New questions with the correlation coefficient

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New questionswith the correlation coefficient
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Testing the H0 r r0 hypothesis

• Under H0 r 0

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In general
Fishers Z-transformation Z(r) will be normal
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E.g., if r 0.80see MiniStat
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Z(r) N(Z(r), sz) (sz )2 1/(n - 3) E.g.,
Z(0.80) 1.099 If n 10 (sz )2 1/7
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H0 r r0
Under H0 Z will be N(0, 1)
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Decision
-1.96 lt Z lt 1.96 Keep H0
Z -1.96 r lt r0
Z ³ 1.96 r gt r0
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An example
H0 r 0.5 n 28, r 0.8
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2. Interval estimation of r
For Z(r) C0.95 Z(r) 1.96sz (z1 z2)
For r Transform back C0.95 (r1 r2)
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An example
n 28, r 0.8
C0.95(Z(r)) Z(0.8) 1.96/sz 1.099
1.96/5 (0.707 1.491) C0.95(r) (0.610
0.905)
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3. H0 r1 r2
Under H0 ZN(0, 1)
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Surprising correlations
(a) Wagners preference and number of socks (b)
Word proficiency and foot size
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4. Partial correlation
X Y
Z
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r 0.85
Y
r3 -0.20
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r2 -0.54
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10
5
0
X
0
5
10
15
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r1 -0.61
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By means of linear regression
X Xz Xres
Y Yz Yres
rXY.Z r(Xres,Yres)
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Formula of theoretical partial correlation
coefficient
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Formula of sample partial correlation coefficient
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Two examples
0.64
0.46
X Y
X Y
0.80
0.80
0.80
0.80
Z
Z
rxy.z 0
rxy.z -0.50
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Another two examples
0
0.10
X Y
X Y
0.60
0.60
-0.60
0.60
Z
Z
rxy.z -0.56
rxy.z 0.72
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Comparing two dependent samples
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Ss X Y Y - X 1. 4 1 -
2. 1 0 - 3. 2 0 -
4. 0 0 0 5. 3 7
6. 3 11 7. 4.5 16
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Means and medians
X
Y
Mean
2.5
5.0
X lt Y
Median
3
1
X gt Y
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Stochastic equality (equality in tendency)
P(X lt Y) P(X gt Y)
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Meaningful null hypotheses
H0 E(X) E(Y)
H0 Med(X) Med(Y)
H0 P(X lt Y) P(X gt Y)
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H0 E(X) E(Y)

• One-sample t-test
• Assumption normality
• Robust alternatives
• Johnson test
• Gayen test

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H0 Med(X) Med(Y)

• Wilcoxon test
• Assumptions
• X and Y are continuous
• Y-X is symmetric

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If X and Y are symmetric Med(X)
Med(Y) and Med(Y-X) 0 are equivalent.
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If X and Y are continuous Med(Y-X) 0 and P(X lt
Y) P(X gt Y) are equivalent.
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H0 P(X lt Y) P(X gt Y)

• Sign test
• Assumptions
• None
• But, it is good if N is large

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How to perform the sign test?

• To be determined
• n of X gt Y occurrences
• n- of X lt Y occurrences
• (t1 t2) region of acceptance

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Decision in the sign test

• t1 lt n lt t2 Keep H0
• n t1 P(X lt Y) lt P(X gt Y) (Y lt X
  stochastically)
• n ³ t2 P(X lt Y) gt P(X gt Y)
• (Y gt X stochastically)

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An example to the sign test
N 50 X Pulse rate before experiment Y Pulse
rate during experiment n 33 (incr.) n- 15
(decr.) In case of n 3315 48 and a
5 (t1-t2) (16-32) n ³ t2 P(X lt Y) gt P(X gt
Y)
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Comparing two independent samples
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X-sample Y-sample 0 1 1 2 8 3 X lt
Y (0 1), (0 2), (0 3), (12), (1 3)
X gt Y (8 1), (8 2), (8 3) n 5 (incr.)
n- 3 (decr.)
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Means and medians
X
Y
Mean
3
2
X gt Y
Median
1
2
X lt Y
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Stochastic equality (equality in tendency)
P(X lt Y) P(X gt Y)
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Meaningful null hypotheses
H0 E(X) E(Y)
H0 Med(X) Med(Y)
H0 P(X lt Y) P(X gt Y)
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H0 E(X) E(Y)

• Two-sample t-test
• Assumptions
• normaliy, s1 s2
• Robust variant
• Welch-test

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H0 P(X lt Y) P(X gt Y)

• Mann-Whitney test (MW)
• Assumption
• s1 s2
• Robust variants
• Brunner-Munzel test (BM)
• Fligner-Policello test (FP)

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How to perform MW
xi rank yj rank 0 1 1
2.5 1 2.5 2 4 8 6 3
5 R1 9.5 R2 11.5
(t1 t2) Region of acceptance
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Decision in MW

• t1 lt R1 lt t2 Keep H0
• R1 t1 Xlt Y stochastically
• R1³ t2 X gtY stochastically

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Monotonic relationship of two variables, X and Y
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Deterministic monotonicity
Y
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If X grows then Y grows too
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8
4
0
X
0
1
2
3
4
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Stochastic monotonicity
Y
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If X grows then likely Y grows too

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8



4








0
X
0
1
2
3
4
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An example
Ss X Y 1 1 35 2
1.5 34 3 2 36 4 3 37 5
7 38 6 10 39
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Rank data separately for X and Y
Ss X rank Y rank 1 1 1
35 2 2 1.5 2 34 1 3 2
3 36 3 4 3 4 37 4 5
7 5 38 5 6 10 6 39 6
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Spearmans rank correlation (rS)Correlation
between ranksIn the above example r 0.91, rS
0.94
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Discordancy
Concordancy
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Concordancy and discordancy
Y
B

C
-
A
X
D
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Kendall-s tau
p Proportion of concordant pairs in the
population p- Proportion of discordant pairs in
the population
t p - p-