The reduction in uncertainty from ssy to ssresidual

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Unformatted text preview: estimate of Y, again by adding up the squared error over all subjects. The result is called the residual sum of squares, because it represents the uncertainty in Y that’s left over after we do the regression. Notice that the residual sum of squares is the same as mean squared error except that once again it’s a sum instead of a mean, because we don’t divide by degrees of freedom. ˆ SSresidual = ∑(Y – Y )2 (3) As stated above, the goal of regression is to find the values of the regression coefficients (bi) that lead to the best predictions. By best predictions, we mean minimizing the squared ˆ difference between Y and Y . In other words, the goal is to minimize SSresidual. This is how we determine the regression coefficients (or, usually, how a computer determines them for us). The question now is how well the regression did. If the predictors do a really good job of explaining the outcome, SSresidual will be close to zero. If the predictors tell us little or ! nothing about the outcome, then SSresidual will be close to the original, total sum of squares, SSY. SSresidual is always less than or equal to SSY (because SSY is the error of the naive ˆ prediction, MY, and SSresidual is the error of the best possible prediction, Y ), but the question is how much less. The reduction in uncertainty from SSY to SSresidual is called SSregression, because it’s the amount of variability in the outcome (Y) that the regression can explain. SSregression = SSY – SSresidual SSY = SSregression + SSresidual (4) The two versions of Equation 4 say the same thing. The first version shows how we get SSregression by subtracting the residual sum of squares from the total sum of squares. The second version shows how the total sum of squares (i.e., the original variability...
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This document was uploaded on 02/25/2014 for the course PSYC 3101 at Colorado.

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