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Hello, I'm having difficulty in answering this question, any help will be very much appreciated thanks!
Explain why, in the simple linear regression model, r^2is the proportion ofvariation of the response y explained by the linear predictor
Original post by maths10101
Hello, I'm having difficulty in answering this question, any help will be very much appreciated thanks!
Explain why, in the simple linear regression model, r^2is the proportion ofvariation of the response y explained by the linear predictor


The answer you give to this question depends upon the level of course in which it is set! I'll assume that this is a straightforward course on regression analysis. If you're looking for a more sophisticated information-based criterion answer, we can return to it.

Do you remember, among the equations that you've come across, a partitioning of the total sum of squares into the regression (or explained) sum of squares and the residual sum of squares?

∑(yi−yˉ)2=∑(y^i−yˉ)2+∑(yi−y^i)2 \displaystyle \sum \left(y_i - \bar{y}\right)^2 = \sum \left(\hat{y}_i - \bar{y}\right)^2 + \sum \left(y_i - \hat{y}_i\right)^2

This decomposition represents the amount of variation in the data away from the mean partitioned into what the regression can explain and what it cannot explain. The coefficient of explained variation is simply the second term divided by the first, representing the proportion of variation explained.

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