Chapter 10 Regression

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damea134
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Chapter 10 Regression
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2015-08-11 00:09:48
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Regression
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Regression line
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  1. Regression
    Regression method describes how one variable depends on another

    Regression is used to PREDICT the value of dependent variable.  

    It answer the question of how of an unit increase (1 SD)in x (Horizontal axis) is associated with an increase (r=.40) in Y Vertical axis).

    Associated with each increase of one SD in x there is an increase of only r SDs in y, on the average.

    • Example:
    • SD=15 
    • x Ave=50 
    • y Ave=70
    • R= .04

    if there is a 1 SD (15) in X value there will be a (r=0.40) increase in the Y value. (15x.04=6, 6+70 =76)
  2. If SD goes in negative direction of Average (Important)
    If SD in negative direction of average, then Remember to subtract the regression from the average [-X(SD)R + AVE] or [AVE - X(SD)R] 

    • example:
    • Ave Height= 69" SD = 3
    • Ave Weight = 190 Pounds SD = 42

    • Find SD of Weight (important!) for Heights:
    • 69, &  24

    69 (if height is 69" SD = 0. There is no movement in Height)

    24, (69-24 = -45), (45/3 = -15SD), (-15 x 42 = -630), (-630 x r or .41 = -258.3), (-258.3 + Ave or 190) = {Important change order} (190 - 258.3)  = -68.3
  3. Calculating Regression
    How much of an increase in height (X-Value) is associated with an increase in Weight (Y-Value). 

    • average height =70 inches (x Value)
    • average weight = 70 inches (y Value)
    • Height SD = 3
    • Weight SD = 15
    • R=.40

    If height is one SD above average how much will the weight be above average?

    70 (Ave Height) + 3 (1 SD) = 73


    • 180 (Ave Weight) + 45 (1 SD) x .40 (r) = 18,
    • 18 +180 = 198 LBS

    1.5 SD in Height

    Height = 70 + 4.5 (3 x 1.5 SD) = 74.5

    REMEMBER TO ALWAYS MULTIPLY r by SD value of X. So (R x 1.5 =.60). Than Multiply by Y SD

    180 (Ave Weight) + 45 (1 SD) x .60 (r=.4 x 1.5) = 27,



    27 +180 = 207 LBS

    Height is -2 SD

    Height = 70 - 6 (3 x -2 SD) = 63

    • 180 (Ave Weight) - 45 (SD) x .80 (r=.4 x 2) = 36,
    • 36 - 180 = 144 LBS

    [(SU of x)] x (r) +or- (Ave)]
  4. Graph of Averages
    Graph of averages is not the same as individual values. it is a graph that is made up of many averages from many samples. (i.e. averages of states income)

    The regression is a smoothed version of the graph of averages if the Graph of averages follows a straight line, that line is the regression line.

    In some situations, the regression line smooths away too much if there is a non-linear association between the two variables.
  5. Chance Variation,
    Chance Error
    When samples tend to not follow the regression line. Chance variation is at works.

    When there point(s) are way outside of the regression line

    By the luck of the Draw.

    Meaning samples that were taken were not true representation of the population.

    • Example:
    • Average 15 subjects measuring 6-1' could be taken. The average weight in the group could be 100lbs

    Average of 8 subjects measuring 6-2' could be taken and the Average weight in this group could be 80lbs because it was taken from a smaller sample. We would say that this is chance variation.
  6. Regression line
    • The regression line isn't as steep as the SD line. 
    • SD Line will always be steeper. (More Sloped)
    • The regression line is a smoothed version of the graph of averages.
    • None linear association. Regression should not be used when there is a none linear association.
    • (Note: If the graph of averages follows a straight line than that line is the regression line)
  7. Regression Method For Individuals:
    The Rule if you have to predict one variable from another use the new average.

    The regression method give a sensible way of estimating the new average.

    If There is a none linear association between the the variables the regression method would not apply
  8. Calculating for individuals:
    • Ave Hieght = 70, SD =3
    • Ave Weight = 180, SD 45  
    • r = 0.40

    estimate wieght of a man with a height of 73 inches.

    r=.40, .40 x 45lbs = 18 , 18 + 180 = 198


    • Predict first year GPA, Based on SAT score
    •  
    • Ave Sat = 550, SD = 80
    • Ave 1st Year GPA = 2.6, SD = .60, R=.40

    SAT = 650

    • 650-550= 100, 100/80 = 1.25 SD,
    • .40 x 1.25 = .50, .50 x .60 = .3 GPA points
    • .30 + 2.6 = 2.9 GPA
  9. Calculating for individuals:

    Percentages
  10. Regression fallacy
    In virtually all test-retest situations, the bottom group on the first test will on average show some improvement on the second test and the top group will on average fall back. this the regression effect.

    It is this effect that makes spread around the SD line

    Also termed Regression to Mediocrity 

    Or Regression Effect



  11. Regression Effect
    Observed test score = true score + chance error

    Chance error can be +-5 (Chance error can be any value (+-4) or (+-6) and so on)

    True score below 140 with a positive Chance error.

    True score above 140 with a negative chance error.

    The example for regression effect. If some scores above average on the first test, the true score is probably a bit lower than the observed score. If the person takes the test again we predict second test scores to a bit lower than the first test scores and vice versa

    Note: Regression Effect take values/scores closer to the average not below or above.
  12. Two Regression Lines
    One regression line can be y on x (Common) SD line is Steeper

    The other is x on y (Less Common) Regression is steeper

    • Note: Because x of y = Z lbs doesn't mean Y of X = Z hieght

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