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Plant Competition
Correlation
We are sometimes interested in how two variables are linearly related to one another,
particularly with observational data for which we are unsure of causal relationships
between the variables. The sample correlation coefficient (r) is a value that indicates
how strongly two variables are related to one another. This value ranges from -1,
indicating a strong negative relationship, to 1, indicating a strong positive
relationship. A value of zero indicates that the variables are not linearly related.
r=
Sxy
Sxx · Syy
é (å xi) 2 ù
ú
Sxx = å[xi ] - ê
êë n úû
é (å yi) 2 ù
2ù
é
ú = (0.79 2 + 0.59 2 + 0.48 2 ) - ê (1.86) ú = 1.206 -1.1532 = 0.0494
Syy = å[yi 2 ] - ê
ë 3 û
êë n úû
2
é (å xi)(å yi) ù
é
ù
ú = ((4 · 0.79) + (7 · 0.59) + (8.5 · 0.48) - ê (19.5)(1.86) ú = 11.37 -12.09 = -0.72
Sxy = å[xi · yi] - ê
n
3
ë
û
êë
úû
r=
Sxy
-0.72
=
= -0.9997
Sxx · Syy
(10.5)(0.0494)
xi = variable 1
yi = variable 2
n = sample size or number of samples taken or number of habitats
Example using the dissolved oxygen (x) and species diversity (y) from three sample sites:
Table 1. Dissolved oxygen (DO) and species diversity (H’) measurements.
r=
Habitat Type
DO (mg/L)
H’
Riffle
Run
Run
Total
4.0
7.0
8.5
19.5
0.79
0.59
0.48
1.86
Sxy
Sxx · Syy
é (å xi) 2 ù
2ù
é
ú = (4 2 + 7 2 + 8.5 2 ) - ê (19.5) ú = 137.25 -126.75 = 10.5
Sxx = å[xi 2 ] - ê
ë 3 û
êë n úû
é (å yi) 2 ù
2ù
é
ú = (0.79 2 + 0.59 2 + 0.48 2 ) - ê (1.86) ú = 1.206 -1.1532 = 0.0494
Syy = å[yi 2 ] - ê
ë 3 û
êë n úû
é (å xi)(å yi) ù
é
ù
ú = ((4 · 0.79) + (7 · 0.59) + (8.5 · 0.48) - ê (19.5)(1.86) ú = 11.37 -12.09 = -0.72
Sxy = å[xi · yi] - ê
n
3
ë
û
êë
úû
r=
Sxy
-0.72
=
= -0.9997
Sxx · Syy
(10.5)(0.0494)
So, as the dissolved oxygen increases we see a decrease in the species diversity at each site.
Thus, dissolved oxygen and species diversity at this study site are strongly negatively
correlated (r = 0.9997). However we must be careful, just because two variables are
correlated does not mean that a causal relationship exists; other variables are also
important in determining species diversity. Can you think of what may also affect species
diversity in the river? Another thing to note is that correlation is a measure of the linear
relationship between variables; two variables may be highly related to one another in a
nonlinear fashion as well.
Plant Competition Lab Activity
In this lab session, we will conduct an observational field study to detect interference
competition between species by correlating their size and interplant distance. You can
select the species as a group, but select common and similarly sized species such as sugar
maple, hemlock, or beech. We will use correlations to detect both the presence and
intensity of intraspecific and interspecific competition. Basically, we will measure the size
of a randomly located plant of species 1 and the distance to its nearest desired neighbor of
species 2.
Each group will sample one 30-meter long transect at least 5m apart from others’ transects.
Randomly locate the beginning point for your transect. Run a measuring tape 30m to the
end of your transect.
At each 6m transect interval, select the nearest tree of species 1. Record the species and
diameter at breast height (DBH) with the diameter tape for each overstory (canopy) tree
or sampling. DBH is defined as the outside bark diameter at breast height, 1.37m above the
forest floor, on the uphill side of a tree. For this measurement, the forest floor includes the duff
layer that may be resent but does not include woody debris that may rise above the ground
line. DBH is used as a measurement of tree growth, volume, yield, and forest potential. Next
locate and measure the distance to its closest intraspecific and interspecific neighbors.
Measure the DBH for these two trees as well. Record your data on the data sheet provided.
Make an effort to obtain equal numbers of intraspecific and interspecific airs, and do not
sample the same tree/sapling twice.
Species 1 DBH
Nearest Species 1
Nearest Species 2
Distance, DBH
Distance, DBH