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Correlation. Hal Whitehead BIOL4062/5062. The correlation coefficient Tests Non-parametric correlations Partial correlation Multiple correlation Autocorrelation Many correlation coefficients. The correlation coefficient. Linked observations: x 1 , x 2 ,..., x n y 1 , y 2 ,..., y n

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correlation

Correlation

Hal Whitehead

BIOL4062/5062

slide2
The correlation coefficient
  • Tests
  • Non-parametric correlations
  • Partial correlation
  • Multiple correlation
  • Autocorrelation
  • Many correlation coefficients
slide4
Linked observations:

x1,x2,...,xny1,y2,...,yn



Mean:x = Σxi / ny = Σyi / n



Variance:

S²(x)= Σ(xi-x)²/(n-1) S²(y)= Σ(yi-y)²/(n-1)



Standard Deviation:

S(x) S(y)

Covariance: S²(x,y) = Σ(xi-x) ∙ (yi-y) / (n-1)

slide5
Covariance: S²(x,y) = Σ(xi-x) ∙ (yi-y) / (n-1)

Correlation coefficient

(“Pearson” or “product-moment”):

r = {Σ(xi-x) ∙ (yi-y) / (n-1) } / {S(x) ∙ S(y)}

r = S²(x,y) / {S(x) ∙ S(y)}

the correlation coefficient6
The correlation coefficient:

r = S²(x,y) / {S(x) ∙ S(y)}

-1 ≤r≤ +1

If no linear relationship: r = 0

r2:

proportion of variance accounted for by linear regression

tests on correlation coefficients26
Tests on Correlation Coefficients
  • Assume:
    • Independence
    • Bivariate Normality
tests on correlation coefficients27
Tests on Correlation Coefficients
  • Assume:
    • Independence
    • Bivariate Normality
tests on correlation coefficients28
Tests on Correlation Coefficients
  • Assume:
    • Independence
    • Bivariate Normality
  • Then:

z = Ln [(1+r)/(1-r)]/2 is normally distributed

with variance 1/(n-3)

And, if  (true population value of r) = 0 :

r∙√(n-2) / √(1-r²) is distributed as Student's t with n-2 degrees of freedom

we can test
We can test:

a) r≠ 0

b) r > 0 or r < 0

c) r = constant

d) r(x,y) = r(z,w)

Also confidence intervals for r

are whales battering rams carrier et al j exp biol 200231
Are Whales Battering Rams?(Carrier et al. J. Exp. Biol. 2002)

r= 0.75

(SE = 0.15)

(95% C.I. 0.47-0.89)

Tests:

r≠ 0 : P = 0.0001

r > 0 : P = 0.00005

More sexually dimorphic species

have relatively larger melons

why do large animals have large brains schoenemann brain behav evol 2004
Why do Large Animals have Large Brains?(Schoenemann Brain Behav. Evol. 2004)
  • Correlations among mammals
    • Log brain size with
      • Log muscle mass

r=0.984

      • Log fat massr=0.942
  • Are these significantly different?

t=5.50; df=36; P<0.01

Hotelling-William test

  • Brain mass is more closely related to muscle than fat
non parametric correlation34
Non-Parametric Correlation
  • If one variable normally distributed
    • can test r=0 as before.
  • If neither normally distributed:
    • Spearman's rS rank correlation coefficient

(replace values by ranks)

or:

    • Kendall's τcorrelation coefficient
  • Use Spearman's when there is less certainty about the close rankings
partial correlation37
Partial Correlation
  • Correlation between X and Y controlling for Z

r (X,Y|Z) = {r(X,Y) - r(X,Z)∙r(Y,Z)}

√{(1 - r(X,Z)²)∙(1 - r(Y,Z)²)}

  • Correlation between X and Y controlling for W,Z

r (X,Y|W,Z) = {r(X,Y|W) - r(X,Z|W)∙r(Y,Z|W)}

√{(1 - r(X,Z|W)²)∙(1 - r(Y,Z|W)²)}

n-2-c degrees of freedom

(c is number of control variables)

why do large animals have large brains schoenemann brain behav evol 200438
Why do Large Animals have Large Brains?(Schoenemann Brain Behav. Evol. 2004)
  • Correlations among mammals
    • Log brain size with

Log musclemass

Controlling for Log bodymass

r=0.466

Log fat mass

Controlling for Log body mass

r=-0.299

  • Fatter species have relatively smaller brains and more muscular species relatively larger brains
semi partial correlation coefficient
Semi-partial Correlation Coefficient
  • Correlation between X & Y controlling Y for Z

r (X,(Y|Z)) = {r(X,Y) - r(X,Z)∙r(Y,Z)}

√(1 - r(Y,Z)²)

are whales battering rams carrier et al j exp biol 200240
Are Whales Battering Rams?(Carrier et al. J. Exp. Biol. 2002)

Correlation

r= 0.75

Partial Correlation

r (SSD,MA|L) = 0.73

Semi-partial Correlations

r (SSD,(MA|L)) = 0.69

r ((SSD |L),MA) = 0.71

multiple correlation coefficient
Multiple Correlation Coefficient
  • Correlation between one dependent variable and its best estimate from a regression on several independent variables:

r(Y∙X1,X2,X3,...)

  • Square of multiple correlation coefficient is:
    • proportion of variance accounted for by multiple regression
autocorrelation45
Autocorrelation
  • Purposes
    • Examine time series
    • Look at (serial) independence
slide46
Data

(e.g. Feeding rate on consecutive days,

plankton biomass at each station on a transect):

1.5 1.7 4.3 5.4 5.7 6.2 3.9 4.4 5.2 4.8 3.9 3.7 3.6

Autocorrelation of lag=1 is correlation between:

1.5 1.7 4.3 5.4 5.7 6.2 3.9 4.4 5.2 4.8 3.9 3.7

1.7 4.3 5.4 5.7 6.2 3.9 4.4 5.2 4.8 3.9 3.7 3.6

r = 0.508

Autocorrelation of lag=2 is correlation between:

1.5 1.7 4.3 5.4 5.7 6.2 3.9 4.4 5.2 4.8 3.9

4.3 5.4 5.7 6.2 3.9 4.4 5.2 4.8 3.9 3.7 3.6

r = -0.053

…….

many correlation coefficients behaviour of sperm whale groups
Many Correlation Coefficients:[Behaviour of Sperm Whale Groups]

Listwise deletion, n=40; P<0.10; P<0.05; uncorrected

NGR25L SST SHITR LSPEED APROP SOCV SHR2 LFMECS LAERR

NGR25L 1.00

SST 0.12 1.00

SHITR -0.21 -0.33* 1.00

LSPEED 0.10 -0.28+ 0.06 1.00

APROP -0.15 -0.34* 0.07 0.18 1.00

SOCV -0.05 0.08 -0.16 -0.01 -0.33* 1.00

SHR2 -0.18 -0.12 0.01 -0.20 0.19 -0.03 1.00

LFMECS 0.08 0.14 -0.13 -0.12 -0.22 0.29+ -0.18 1.00

LAERR -0.10 0.03 -0.21 -0.24 -0.02 0.24 -0.08 0.23 1.00

Expected no. with P<0.10 = 3.6; with P<0.05 = 1.8

many correlation coefficients behaviour of sperm whale groups50
Many Correlation Coefficients:[Behaviour of Sperm Whale Groups]

Listwise deletion, n=40; P<0.10; P<0.05; Bonferronicorrected

NGR25L SST SHITR LSPEED APROP SOCV SHR2 LFMECS LAERR

NGR25L 1.00

SST 0.12 1.00

SHITR -0.21 -0.33 1.00

LSPEED 0.10 -0.28 0.06 1.00

APROP -0.15 -0.34 0.07 0.18 1.00

SOCV -0.05 0.08 -0.16 -0.01 -0.33 1.00

SHR2 -0.18 -0.12 0.01 -0.20 0.19 -0.03 1.00

LFMECS 0.08 0.14 -0.13 -0.12 -0.22 0.29 -0.18 1.00

LAERR -0.10 0.03 -0.21 -0.24 -0.02 0.24 -0.08 0.23 1.00

P=1.0 for all coefficients

many correlation coefficients behaviour of sperm whale groups51
Many Correlation Coefficients:[Behaviour of Sperm Whale Groups]

Listwise deletion, n=40; P<0.10; P<0.05; uncorrected

NGR25L SST SHITR LSPEED APROP SOCV SHR2 LFMECS LAERR

NGR25L 1.00

SST 0.12 1.00

SHITR -0.21 -0.33* 1.00

LSPEED 0.10 -0.28+ 0.06 1.00

APROP -0.15 -0.34* 0.07 0.18 1.00

SOCV -0.05 0.08 -0.16 -0.01 -0.33* 1.00

SHR2 -0.18 -0.12 0.01 -0.20 0.19 -0.03 1.00

LFMECS 0.08 0.14 -0.13 -0.12 -0.22 0.29+ -0.18 1.00

LAERR -0.10 0.03 -0.21 -0.24 -0.02 0.24 -0.08 0.23 1.00

Pairwise deletion, n=59-118; P<0.10; P<0.05; uncorrected

NGR25L SST SHITR LSPEED APROP SOCV SHR2 LFMECS LAERR

NGR25L 1.00

SST 0.11 1.00

SHITR -0.17+-0.46* 1.00

LSPEED 0.05 -0.17 0.05 1.00

APROP -0.05 -0.20+ 0.04 0.31* 1.00

SOCV -0.00 -0.05 -0.06 -0.02 -0.25* 1.00

SHR2 -0.15 -0.13 0.07 -0.14 0.05 0.01 1.00

LFMECS 0.01 0.07 -0.02 -0.14 -0.25* 0.43*-0.26+ 1.00

LAERR -0.06 0.06 0.09 -0.27*-0.20+ 0.06 -0.06 0.21+ 1.00

many correlation coefficients52
Many Correlation Coefficients
  • Missing values:
    • Listwise deletion (comparability), or
    • Pairwise deletion (power)
  • P-values:
    • Uncorrected: type 1 errors
    • Bonferroni, etc.: type 2 errors
beware

Beware!

Y2

Y1 Y3

Y4

Y1 Y2

Correlation Causation

Y1Y3

Y4

Y2Y5

Y1

Y3

Y2

Y1Y3

Y4

Y5

Y2 Y6

Y1Y3

Y4

Y2Y5