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Chapter 5

Chapter 5.odt

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STAT 101
Qian( Michelle) Zhou

Chapter 5 – Regression Regression Lines • Regression Lines – a straight line that describes relationship b/w explanatory and response variables ◦ we use to to predict value of y for given value of x • a review of straight lines: ◦ y = a + bx ▪ a = intercept point • a = ybar - b(xbar) • when x and y are 0 • x is whatever the 'x' is that lines up with 'a' ▪ b = slope (amount by which y changes when x increases by one unit); change of one SD in x corresponds to change of r SD in y • b = r(sy/sx) • also known as “rate of change” • when units changes, slope changes Example 5.2) Using a Regression line Refer to figure 5.1. x is non exercise activity and y is fat gain. “b” = -0.00344 tells us, on avg, fat gained goes down by 0.00344 kg for each added calorie to the NEAchange a = 3.505 it's the predicted fat gain when x is 0 Fat gain = a + (b x “non-exercise” activity change) Fat gain = 3.505 – 0.00344 x NEAchange = 3.505 – 0.00344 x 400 = 2.13 kg • size of slope depends on units which we measure two variables ◦ also, slope is a numerical description of the relationship b/w two variables, it does not mean change in NEA has little effect on fat gain ◦ you can't determine how important a relationship is by looking at the size of the slope of the regression line The Least-Squares Regression Line • prediction errors – we make are in y ◦ good regression line makes vertical distances of points from the line as small as possible ◦ error = observed response – predicted response • least-regression line – of y on x is the line that makes the sum of the squares of vertical distances of data points, as small as possible ◦ ŷ = a + bx ▪ the line gives a predicted response, ŷ, for any x ◦ ŷ - ybar / sy = r ( x-xbar / sx ) ◦ least-squares regression always passes the point (xbar, ybar) Facts about Least-Squares Regression 1. Distinction b/w explanatory and response variable is essential in regression. 2. Slope and correlation always have same sign. If scatterplot has positive association, b and r are both positive. If r=1 or -1, change in predicted response, ŷ, is same (in standard deviation units) as change in x. If correlation is less strong, ŷ changes less. 3. The least-squares regression line always pass through point (x bar, y bar). 4. The square of the correlation, r^2, is fraction of the variation in the values of y that is explain by least- squares regression of y on x. • r^2 = variation in ŷ along regression line as x varies/ total variation in observed values of y ◦ what it means is that about x% of a variation is explained by linear relationship/regression • the main idea is that, as x changes, y changes Residuals • residuals – difference b/w observed value for y and value predicted by regression line. It's a prediction error that remains after we have chose the regression line: ◦ residual = observed y – predicted y = y – ŷ Example 5.5 – I feel your pain) r = 0.515. ŷ = -0.0578 + 0.00761x For empathy score of 38, (empathy on x scale so it is x).. ŷ = -0.0578 + 0.00761(38) = 0.231 At subjects brain activity level was as -0.120. The residual is = y – ŷ
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