Dr. Allen Back. Sep. 23, 2016
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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!)
48
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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