CorrelationHal WhiteheadBIOL4062/5062

The correlation coefficientTestsNon-parametric correlationsPartial correlationMultiple correlationAutocorrelationMany correlation coefficients

The correlation coefficient

Linked observations: x1,x2,...,xn y1,y2,...,yn Mean: x = xi / n y = yi / n Variance: S(x)= (xi-x)/(n-1) S(y)= (yi-y)/(n-1) Standard Deviation: S(x) S(y) Covariance: S(x,y) = (xi-x) (yi-y) / (n-1)

Covariance: S(x,y) = (xi-x) (yi-y) / (n-1)

Correlation coefficient(Pearson or product-moment):

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

r = S(x,y) / {S(x) S(y)}

The correlation coefficient:

r = S(x,y) / {S(x) S(y)}

-1 r +1

If no linear relationship: r = 0

r2: proportion of variance accounted for by linear regression

r = -0.01

r = 0.38

r = -0.31

r = 0.95

r = 0.04

r = 0.64

r = -0.46

r = 0.99

r = -0.0

Tests on Correlation Coefficients

Tests on Correlation CoefficientsAssume:IndependenceBivariate Normality

Tests on Correlation CoefficientsAssume:IndependenceBivariate Normality

Tests on Correlation CoefficientsAssume:IndependenceBivariate NormalityThen: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-r) is distributed as Student's t with n-2 degrees of freedom

We can test:a) r 0b) r > 0 or r < 0c) r = constantd) r(x,y) = r(z,w)

Also confidence intervals for r

Are Whales Battering Rams?(Carrier et al. J. Exp. Biol. 2002)

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 = 0.0001r > 0 : P = 0.00005

More sexually dimorphic specieshave relatively larger melons

Why do Large Animals have Large Brains?(Schoenemann Brain Behav. Evol. 2004)Correlations among mammalsLog brain size withLog muscle massr=0.984Log fat mass r=0.942Are these significantly different?t=5.50; df=36; P

Non-Parametric Correlation

Non-Parametric CorrelationIf one variable normally distributedcan test r=0 as before.If neither normally distributed:Spearman's rS rank correlation coefficient(replace values by ranks)or:Kendall's correlation coefficientUse Spearman's when there is less certainty about the close rankings

Are Whales Battering Rams?(Carrier et al. J. Exp. Biol. 2002)r = 0.75rS = 0.62= 0.47

Partial Correlation

Partial CorrelationCorrelation between X and Y controlling for Zr (X,Y|Z) = {r(X,Y) - r(X,Z)r(Y,Z)} {(1 - r(X,Z))(1 - r(Y,Z))}

Correlation between X and Y controlling for W,Zr (X,Y|W,Z) = {r(X,Y|W) - r(X,Z|W)r(Y,Z|W)} {(1 - r(X,Z|W))(1 - r(Y,Z|W))}

n-2-c degrees of freedom(c is number of control variables)

Why do Large Animals have Large Brains?(Schoenemann Brain Behav. Evol. 2004)Correlations among mammalsLog brain size withLog muscle massControlling for Log body massr=0.466Log fat massControlling for Log body mass r=-0.299Fatter species have relatively smaller brains and more muscular species relatively larger brains

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))

Are Whales Battering Rams?(Carrier et al. J. Exp. Biol. 2002)Correlationr = 0.75

Partial Correlationr (SSD,MA|L) = 0.73

Semi-partial Correlationsr (SSD,(MA|L)) = 0.69r ((SSD |L),MA) = 0.71

Multiple Correlation

Multiple Correlation CoefficientCorrelation between one dependent variable and its best estimate from a regression on several independent variables:r(YX1,X2,X3,...)

Square of multiple correlation coefficient is:proportion of variance accounted for by multiple regression

Multiple Partial Correlation Coefficient!

Autocorrelation

AutocorrelationPurposesExamine time seriesLook at (serial) independence

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.

Autocorrelation Plot (Correlogram)

Many Correlation Coefficients

Many Correlation Coefficients:[Behaviour of Sperm Whale Groups]

NGR25LSSTSHITRLSPEEDAPROPSOCVSHR2LFMECSLAERRNGR25L1.00SST0.121.00SHITR-0.21-0.33*1.00LSPEED0.10-0.28+0.061.00APROP-0.15-0.34*0.070.181.00SOCV-0.050.08-0.16-0.01-0.33*1.00SHR2-0.18-0.120.01-0.200.19-0.031.00LFMECS0.080.14-0.13-0.12-0.220.29+-0.181.00LAERR-0.100.03-0.21-0.24-0.020.24-0.080.231.00Listwise deletion, n=40; P

Many Correlation Coefficients:[Behaviour of Sperm Whale Groups]

NGR25LSSTSHITRLSPEEDAPROPSOCVSHR2LFMECSLAERRNGR25L1.00SST0.121.00SHITR-0.21-0.331.00LSPEED0.10-0.280.061.00APROP-0.15-0.340.070.181.00SOCV-0.050.08-0.16-0.01-0.331.00SHR2-0.18-0.120.01-0.200.19-0.031.00LFMECS0.080.14-0.13-0.12-0.220.29-0.181.00LAERR-0.100.03-0.21-0.24-0.020.24-0.080.231.00Listwise deletion, n=40; P

Many Correlation Coefficients:[Behaviour of Sperm Whale Groups]

NGR25LSSTSHITRLSPEEDAPROPSOCVSHR2LFMECSLAERRNGR25L1.00SST0.121.00SHITR-0.21-0.33*1.00LSPEED0.10-0.28+0.061.00APROP-0.15-0.34*0.070.181.00SOCV-0.050.08-0.16-0.01-0.33*1.00SHR2-0.18-0.120.01-0.200.19-0.031.00LFMECS0.080.14-0.13-0.12-0.220.29+-0.181.00LAERR-0.100.03-0.21-0.24-0.020.24-0.080.231.00Listwise deletion, n=40; P

Many Correlation CoefficientsMissing values:Listwise deletion (comparability), orPairwise deletion (power)P-values:Uncorrected: type 1 errorsBonferroni, etc.: type 2 errors

Beware!Correlation Causation Y1 Y2Y1 Y3 Y4Y2 Y5Y1 Y3Y2 Y2Y1 Y3 Y4Y1 Y3 Y4Y2 Y5Y1 Y3 Y4 Y5Y2 Y6