Reminders. Quiz today. Please bring a calculator to the quiz

Transcription

1 Reminders Quiz today Please bring a calculator to the quiz 1

2 Regression Review (sort of Ch. 15) Warning: Outside of known textbook space Aaron Zimmerman STAT Summer 2014 Department of Statistics University of Washington - Seattle 2

3 Plotting Lines Any line I draw through the data can be written in slope-intercept form: y = a + b x a = 1.0 is the y-intercept of the line while b = 0.35 is the slope of the line 3

4 Plotting Lines The intercept is the value of the response variable when the explanatory has value zero. The slope is how much the response variable changes if the explanatory variable increases by one 4

5 The SD Line The points in a scatter diagram generally seem to cluster around the SD line. This line goes through the point of averages And it goes through all the points which are an equal number of SDs away from the average, for both variables. What s the slope of the SD line? 5

6 The SD Line The slope of the SD line is (SD of Y )/(SD of X ) if the correlation is positive The slope of the SD line is (SD of Y )/(SD of X ) if the correlation is negative 6

7 The Least Squares Regression Line The least squares regression line for y on x estimates the average value of y corresponding to each value of x. It is the solid line The vertical bars shows fathers who were about 1.5 SDs above/below the average. The dashed line is the SD line and sons who were also 1.5 SDs above/below average would be on the line between the vertical bars. Note that most points in the strip at 72 below the SD line. Most points in the strip at 64 are above the SD line 7

8 The Least Squares Regression Line The dots show the average height of the sons, for each value of father s height. This is called the graph of averages. It s NOT all the data! (See last or next slide for all the data) The SD line is dashed, the least squares regression line is solid and smooths the averages The averages rise less steeply than the SD line. 8

9 The Least Squares Regression Line This line also goes through the point of averages It turns out that the slope of the least squares regression line is r SD of y SD of x Because r is always between -1 and 1, this means that the least squares regression line is always less steep than the SD line This is the mathematical equivalent of regression to the mean 9

10 Writing down formulas for the line The slope of the SD line is b = SD of y SD of x The slope of the least squares line is b = r SD of y SD of x In both cases we find the intercept by calculating: a = y b x = mean(y) b mean(x) Then you use y = a + b x 10

11 Prediction and Prediction Error PREDICTION Regression lines are often used for prediction as well as association Points on the line are predictions for the response given the value of the explanatory You can use the equation of the line to more accurately make predictions RESIDUALS Are prediction error residual = observed-predicted observations above the line have positive residuals observations below the line have negative residuals 11

12 Summary of Lines SD LINE SD of y Slope: b = SD of x Intercept: a = y b x Formula/Eq n: y = a + b x Relates percentiles of the variables Not as good for prediction Affected by outliers LEAST SQUARES LINE Slope: b = r SD of y SD of x Intercept: a = y b x Formula/Eq n: y = a + b x Predicts using average response for given explanatory value Affected by outliers r 2 can be interpreted as percent variability in response explained by line Minimizes sum of squared residuals 12

13 Education Examples An education researcher is interested in the relationship between the percent of students who pass the state exam in reading in grade 5 in various elementary schools in Seattle and other characteristics of those schools For each potential explanatory variable, remember our strategy! Make a scatterplot If it looks linear, calculate a correlation To understand how the response variable responds to changes in the explanatory variable, we use regression Think critically about outliers 13

14 Example # 1 The first explanatory variable the researcher considers is the percent of students in the school eligible for free or reduced priced meals (PercentFRPM). Again, the dependent variable is the percent of students who passed the 5th grade reading exam (ReadGrade5) 14

15 Example # 1 r = Which of the following statements is true? 85% of the variation in ReadGrade5 can be explained by the linear regression on PercentFRPM 72% of the variation in ReadGrade5 can be explained by the linear regression on PercentFRPM A one percentage increase in PercentFRPM is correlated with a 0.85 percentage decrease in ReadGrade5 A one percentage increase in PercentFRPM is correlated with a 0.72 percentage decrease in ReadGrade5 15

16 Example # 1 r = 0.85, and the formula for the line is ReadGrade5 = (PercentFRPM) What s the slope of the SD line? If the the SD of PercentFRPM is 20, what s the SD of ReadGrade5? If the mean of ReadGrade5 is 68, what s the mean of PercentFRPM? 16

17 Example # 1 r = 0.85, and the formula for the line is ReadGrade5 = (PercentFRPM) What s the slope of the SD line? SD slope = -LS slope/r = -(-0.55/-0.85) = If the the SD of PercentFRPM is 20, what s the SD of ReadGrade5? SD slope = SD(read)/SD(FRPM) SD(Read) = SD slope SD(FRPM) = -(-0.65*20)=13 If the mean of ReadGrade5 is 68, what s the mean of PercentFRPM? Intercept = mean(read) - LS slope mean(frpm) mean(frpm) = (Int - mean(read))/-ls slope = ( )/-(-0.55))=

18 Example # 1 r = 0.85, and the formula for the line is ReadGrade5 = (PercentFRPM) What is the interpretation of the slope of the least-squares regression line? At Van Asselt Elementary School, 84.0% of students are eligible for free/reduced priced lunch. What is the predicted percent of students at this school who pass the reading exam in grade 5? 78.80% of students at Van Asselt Elementary School actually passed the reading test in grade 5. What is the residual for this school? 18

19 Example # 1 r = 0.85, and the formula for the line is ReadGrade5 = (PercentFRPM) What is the interpretation of the slope of the least-squares regression line? A one percentage increase in PercentFRPM is associated with a 0.55 decrease in ReadGrade5 At Van Asselt Elementary School, 84.0% of students are eligible for free/reduced priced lunch. What is the predicted percent of students at this school who pass the reading exam in grade 5? prediction = = % of students at Van Asselt Elementary School actually passed the reading test in grade 5. What is the residual for this school? =

20 Have a great weekend! 20