Dr. Allen Back. Sep. 23, 2016

Transcription

1 Dr. Allen Back Sep. 23, 2016

2 Look at All the Data Graphically A Famous Example: The Challenger Tragedy

3 Look at All the Data Graphically A Famous Example: The Challenger Tragedy Type of Data Looked at the Night Before: Conclusion: Failures at all temperatures so Temp not the issue.

4 Look at All the Data Graphically A Famous Example: The Challenger Tragedy What Wasn t Looked At: Conclusion: Clear Pattern of Failures More Likely at Low Temp.

5 Given: paired data (x 1, y 1 ),..., (x n, y n )

6 Given: paired data (x 1, y 1 ),..., (x n, y n ) Scatterplot: Plot x horizontally, y vertically.

7 Given: paired data (x 1, y 1 ),..., (x n, y n ) If one variable is potentially explanatory for the response of the other, choose the explanatory variable as x.

8 Given: paired data (x 1, y 1 ),..., (x n, y n ) is what we care about. If we know the x value of a point, does it tell us something about the likely y value?

9 Given: paired data (x 1, y 1 ),..., (x n, y n ) is what we care about. If we know the x value of a point, does it tell us something about the likely y value? (a number) and regression (a line) are just techniques to study association.

10 Given: paired data (x 1, y 1 ),..., (x n, y n ) Principal Aspects of : Direction: Strength: Form:

11 Given: paired data (x 1, y 1 ),..., (x n, y n ) Principal Aspects of : Direction: positive or negative Strength: Form:

12 Given: paired data (x 1, y 1 ),..., (x n, y n ) Principal Aspects of : Direction: positive or negative Strength: Form: Negative means as x increases, y generally decreases.

13 Given: paired data (x 1, y 1 ),..., (x n, y n ) Principal Aspects of : Direction: Strength: strong, moderate, or weak Form:

14 Given: paired data (x 1, y 1 ),..., (x n, y n ) Principal Aspects of : Direction: Strength: Form: linear, curved, or clustered

15 Example b

16 Example b My call: Direction: positive Strength: moderate Form: curved

17 Example b using Data Desk

18 Example c

19 Example c My call: Direction: negative Strength: strong Form: linear

20 Example d

21 Example d My call: Direction: negative Strength: moderate Form: (perhaps some outliers)

22 Example d using Data Desk

23 Example e

24 Example e My call: Direction: positive Strength: strong Form: linear

25 Example f

26 Example f My call: Direction: negative Strength: weak Form: curved, maybe an outlier

27 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y (Without the s x and s y, this would be the statistics (i.e. data related) version of the covariance of two random variables which was briefly mentioned in chapter 16 of your textbook.)

28 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y Positive association means as x increases, so does y generally. (And similarly when x decreases.)

29 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y Positive association means as x increases, so does y generally. (And similarly when x decreases.) So for pos. association, most terms in the sum are either (+) (+) or ( ) ( ).

30 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ Positive association means as x increases, so does y generally. (And similarly when x decreases.) So for pos. association, most terms in the sum are either (+) (+) or ( ) ( ). Thus with pos. association, r tends to be positive. s y

31 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y pos r pos. association neg. r neg. association

32 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y 1 <= r <= 1 (= ±1 only for perfect linear association)

33 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y 1 <= r <= 1 (= ±1 only for perfect linear association)

34 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y 1 <= r <= 1 (= ±1 only for perfect linear association) To see = ±1 for perfect linear association y = β 1 x + β 0 means s y = β 1 s x and ȳ = β 1 x + β 0.

35 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ To see = ±1 for perfect linear association y = β 1 x + β 0 means s y = β 1 s x and ȳ = β 1 x + β 0. r = 1 ( ) ( ) n 1 Σ xi x yi ȳ s x s y = 1 ( ) ( ) n 1 Σ xi x xi x β 1 = 1 n 1 Σ s x ( xi x s x ) 2 β 1 β 1 s y β 1 s x = β 1 β 1 where the last line used the definition of the variance.

36 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ If you ve studied vectors, the fact 1 <= r <= 1 comes from the same mathematics which explains why s y cos θ = v w v w has right hand side between 1 and +1.

37 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y r is unchanged if x and y are exchanged.

38 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y invariant under rescaling

39 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y Curved association and r=0 are consistent!

40 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ Curved association and r=0 are consistent! Five points along y = x 2. ( x = 0 and ȳ = 2.) s y x y (x-0) (y-2) (x-0)(y-2)

41 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ Curved association and r=0 are consistent! Five points along y = x 2. ( x = 0 and ȳ = 2.) s y x y (x-0) (y-2) (x-0)(y-2) r is exactly zero even though the association is very strong.

42 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y r is strongly affected by outliers.

43 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y r is strongly affected by outliers.

44 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ This shows the effect of a single outlier. (All 10 points but the last fit y = 2x + 3.) s y x y

45 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ This shows the effect of a single outlier. (All 10 points but the last fit y = 2x + 3.) s y ɛ

46 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y samples from independent RV s r 0

47 Properties r = 1 ( n 1 Σ xi x s x ) ( ) yi ȳ s y X,Y indep std normal RV s ; set Y = ρx + 1 ρ 2 Y. Then (X, Y ) will tend to generate data with r ρ. (e.g. ρ =.99 (1 ρ => 2 ) ρ =.14!)

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49 X,Y indep std normal RV s ; set Y = ρx + 1 ρ 2 Y. Then (X, Y ) will tend to generate data with r ρ.

50 X,Y indep std normal RV s ; set Y = ρx + 1 ρ 2 Y. Then (X, Y ) will tend to generate data with r ρ. 1 ρ 2 Or Y = X + ρ Y with the coefficient of Y interpreted as the magnitude of the deviation from perfect association.

51 X,Y indep std normal RV s ; set Y = ρx + 1 ρ 2 Y. Then (X, Y ) will tend to generate data with r ρ. 1 ρ 2 Or Y = X + ρ Y with the coefficient of Y interpreted as the magnitude of the deviation from perfect association. 1 ρ 2 ρ ρ

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