsilon}_i x_i=0.$$

(Here $$\varepsilon_i$$ is the ith error, and $$\widehat{\varepsilon}_i$$ is the ith residual.) Moreover, the residuals, unlike the errors, do not all have the same variance: the variance increases as the corresponding x-value gets farther from the average x-value. 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.

How to studentize
For this simple model, the design matrix is


 * $$X=\left[\begin{matrix}1 & x_1 \\ \vdots & \vdots \\ 1 & x_n \end{matrix}\right]$$

and the "hat matrix" H is the matrix of the orthogonal projection onto the column space of the design matrix:


 * $$H=X(X^T X)^{-1}X^T.$$

The "leverage" hii is the ith diagonal entry in the hat matrix. The variance of the ith residual is


 * $$\mbox{var}(\widehat{\varepsilon}_i)=\sigma^2(1-h_{ii}).$$

The corresponding studentized residual is then


 * $${\widehat{\varepsilon}_i\over \widehat{\sigma} \sqrt{1-h_{ii}\ }}$$

where $$\widehat{\sigma}$$ is an appropriate estimate of σ.

Internal and external studentization
The estimate of σ2 is


 * $$\widehat{\sigma}^2={1 \over n-m}\sum_{j=1}^n \widehat{\varepsilon}_j^2.$$

where m is the number of parameters in the model (2 in our example). But it is desirable to exclude the ith observation from the process of estimating the variance when one is considering whether the ith case may be an outlier. Consequently one may use the estimate


 * $$\widehat{\sigma}_{(i)}^2={1 \over n-m-1}\sum_{\begin{smallmatrix}j = 1\\j \ne i\end{smallmatrix}}^n \widehat{\varepsilon}_j^2,$$

based on all but the ith case. If the latter estimate is used, excluding the ith case, then the residual is said to be externally studentized; if the former is used, including the ith case, then it is internally studentized.

If the errors are independent and normally distributed with expected value 0 and variance σ2, then the probability distribution of the ith externally studentized residual is a Student's t-distribution with n &minus; m &minus; 1 degrees of freedom, and can range from $$-\infty$$ to $$+\infty$$.

On the other hand, the internally studentized residuals are in the range $$0 \pm \sqrt{\mathrm{r.d.f.}}$$, where r.d.f. is the number of residual degrees of freedom, namely n &minus; m. If "i.s.r." represents the internally studentized residual, and again assuming that the errors are independent identically distributed Gaussian variables, then


 * $$\mathrm{i.s.r.}^2 = \mathrm{r.d.f.}{t^2 \over t^2+\mathrm{r.d.f.}-1}$$

where t is distributed as Student's t-distribution with r.d.f. &minus; 1 degrees of freedom. In fact, this implies that i.s.r.2/r.d.f. follows the beta distribution B(1/2,(r.d.f. &minus; 1)/2). When r.d.f. = 3, the internally studentized residuals are uniformly distributed between $$-\sqrt{3}$$ and $$+\sqrt{3}$$.

If there is only one residual degree of freedom, the above formula for the distribution of internally studentized residuals doesn't apply. In this case, the i.s.r.'s are all either +1 or &minus;1, with 50% chance for each.

The standard deviation of the distribution of internally studentized residuals is always 1, but this does not imply that the standard deviation of all the i.s.r.'s of a particular experiment is 1.