What is the difference between studentized and standardized residuals

So far, we have learned various measures for identifying extreme x values high leverage observations and unusual y values outliers. When trying to identify outliers, one problem that can arise is when there is a potential outlier that influences the regression model to such an extent that the estimated regression function is "pulled" towards the potential outlier, so that it isn't flagged as an outlier using the standardized residual criterion.

To address this issue, studentized residuals offer an alternative criterion for identifying outliers. The basic idea is to delete the observations one at a time, each time refitting the regression model on the remaining n —1 observations. Then, we compare the observed response values to their fitted values based on the models with the i th observation deleted.

This produces deleted residuals. Standardizing the deleted residuals produces studentized residuals. Why this measure?

Well, data point i being influential implies that the data point "pulls" the estimated regression line towards itself. In that case, the observed response would be close to the predicted response. But, if you removed the influential data point from the data set, then the estimated regression line would "bounce back" away from the observed response, thereby resulting in a large deleted residual. That is, a data point having a large deleted residual suggests that the data point is influential.

An example. The solid line represents the estimated regression line for all four data points, while the dashed line represents the estimated regression line for the data set containing just the three data points — with the red data point omitted. Observe that, as expected, the red data point "pulls" the estimated regression line towards it.

When the red data point is omitted, the estimated regression line "bounces back" away from the point. Let's determine the deleted residual for the fourth data point — the red one. The estimated regression equation for the data set containing just the first three points is:.

Is this a large deleted residual? The standard deviation of the residuals at different values of the predictors can vary, even if the variances are constant. An alternative is to use studentized residuals.

A studentized residual is calculated by dividing the residual by an estimate of its standard deviation. The standard deviation for each residual is computed with the observation excluded. For this reason, studentized residuals are sometimes referred to as externally studentized residuals. Studentized residuals are more effective in detecting outliers and in assessing the equal variance assumption.

The Studentized Residual by Row Number plot essentially conducts a t test for each residual. Studentized residuals falling outside the red limits are potential outliers. This plot does not show any obvious violations of the model assumptions. We also do not see any obvious outliers or unusual observations. An observation is considered an outlier if it is extreme, relative to other response values. In contrast, some observations have extremely high or low values for the predictor variable, relative to the other values.

These are referred to as high leverage observations. The fact that an observation is an outlier or has high leverage is not necessarily a problem in regression. But some outliers or high leverage observations exert influence on the fitted regression model, biasing our model estimates. Active Oldest Votes. Improve this answer. Sergio Sergio 5, 2 2 gold badges 11 11 silver badges 26 26 bronze badges. There is no definition of a corresponding standardized residual. The regression framework doesn't seem to apply to the question asked.

I have noticed my oversight only after the last click. This takes the form of a technical, terminological point of distinction rather than a misleading statement about the more general, broadly-used term. Sometimes you do actually have the population standard deviation e. It covers just about all conceivable fields, not just public health. On the other hand, one of its strengths is to avoid emphasizing small, meaningless, or overly technical distinctions, so although it is a good guide to statistics generally, it cannot be relied on for settling arcane matters.

Show 14 more comments. We use this to investigate outliers in model. Studentized Residual: We use this to study stability of model. NBhoyar NBhoyar 31 2 2 bronze badges. Sign up or log in Sign up using Google.

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