Mark Breunig
Applied Multivariate Statistics in Fisheries and Wildlife Research
Canonical Correlation – Variable Group B
Expectations
This exercise will involve generating and analyzing one or more canonical correlations. Canonical correlations optimize the correlation between two sets of independent variables. I expect that habitats that have suitable cover such as woody debris and large woody debris will have the most diversity in the fish community.
Summary Statistics
Summary statistics can be found in Appendix A. This table was used to insure the data was imported into SAS correctly, and to identify any potential issues with the data. Based on this, it appears clearance has 1 null value.
Data Standardization
Data was standardized using SAS to have a mean of zero and a standard deviation of 1.
Correlation Matrix
The pearson correlation matrix can be found in Appendix B. Cross correlation is present for species such as small mouth bass and walleye, small mouth bass and rock bass, small mouth bass and blue gill, and small mouth bass and cyprinids. This may indicate these fish are members of the same functional guilds, or could be an artifact of the study design.
Results
Only one significant canonical correlation was found in the analysis. This explains only 68% of the variation in the dataset. The reminder of the observed variation is not explainable by any statistically significant canonical correlations.
Table 1. Significant (α=0.05) Canonical correlations
# / Canonical Correlation / Eigenvalue / Proportion / Cumulative / Ratio / Pr > F1 / 0.970 / 15.954 / 0.684 / 0.684 / 0.002 / 0.0442
Blue gills, cyprinids, walleye, and small mouth bass all loaded strongly into the canonical correlation (Table 2). None of the fish species were negatively correlated. Clearance, bole diameter, length of submerged portion of tree bole in water, and branching complexity were had strong positive correlations. Distance to shore and minimum water depth were negatively correlated.
Discussion
This canonical correlation is an indication that all of the fish species, especially blue gills, cyprinids, walleye, and small mouth bass are positively correlated to habitats that have deeper shoreline depths and large woody debris (high clearance and large bole diameter with most of the bole submerged) such as a riparian forest.
Table 2. Structure coefficients for each significant canonical correlation
Correlations Between the Fish Community and Their Canonical VariablesFish 1
LNBLGILL / 0.79
LNCYPRN / 0.78
LNWALL / 0.76
LNSMB / 0.70
LNPERCH / 0.52
LNRBASS / 0.30
Correlations Between the Habitat and Their Canonical Variables
Habitat 1
CLEAR / 0.89
BOLEDIA / 0.83
WETLEN / 0.78
LNCMPLX / 0.76
MAXWAT / 0.61
SLOPE / 0.17
DOMSUB / 0.12
FREEBR / 0.07
TBOLNUM / -0.24
DISTCWS / -0.32
MINWAT / -0.47
DISTSHR / -0.52
Conlcusion
As expected, large woody debris plays an important role in providing adequate fish habitat for a variety of species.
Appendix A: Summary Statistics
Variable N Mean Std Dev Minimum Maximum
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LNSMB 28 0.71 0.73 0.00 2.14
LNWALL 28 0.51 1.04 0.00 3.60
LNPERCH 28 0.28 0.65 0.00 2.40
LNRBASS 28 0.51 0.90 0.00 3.78
LNBLGILL 28 1.45 1.79 0.00 4.31
LNCYPRN 28 2.48 3.37 0.00 8.01
LNCMPLX 28 3.80 1.67 0.00 6.11
BOLEDIA 28 33.79 15.51 10.00 73.00
FREEBR 28 1.69 1.06 0.20 4.40
CLEAR 27 8.65 14.90 0.00 59.40
WETLEN 28 9.70 5.86 1.70 27.00
DISTCWS 28 0.94 1.39 0.00 5.00
DISTSHR 28 4.86 7.63 0.00 29.00
MINWAT 28 0.85 0.97 0.00 2.90
MAXWAT 28 2.61 1.13 0.60 5.20
TBOLNUM 28 7.36 6.49 1.00 28.00
SLOPE 28 1.03 0.19 0.59 1.32
DOMSUB 28 3.29 1.18 1.00 5.00
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Appendix B: Correlation Matrix
Pearson Correlation Coefficients
Prob > |r| under H0: Rho=0
Number of Observations
LNSMB LNWALL LNPERCH LNRBASS LNBLGILL LNCYPRN LNCMPLX BOLEDIA FREEBR
LNSMB 1.00000 0.52572 0.20319 0.46196 0.50334 0.37839 0.65295 0.52232 -0.05287
0.0041 0.2997 0.0133 0.0063 0.0471 0.0002 0.0044 0.7893
28 28 28 28 28 28 28 28 28
LNWALL 0.52572 1.00000 0.46819 0.54658 0.73044 0.68606 0.54621 0.68562 0.03671
0.0041 0.0120 0.0026 <.0001 <.0001 0.0026 <.0001 0.8529
28 28 28 28 28 28 28 28 28
LNPERCH 0.20319 0.46819 1.00000 0.66049 0.61090 0.51543 0.48389 0.64274 -0.25357
0.2997 0.0120 0.0001 0.0006 0.0050 0.0091 0.0002 0.1929
28 28 28 28 28 28 28 28 28
LNRBASS 0.46196 0.54658 0.66049 1.00000 0.60121 0.52817 0.47731 0.45486 -0.34853
0.0133 0.0026 0.0001 0.0007 0.0039 0.0102 0.0150 0.0691
28 28 28 28 28 28 28 28 28
LNBLGILL 0.50334 0.73044 0.61090 0.60121 1.00000 0.89236 0.74925 0.67393 -0.00905
0.0063 <.0001 0.0006 0.0007 <.0001 <.0001 <.0001 0.9636
28 28 28 28 28 28 28 28 28
LNCYPRN 0.37839 0.68606 0.51543 0.52817 0.89236 1.00000 0.65832 0.63410 -0.06234
0.0471 <.0001 0.0050 0.0039 <.0001 0.0001 0.0003 0.7526
28 28 28 28 28 28 28 28 28
LNCMPLX 0.65295 0.54621 0.48389 0.47731 0.74925 0.65832 1.00000 0.57750 0.07981
0.0002 0.0026 0.0091 0.0102 <.0001 0.0001 0.0013 0.6864
28 28 28 28 28 28 28 28 28
BOLEDIA 0.52232 0.68562 0.64274 0.45486 0.67393 0.63410 0.57750 1.00000 -0.03654
0.0044 <.0001 0.0002 0.0150 <.0001 0.0003 0.0013 0.8535
28 28 28 28 28 28 28 28 28
Appendix B: Correlation Matrix
Pearson Correlation Coefficients
Prob > |r| under H0: Rho=0
Number of Observations
LNSMB LNWALL LNPERCH LNRBASS LNBLGILL LNCYPRN LNCMPLX BOLEDIA FREEBR
FREEBR -0.05287 0.03671 -0.25357 -0.34853 -0.00905 -0.06234 0.07981 -0.03654 1.00000
0.7893 0.8529 0.1929 0.0691 0.9636 0.7526 0.6864 0.8535
28 28 28 28 28 28 28 28 28
CLEAR 0.49045 0.70213 0.43965 0.17536 0.62785 0.66229 0.58239 0.70506 0.00418
0.0094 <.0001 0.0218 0.3816 0.0005 0.0002 0.0014 <.0001 0.9835
27 27 27 27 27 27 27 27 27
WETLEN 0.51466 0.56306 0.58234 0.41043 0.62840 0.64879 0.69966 0.75024 0.02009
0.0051 0.0018 0.0011 0.0301 0.0003 0.0002 <.0001 <.0001 0.9192
28 28 28 28 28 28 28 28 28
DISTCWS -0.26319 -0.29361 -0.25061 -0.30659 -0.35679 -0.36416 -0.34261 -0.44256 0.00415
0.1760 0.1294 0.1983 0.1126 0.0624 0.0568 0.0743 0.0184 0.9833
28 28 28 28 28 28 28 28 28
DISTSHR -0.53203 -0.30968 -0.28316 -0.33706 -0.49024 -0.43551 -0.56076 -0.42348 0.31633
0.0036 0.1088 0.1443 0.0794 0.0081 0.0205 0.0019 0.0247 0.1010
28 28 28 28 28 28 28 28 28
MINWAT -0.49162 -0.17289 -0.27064 -0.27066 -0.40175 -0.37717 -0.43454 -0.34107 0.51013
0.0079 0.3790 0.1636 0.1636 0.0341 0.0479 0.0209 0.0757 0.0055
28 28 28 28 28 28 28 28 28
MAXWAT 0.35755 0.54447 0.21843 0.06931 0.43593 0.32059 0.48621 0.43642 0.52804
0.0618 0.0027 0.2641 0.7260 0.0204 0.0963 0.0087 0.0202 0.0039
28 28 28 28 28 28 28 28 28
TBOLNUM -0.32444 -0.13867 -0.02698 -0.13035 -0.20174 -0.17051 -0.02427 -0.12432 -0.23136
0.0921 0.4816 0.8916 0.5085 0.3033 0.3857 0.9024 0.5285 0.2362
28 28 28 28 28 28 28 28 28
SLOPE 0.07281 0.22631 0.02735 0.12041 0.22752 0.26708 -0.06094 0.25616 -0.20751
0.7127 0.2468 0.8901 0.5416 0.2443 0.1695 0.7580 0.1883 0.2893
28 28 28 28 28 28 28 28 28
DOMSUB 0.08431 0.13853 -0.15119 -0.13635 -0.00999 0.02543 -0.06304 -0.23893 0.06858
0.6697 0.4821 0.4425 0.4890 0.9598 0.8978 0.7500 0.2208 0.7288
28 28 28 28 28 28 28 28 28
CLEAR WETLEN DISTCWS DISTSHR MINWAT MAXWAT TBOLNUM SLOPE DOMSUB
LNSMB 0.49045 0.51466 -0.26319 -0.53203 -0.49162 0.35755 -0.32444 0.07281 0.08431
0.0094 0.0051 0.1760 0.0036 0.0079 0.0618 0.0921 0.7127 0.6697
27 28 28 28 28 28 28 28 28
LNWALL 0.70213 0.56306 -0.29361 -0.30968 -0.17289 0.54447 -0.13867 0.22631 0.13853
<.0001 0.0018 0.1294 0.1088 0.3790 0.0027 0.4816 0.2468 0.4821
27 28 28 28 28 28 28 28 28
LNPERCH 0.43965 0.58234 -0.25061 -0.28316 -0.27064 0.21843 -0.02698 0.02735 -0.15119
0.0218 0.0011 0.1983 0.1443 0.1636 0.2641 0.8916 0.8901 0.4425
27 28 28 28 28 28 28 28 28
LNRBASS 0.17536 0.41043 -0.30659 -0.33706 -0.27066 0.06931 -0.13035 0.12041 -0.13635
0.3816 0.0301 0.1126 0.0794 0.1636 0.7260 0.5085 0.5416 0.4890
27 28 28 28 28 28 28 28 28
Appendix B: Correlation Matrix
Pearson Correlation Coefficients
Prob > |r| under H0: Rho=0
Number of Observations
CLEAR WETLEN DISTCWS DISTSHR MINWAT MAXWAT TBOLNUM SLOPE DOMSUB
LNBLGILL 0.62785 0.62840 -0.35679 -0.49024 -0.40175 0.43593 -0.20174 0.22752 -0.00999
0.0005 0.0003 0.0624 0.0081 0.0341 0.0204 0.3033 0.2443 0.9598
27 28 28 28 28 28 28 28 28
LNCYPRN 0.66229 0.64879 -0.36416 -0.43551 -0.37717 0.32059 -0.17051 0.26708 0.02543
0.0002 0.0002 0.0568 0.0205 0.0479 0.0963 0.3857 0.1695 0.8978
27 28 28 28 28 28 28 28 28
LNCMPLX 0.58239 0.69966 -0.34261 -0.56076 -0.43454 0.48621 -0.02427 -0.06094 -0.06304
0.0014 <.0001 0.0743 0.0019 0.0209 0.0087 0.9024 0.7580 0.7500
27 28 28 28 28 28 28 28 28
BOLEDIA 0.70506 0.75024 -0.44256 -0.42348 -0.34107 0.43642 -0.12432 0.25616 -0.23893
<.0001 <.0001 0.0184 0.0247 0.0757 0.0202 0.5285 0.1883 0.2208
27 28 28 28 28 28 28 28 28
FREEBR 0.00418 0.02009 0.00415 0.31633 0.51013 0.52804 -0.23136 -0.20751 0.06858
0.9835 0.9192 0.9833 0.1010 0.0055 0.0039 0.2362 0.2893 0.7288
27 28 28 28 28 28 28 28 28
CLEAR 1.00000 0.71881 -0.21509 -0.32581 -0.30739 0.68055 -0.18304 0.26437 0.15139
<.0001 0.2813 0.0972 0.1188 <.0001 0.3608 0.1827 0.4510
27 27 27 27 27 27 27 27 27
WETLEN 0.71881 1.00000 -0.40509 -0.53417 -0.50468 0.40518 -0.16500 0.12084 -0.12787
<.0001 0.0325 0.0034 0.0062 0.0324 0.4014 0.5402 0.5167
27 28 28 28 28 28 28 28 28
DISTCWS -0.21509 -0.40509 1.00000 0.25907 0.32992 -0.28472 -0.06906 -0.18830 0.03878
0.2813 0.0325 0.1831 0.0864 0.1420 0.7269 0.3373 0.8447
27 28 28 28 28 28 28 28 28
DISTSHR -0.32581 -0.53417 0.25907 1.00000 0.80997 0.01531 0.18587 -0.14105 0.17002
0.0972 0.0034 0.1831 <.0001 0.9384 0.3436 0.4740 0.3870
27 28 28 28 28 28 28 28 28
MINWAT -0.30739 -0.50468 0.32992 0.80997 1.00000 0.15556 0.09769 -0.09932 0.17121
0.1188 0.0062 0.0864 <.0001 0.4293 0.6209 0.6151 0.3837
27 28 28 28 28 28 28 28 28
MAXWAT 0.68055 0.40518 -0.28472 0.01531 0.15556 1.00000 -0.25799 0.04456 0.17648
<.0001 0.0324 0.1420 0.9384 0.4293 0.1850 0.8219 0.3690
27 28 28 28 28 28 28 28 28
TBOLNUM -0.18304 -0.16500 -0.06906 0.18587 0.09769 -0.25799 1.00000 -0.03020 -0.03795
0.3608 0.4014 0.7269 0.3436 0.6209 0.1850 0.8788 0.8479
27 28 28 28 28 28 28 28 28
SLOPE 0.26437 0.12084 -0.18830 -0.14105 -0.09932 0.04456 -0.03020 1.00000 0.05026
0.1827 0.5402 0.3373 0.4740 0.6151 0.8219 0.8788 0.7995
27 28 28 28 28 28 28 28 28
DOMSUB 0.15139 -0.12787 0.03878 0.17002 0.17121 0.17648 -0.03795 0.05026 1.00000
0.4510 0.5167 0.8447 0.3870 0.3837 0.3690 0.8479 0.7995
27 28 28 28 28 28 28 28 28