![]() ![]() Points that have the most influence produce the largest change in the equation of the regression line. For our purposes now, we need to look at the version of the studentized residual when the ith observation is removed from the model, i.e.ĭefinition 2: If we remove a point from the sample, then the equation for the regression line changes. We will use this measure when we define Cook’s distance below. ![]() ![]() A rule of thumb (Steven’s) is that values 3 times this mean value are considered large.Īs we saw in Residuals, the standard error of the residual e i isĪnd so the studentized residuals s ihave the following property: Where there are k independent variables in the model, the mean value for leverage is ( k+1)/ n. Leverage measures how far away the data point is from the mean value. Where there is only one independent variable, we have Thus the strength of the contribution of sample value y i on the predicted value ŷ i is determined by the coefficient h ii, which is called the leverage and is usually abbreviated as h i. Where each h ijonly depends on the x values in the sample. Leverage – By Property 1 of Method of Least Squares for Multiple Regression, Y-hat = HY where H is the n × n hat matrix =. Points with large residuals are potential outliers. Is the measure of the distance of the ith sample point from the regression line. Leverageĭefinition 1: The following parameters are indicators that a sample point ( x i1, …, x ik, y i) is an outlier: for the general population, there is nothing unusual about a 6-foot man or a 125-pound man, but a 6-foot man that weighs 125 pounds is unusual. Keep in mind that since we are dealing with a multi-dimensional model, there may be data points that look perfectly fine in any single dimension but are multivariate outliers. We now look at how to detect potential outliers that have an undue influence on the multiple regression model. ![]()
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