Regression

Sunday, 18 August 2019

8:42 PM

10.6 Adequacy of the regression model Variability decomposition Similarly to the notations on Slide 11, we can define n E(Yi - Syy i=l this measures the total amount of variability in the response values, and is sometimes denoted sst (for 'total sum of squares') Now, this variability in the observed values Yi arises from two factors: O because the Xi values are different, all Yi have different means. This variability is quantified by the 'regression sum of squares': n ssr O each value Yi has variance 02 around its mean. This variability is quantified by the 'error sum of squares': n sse We can always write: MATH2099/2859 (Statistics) i=l sst = ssr + sse Dr Jia Deng n i=l Term 2019 - Lecture g 44/50

 

10.6 Adequacy of the regression model Coefficient of determination Suppose sst ssr and sse 0: the variability in the responses due to the effect of the predictor is almost the total variability in the responses —+ all the dots are very close to the straight line, the predictions are very accurate: the linear regression model fits the data very well Now suppose sst sse and ssr 0: almost the whole variation in the responses is due to the error terms *the dots are very far away from the fitted straight line, the predictions are very imprecise: the regression model is useless + comparing ssr to sst allows us to judge the model adequacy The quantity r2, called the coefficient of determination, defined as 2 ssr sst represents the proportion of the variability in the responses that is explained by the predictor and hence taken into account in the model. MATH2099/2859 (Statistics) Dr Jia Deng Term Lecture g 45/50

 

 

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