prediction ii assumptions and interpretive aspects

Prediction II

Assumptions and Interpretive Aspects

Assumptions of Regression

• Normal Distribution– Both variables should be normally distributed– For non-normal distributions we use non-parametric tests

• Continuous Variables– Variables must be measured with a interval or ratio scale– Non-parametric tests are better for the scores collected with a

nominal and ordinal scales• Linearity

– The relation between two variables should be linear• Homoscedasticity

– The variability of of actual Y values about YI must be the same for all values of X.

Linearity

• In unlinear distributions, r is lower than its real value– So, prediction is less successful

• Some characteristics in nature are curvilinearly related. • For such variables, we need to use some

advanced tecniques• For instance, the relationship between anxiety

and success is curvilinear• When anxiety is low, success is low

(motivation is low)• When anxiety is at its medium, success is

high (motivation is high and anxiety does not have a derograting effect)

• When anxiety is high, success is low (the organism is shocked)

Homoscedasticity

Interpretive AspectsFactors Influencing r

• Range of Talent– When Y, X or both are restricted the r is lower than its real value– Because, r is a byproduct of both S2

YX and S2Y

• That is S2YX / S2

Y in formula B• If we restrict the variance of Y, for instance, standart error of prediction would stay

same. So, the r would get lower– See figure 11.1 on page 195

– This is what we called ceiling and floor effect


• Range of Talent Item per minute

Worker Payment $ Apprerent Real

A 12 8 8B 13 11 11C 14 12 12D 15 12 12E 16 13 13F 17 15 14G 18 15 15I 19 15 15J 20 15 16K 21 15 17L 22 15 19M 23 15 20N 24 15 21

r= 0,84 0,9810 12 14 16 18 20 22 24 26

0

5

10

15

20

25

ApprerentLinear (Apprerent)RealLinear (Real)


• Heterogeneity of Samples– When samples are pooled, the correlation for

aggregated data depends on where the sample values lie relative to one another in both the X and Y dimensions • Let’s say professor Aktan and Göktürk prepared final

exams for two courses: Statistics and Int. Resch. Methd.


• Heterogeneity of Samples– Students always gets 20 points higher in Göktürk’s

exams Statistics ResearchGöktürk 65 67 72 75 64 63 80 81 74 82 67 71 88 92 75 82 r= 0,95Aktan 45 47 52 55 44 43 60 61 54 62 47 51 68 72 55 62 r= 0,95 r= 0,98

40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Series1Linear (Series1)Series3Linear (Series3)


• Heterogeneity of Samples– Aktan insist on giving his own Statistics exam

40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Series1Linear (Series1)Series3Linear (Series3)Series5Linear (Series5)

Statistics Research

Göktürk 65 67

72 75

64 63

80 81

74 82

67 71

88 92

75 82 r= 0,95

Aktan 45 67

52 75

44 63

60 81

54 82

47 71

68 92

55 82 r= 0,95

r= 0,58

Interpretive AspectsRegression Equation

• β coefficient shows the slope of the regression line.– General equation of a straight line• Y=bX + c

– Regression of Y on X

β c


• β coefficient shows the slope of the regression line.– To see that let’s use two z score distribution in

which mean is 0 and SD is 1

– Now, Zx-mean and Zy-mean becomes 0. So, c=0– Zsy/Zsx is equal to 1/1. So, B=(r1/1)Zx= rZx– As you can see, beta is equal to r in z distributions


• Now, let’s say we calculated r between statistics and research scores for students of Çağ, ODTÜ and Mersin University– For Çağ University r= .82– For Mersin University r= .62– For ODTÜ r= .35

Statistics ResearchÇağ -3 -2,46 -2 -1,64 -1 -0,82 0 0 1 0,82 2 1,64 3 2,46Mersin -3 -1,86 -2 -1,24 -1 -0,62 0 0 1 0,62 2 1,24 3 1,86ODTÜ -3 -1,05 -2 -0,7 -1 -0,35 0 0 1 0,35 2 0,7 3 1,05

-4 -3 -2 -1 0 1 2 3 4

-3

-2

-1

0

1

2

3

ÇağMersinODTÜ

Interpretive AspectsProportion of Variance in Y Associated with Variance in X

• Correlation coefficient has a special meaning– The squared correlation coefficient is equal to the

proportion of variance in Y which is explained by the variance in X• That is explained variance

– r2 = proportion of explained variance– 1- r2 = proportion of unexplained variance

– Let’s say correlation between depression and GPA is .67• So, change in depression explains 45% of change in GPA

– r= .67, so r2 = .45

Interpretive AspectsProportion of Variance in Y Associated with Variance in X

• We can see the meaning of this in the Figure below

prediction ii assumptions and interpretive aspects

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