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Chap 6. Regression

Chap 6. Regression. The Relation of Two Variables The Graph of Averages The Regression Method The Regression Effect The Regression Line for x on y and y on x. 1. 2. 3. 4. 5. The Regression Method. The Regression Effect. The Regression Line for x on y and y on x. INDEX.

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Chap 6. Regression

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  1. Chap 6. Regression The Relation of Two Variables The Graph of Averages The Regression Method The Regression Effect The Regression Line for x on y and y on x

  2. 1 2 3 4 5 The Regression Method The Regression Effect The Regression Line for x on y and y on x INDEX The Relation of Two Variables The Graph of Averages

  3. 1. Relation of Two Variables Scatter Plot Weight (kg) SD line Regression and SD line go through the point of averages regression line Associated with an increase of 1 SD in height there is an increase of only 0.67 SD in weight, on the average. Height (cm)

  4. y estimation rSDy average SDx x 1. Relation of Two Variables Regression • The regression line for y on x estimates the average value for y corresponding to each value of x. r: correlation coefficient Associated with each increase of 1SD in x there is an increase of only r SDs in y, on the average.

  5. 1 2 3 4 5 The Regression Method The Regression Effect The Regression Line for x on y and y on x INDEX The Relation of Two Variables The Graph of Averages

  6. 2. Graph of Averages Graph of Averages regression Graph of averages 각 점들은 각각의 키에 대하여 그 키에 해당하는 집단의 평균 몸무게를 보여준다. 점 위에 표시된 숫자는 해당집단의 크기를 나타낸다. Many points on the graph of averages are near the line. This line is regression line.

  7. 2. Graph of Averages Improper case of regression line – nonlinear association Regression line When the graph of averages is nonlinear, the regression line does not reflect the exact association.

  8. 1 2 3 4 5 The Regression Method The Regression Effect The Regression Line for x on y and y on x INDEX The Relation of Two Variables The Graph of Averages

  9. 3. Regression Method Guess the weight <Health and Nutrition Examination Survey> Without being told anything else about him, the best guess on the weight of one of these men is the overall average weight 63.5kg. weight (kg) If you are told the man’s height: 179.5cm then your best guess for his weight is the average for all the 179.5cm men in the study , 74.7kg. height(cm)

  10. 3. Regression Method Example 1 • Followings are the results of 100 students majoring in economics in a certain college. Economics 1 average = 3.00 SDX = 0.87 Statistics average = 2.80 SDY = 0.86 r = 0.36 EX) Predict the statistics GPA of a student whose GPA of Economics I is 3.70 .

  11. 3. Regression Method Example1 Corr. coeff 2. 0.36 0.8 = 0.29 0.29 SDy above average on the Statistics. • 0.8SDx above average • on the Economics 1. • 3.7-3.0 = 0.7 = 0.87 0.8 Regression Analysis SDX SDy 4. The predicted GPA of statistics is 2.80 + 0.25 = 3.05 3. That is, 0.29 0.86 = 0.25(GPA) above average on the Statistics.

  12. 3. Regression Method notice ☞ when estimating on new subjects, Foregoing data can represent the new subjects ; reasonable!! Foregoing data and the new subjects are different in groups or in the range ; you have to think about the issue carefully before applying the regression method.

  13. 3. Regression Method In the case of [example 1] … Would be the data from the students majoring in Economics applied directly to Jane whose major is Philosophy? • What if Jane majors in Philosophy not in Economics? • What if Jane’s GPA of Economics I is 4.3 but 3.7? • If most GPAs of the Economics 1 (x-value) are between 2.0 and 4.0 , Jane’s x-value(4.3) is out of the previous x-range. Would the estimated regression line based on the x- value range 2.0~4.0 be applied to Jane’s GPA which is over the range? • [The extrapolation is not a simple problem.]

  14. 3. Regression Method example2- predict percentile ranks • Suppose the percentile rank of one student on the Economics 1 GPA is 90%. Predict her percentile rank on statistics. • This student scored 1.3 • SDx above average on the • Economics1. 2. 0.36 1.3 = 0.47 SDy above average on the statistics. Regression Analysis =90% =68% z = 1.3 z = 0.47 3. The percentile rank on statistics is predicted as 68%.

  15. 1 2 3 4 5 The Regression Method The Regression Effect The Regression Line for x on y and y on x INDEX The Relation of Two Variables The Graph of Averages

  16. 4. Regression Effect Y-values of points are spread symmetrically along the SD line overall. regression to mediocrity containing too many points with small x-value. Most of these points’ y-values are above the SD line. Containing too many points with large x-value. Most of these points’ y-values are below the SD line. Regression line SD line

  17. 4. Regression Effect Regression effect Final average scores corresponding to each midterm scores Regression line – summarizes each average points well SD line

  18. 4. Regression Effect regression fallacy • The regression effect occurs when each point representing data is spread out around the SD line. • The fallacy that the regression effect must be due to something important, not just the spread around the line ☞ Regression Fallacy

  19. Observed score= 140 135 145 4. Regression Effect The model for regression effect If someone scores above average on the first test, the true score is probably a bit lower than the observed score. Suppose that the probability error is 5with probability 0.5. The people scored 140 can be divided into two groups with true score 135 or 145. (observed score) = (true score) + (chance error)

  20. 1 2 3 4 5 The Regression Method The Regression Effect The Regression Line for x on y and y on x INDEX The Relation of Two Variables The Graph of Averages

  21. 5. Regression line for x on y and yon x Two different regression lines Regression of weight on height Regression of weight on height weight weight height height The regression line is solid, and the SD line is dashed

  22. Wife’s IQ Husband’s IQ 140 140 120 140 Husbnad’s IQ 5. Regression line for x on y and y on x exmaple3 • The men whose IQ was 140 had wives whose IQ averaged 120. Look at the wives whose IQ was 120; should the average IQ of their husbands be 140? (IQ scores are scaled to have an average of about 100, and an SD of about 15, both for men and women. The correlation is about 0.50) ?

  23. 5. Regression line for x on y and y on x example 3 Regression line for husband’s IQ on wife’s IQ Wife’s IQ There are two different regression lines; one for predicting the wife’s IQ from her husband’s IQ and the other for vice versa. 120 Regression line for wife’s IQ on husband’s IQ 110 140 Husband’s IQ

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