Chapter 16: CORRELATION Univariate vs Bivariate Distributions

Univariate: a single dependent variable Bivariate: two dependent variables

Correlation and Regression correlation - degree of relationship between two variables regression - prediction of one variable from another variable

Scatterplots or Scatter Diagrams a graphic description of a bivariate distribution a graph of paired X and Y values

Types of Relationships 1. linear vs curvilinear 2. monotonic vs nonmonotonic 3. positive vs negative 4. weak vs strong

Correlation Coefficient a summary number of the magnitude (strength) and direction (positive or negative) of a relationship between 2 variables

1. Pearson product-moment (r) - interval 2. Spearman (rank-order) (rs) - ordinal 3. phi (ø) - nominal 4. others

Pearson product-moment correlation coefficient (r) a measure of linear relationship interval (ratio) scale (mean and variance) the mean of the cross products of the Z scores

1nZZr YX

−−−−ΣΣΣΣ====

-1 ≤≤≤≤ r ≤≤≤≤ +1

S T1 T2 ZX ZY X Y X-X Y-Y

XSXX −−−−

YSYY −−−−

ZXZY

1 20 12 7 2 1.53 0.51 0.78 2 18 16 5 6 1.09 1.53 1.68 3 16 10 3 0 0.66 0.00 0.00 4 15 14 2 4 0.44 1.02 0.45 5 14 12 1 2 0.22 0.51 0.11 6 12 10 -1 0 -0.22 0.00 0.00 7 12 9 -1 -1 -0.22 -0.26 0.06 8 10 8 -3 -2 -0.66 -0.51 0.34 9 8 7 -5 -3 -1.09 -0.77 0.84 10 5 2 -8 -8 -1.75 -2.04 3.58 sum 130 100 0 0 0 0 7.82 mean 13 10 0 0 0 0 0.78 S 4.57 3.92 4.336 3.715 1.00 1.00

Computation Subj T1 T2 X Y X2 Y2 XY 1 20 12 400 144 240 2 18 16 324 256 288 3 16 10 256 100 160 4 15 14 225 196 210 5 14 12 196 144 168 6 12 10 144 100 120 7 12 9 144 81 108 8 10 8 100 64 80 9 8 7 64 49 56 10 5 2 25 4 10 sum 130 100 1878 1138 1440

n)X(X)XX(SS

222

XΣΣΣΣ−−−−ΣΣΣΣ====−−−−ΣΣΣΣ====

n)Y(Y)YY(SS

222

YΣΣΣΣ−−−−ΣΣΣΣ====−−−−ΣΣΣΣ====

[[[[ ]]]]n

YXXY)YY)(XX(SSXYΣΣΣΣΣΣΣΣ−−−−ΣΣΣΣ====−−−−−−−−ΣΣΣΣ====

1nZZr YX

−−−−∑∑∑∑====

YX

YXrSS

)YY)(XX()1n(1nSS

)YY)(XX(−−−−−−−−∑∑∑∑−−−−====

−−−−

−−−−−−−−∑∑∑∑====

YX

XY

YXYX SSSSSS

SSSS)YY)(XX(

1nSS

1nSS

)YY)(XX()1n(r ====−−−−−−−−∑∑∑∑====

−−−−

−−−−

−−−−−−−−∑∑∑∑−−−−====

[[[[ ]]]][[[[ ]]]]222222

22 )Y(Yn )X(Xn

YXXYn

n)Y(Y

n)X(X

nYXXY

r∑∑∑∑−−−−∑∑∑∑∑∑∑∑−−−−∑∑∑∑

∑∑∑∑∑∑∑∑−−−−∑∑∑∑====

∑∑∑∑−−−−∑∑∑∑

∑∑∑∑−−−−∑∑∑∑

∑∑∑∑∑∑∑∑−−−−∑∑∑∑====

]100)1138(10][130)1878(10[)100)(130()1440(10r

22 −−−−−−−−

−−−−====

869.714.1610

1400]1380][1880[

1300014400r ========−−−−====

weight height X Y 120 60 180 72 200 70 160 70 190 74 148 67 155 68 220 76 130 61 145 65

ΣΣΣΣX = 1648 ΣΣΣΣX2 = 280754 ΣΣΣΣY = 683 ΣΣΣΣY2 = 46895 ΣΣΣΣXY = 113951

]100)468951(10][1648)280754(10[)683)(1648()113951(10r

22 −−−−−−−−

−−−−====

927.66.15017

13926]2461][91636[

13926r ============

Interpretation of r

1. linearity

2. causality

3. range of the variables

4. size

Significance of r

sampling distribution of r

mean = mean of the correlation in the

populations

standard deviation = standard error

2nr1s

2

r −−−−−−−−====

Ho: ρρρρ (rho) = 0

H1: ρρρρ ≠≠≠≠ 0 (or one-tail)

222r2n

r12nr

2nr1

r

2nr1

rS

rt−−−−

−−−−====

−−−−−−−−

====

−−−−−−−−

ρρρρ−−−−====ρρρρ−−−−====−−−−

n = 27, r = .5, αααα = .05

Ho: ρρρρ = 0

H1: ρρρρ ≠≠≠≠ 0

887.2866.

5.225.1

255.t l25 ========−−−−

====

critical t25,.05, 2tail = ±±±±.2.060

reject Ho

Table A8 - critical values of r

critical rn-2, αααα = critical r25, .05 = ±±±±.3809

reject H0

Spearman (rs) - rank order correlation coefficient ordinal data

ranks betweer difference the is D where)1n(n

D61r 2

2

S −−−−ΣΣΣΣ−−−−====

-1 < rs < 1

S Anxiety

Rating Noise Rating

Rank

X Y RX RY D=RY-RX D2=(RY-RX)2 1 10 8 10.0 8.0 -2.0 4.00 2 5 5 4.5 4.5 0.0 0.00 3 9 10 8.5 10.0 1.5 2.25 4 4 3 3.0 3.0 0.0 0.00 5 6 7 6.0 7.0 1.0 1.00 6 9 9 8.5 9.0 0.5 0.25 7 8 6 7.0 6.0 -1.0 1.00 8 3 5 2.0 4.5 2.5 6.25 9 5 2 4.5 2.0 -2.5 6.25 10 1 0 1.0 1.0 0.0 0.00 ΣΣΣΣ 0.0 21

)1100(10)21(61rS −−−−

−−−−==== = 9901261−−−− = 1 - .127 = .873

Hypothesis Testing

Ho: ρρρρS = 0 H1: ρρρρS ≠≠≠≠ 0 (or one-tail) Table A9 - critical values of rS critical r = ±±±±.648

Judge A

Judge B

Rank

X Y RX RY D=RX-RY D2=(RY-RX)2 1 4 4 3.5 4 -.5 .25 2 2 2 2 2 0 0 3 4 2 3.5 2 1.5 2.25 4 1 2 1 2 -1 1 5 7 5 7 5 2 4 6 5 6 5 6 -1 1 7 6 7 6 7 -1 1 8 8 8 8 8 0 0 ΣΣΣΣ 9.5

)1n(nD61r 2

2

S −−−−ΣΣΣΣ−−−−====

)63(8)5.9(61rS −−−−==== = .887

Ho: ρρρρS = 0 H1: ρρρρS ≠≠≠≠ 0 critical r = ±±±±.738