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handout_2_4 - 2.4 Cautions about Correlation and Regression...

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2.4 Cautions about Correlation and Regression regression- see scatter of data points about regression line sum or squares of vertical distances from the points to the regression line are as small as possible distances represent “left-over” variation in the response after fitting the regression line- distances are known as residuals Residuals A residual is the difference between an observed value of the response variable and the value predicted by the regression line. residual = observed y – predicted y residual= y – y ˆ Recall the fat gain versus NEA increase data least squares regression line: fat gain = 3.505 – (0.00344 * NEA increase) one subject: NEA increase= 135 calories fat gain= 2.7 kg predicted gain: y ˆ = 3.505 – (0.00344 * NEA increase)= 3.04 kg observed gain: y = 2.7 kg residual = observed y – predicted y residual = 2.7 kg – 3.04 kg = -0.34 kg residuals for 16 data points: 0.37 -0.7 0.1 -0.34 0.19 0.61 -0.26 -0.98 1.64 -0.18 -0.23 0.54 -0.54 -1.11 0.93 -0.03
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To assess the fit of a regression line you could: •look at the vertical deviations of the data points from the regression line •look at a residual plot (easier to study) Residual Plots A residual plot is a scatterplot of the regression residuals against the explanatory variable. ●the mean of the residuals of a least-squares regression is always zero the line (residual = 0) in residual plot corresponds to the fitted regression line (Figure 2.20) the residual plot magnifies the deviations from the line to make the patterns easier to see •if regression line catches the overall pattern of the data, there should be no pattern in the residuals (irregular scatter- randomly distributed above and below zero) residuals in Figure 2.20 have this irregular scatter •don’t have an irregular horizontal pattern in residual plot- this demonstrates regression
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