Download presentation
Presentation is loading. Please wait.
Published byDouglas Wood Modified over 8 years ago
1
Correlation Hal Whitehead BIOL4062/5062
2
The correlation coefficient Tests Non-parametric correlations Partial correlation Multiple correlation Autocorrelation Many correlation coefficients
3
The correlation coefficient
4
Linked observations: x 1,x 2,...,x n y 1,y 2,...,y n Mean: x = Σ x i / n y = Σ y i / n Variance: S²(x)= Σ(x i -x)²/(n-1) S²(y)= Σ(y i -y)²/(n-1) Standard Deviation: S(x) S(y) Covariance: S²(x,y) = Σ(x i -x) ∙ (y i -y) / (n-1)
5
Correlation coefficient (“Pearson” or “product-moment”): r = {Σ(x i -x) ∙ (y i -y) / (n-1) } / {S(x) ∙ S(y)} r = S²(x,y) / {S(x) ∙ S(y)}
6
The correlation coefficient: r = S²(x,y) / {S(x) ∙ S(y)} -1 ≤ r ≤ +1 If no linear relationship: r = 0 r2:r2: proportion of variance accounted for by linear regression
8
r = -0.01
10
r = 0.38
12
r = -0.31
14
r = 0.95
16
r = 0.04
18
r = 0.64
20
r = -0.46
22
r = 0.99
24
r = -0.0
25
Tests on Correlation Coefficients
26
Assume: –Independence –Bivariate Normality
27
Tests on Correlation Coefficients Assume: –Independence –Bivariate Normality
28
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
29
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
30
Are Whales Battering Rams? (Carrier et al. J. Exp. Biol. 2002)
31
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
32
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 mass r=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
33
Non-Parametric Correlation
34
If one variable normally distributed –can test r=0 as before. If neither normally distributed: –Spearman's r S rank correlation coefficient (replace values by ranks) or: –Kendall's τ correlation coefficient Use Spearman's when there is less certainty about the close rankings
35
Are Whales Battering Rams? (Carrier et al. J. Exp. Biol. 2002) r = 0.75 r S = 0.62 τ= 0.47
36
Partial Correlation
37
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)
38
Why do Large Animals have Large Brains? (Schoenemann Brain Behav. Evol. 2004) Correlations among mammals –Log brain size with Log muscle mass Controlling for Log body mass 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
39
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)
40
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
41
Multiple Correlation
42
Multiple Correlation Coefficient Correlation between one dependent variable and its best estimate from a regression on several independent variables: r(Y∙X 1,X 2,X 3,...) Square of multiple correlation coefficient is: –proportion of variance accounted for by multiple regression
43
Multiple Partial Correlation Coefficient !
44
Autocorrelation
45
Purposes –Examine time series –Look at (serial) independence
46
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 …….
47
Autocorrelation Plot (Correlogram)
48
Many Correlation Coefficients
49
Many Correlation Coefficients: [ Behaviour of Sperm Whale Groups] NGR25LSSTSHITRLSPEEDAPROPSOCVSHR2LFMECSLAERR NGR25L1.00 SST0.121.00 SHITR-0.21-0.33*1.00 LSPEED0.10-0.28+0.061.00 APROP-0.15-0.34*0.070.181.00 SOCV-0.050.08-0.16-0.01-0.33*1.00 SHR2-0.18-0.120.01-0.200.19-0.031.00 LFMECS0.080.14-0.13-0.12-0.220.29+-0.181.00 LAERR-0.100.03-0.21-0.24-0.020.24-0.080.231.00 Listwise deletion, n=40; P<0.10; P<0.05; uncorrected Expected no. with P<0.10 = 3.6; with P<0.05 = 1.8
50
Many Correlation Coefficients: [ Behaviour of Sperm Whale Groups] NGR25LSSTSHITRLSPEEDAPROPSOCVSHR2LFMECSLAERR NGR25L1.00 SST0.121.00 SHITR-0.21-0.331.00 LSPEED0.10-0.280.061.00 APROP-0.15-0.340.070.181.00 SOCV-0.050.08-0.16-0.01-0.331.00 SHR2-0.18-0.120.01-0.200.19-0.031.00 LFMECS0.080.14-0.13-0.12-0.220.29-0.181.00 LAERR-0.100.03-0.21-0.24-0.020.24-0.080.231.00 Listwise deletion, n=40; P<0.10; P<0.05; Bonferroni corrected P=1.0 for all coefficients
51
Many Correlation Coefficients: [ Behaviour of Sperm Whale Groups] NGR25LSSTSHITRLSPEEDAPROPSOCVSHR2LFMECSLAERR NGR25L1.00 SST0.121.00 SHITR-0.21-0.33*1.00 LSPEED0.10-0.28+0.061.00 APROP-0.15-0.34*0.070.181.00 SOCV-0.050.08-0.16-0.01-0.33*1.00 SHR2-0.18-0.120.01-0.200.19-0.031.00 LFMECS0.080.14-0.13-0.12-0.220.29+-0.181.00 LAERR-0.100.03-0.21-0.24-0.020.24-0.080.231.00 Listwise deletion, n=40; P<0.10; P<0.05; uncorrected Pairwise deletion, n=59-118; P<0.10; P<0.05; uncorrected NGR25LSSTSHITRLSPEEDAPROPSOCVSHR2LFMECSLAERR NGR25L1.00 SST0.111.00 SHITR-0.17+-0.46*1.00 LSPEED0.05-0.170.051.00 APROP-0.05-0.20+0.040.31*1.00 SOCV-0.00-0.05-0.06-0.02-0.25*1.00 SHR2-0.15-0.130.07-0.140.050.011.00 LFMECS0.010.07-0.02-0.14-0.25*0.43*-0.26+1.00 LAERR-0.060.060.09-0.27*-0.20+0.06-0.060.21+1.00
52
Many Correlation Coefficients Missing values: –Listwise deletion (comparability), or –Pairwise deletion (power) P-values: –Uncorrected: type 1 errors –Bonferroni, etc.: type 2 errors
53
Beware! Correlation Causation Y 1 Y 2 Y 1 Y 3 Y 4 Y 2 Y 5 Y 1 Y 3 Y 2 Y 1 Y 3 Y 4 Y 1 Y 3 Y 4 Y 2 Y 5 Y 1 Y 3 Y 4 Y 5 Y 2 Y 6
Similar presentations
© 2024 SlidePlayer.com Inc.
All rights reserved.