215
No Cla t fo ssr r o Us om e
162 Spearman Rank Correlation Key Idea: The Spearman rank correlation is a test used to determine if there is a statistical dependence (correlation) between two variables. The Spearman rank correlation is appropriate for data that have a non-normal distribution (or where the distribution is not known) and assesses the degree of association between the X and Y variables (if they are correlated). For
PR E V ON IEW LY
the test to work, the values used must be monotonic i.e. the values must increase or decrease together or one increases while the other decreases. A value of 1 indicates a perfect correlation; a value of 0 indicates no correlation between the variables. The example below examines the relationship between precipitation and the number of plant species in southern Africa.
Data based on Global patterns in biodiversity Nature Vol 405, 11 May 2000
Spearman's rank data for number of plant species and precipitation
Site
No. plant species
Rank (R1)
1
60
60
2
30
150
3
40
240
4
70
330
5
120
410
6
50
450
7
160
550
8
280
500
9
150
610
10
320
520
11
340
750
12
140
910
13
400
400
14
550
550
15
570
1
Annual precipitation / mm
Rank (R2)
500
Difference (D) (R1-R2)
D2
Working space
rs value
7.5
Σ(Sum) D2=
Step two: Calculate the difference (D) between each pair of ranks (R1-R2) and enter the value in the table (as a check, the sum of all differences should be 0). Step three: Square the differences and enter them into the table above (this removes any negative values).
Step four: Sum all the the total into the table.
D2
values and enter
Analysing the data
Step five: Use the formula below to calculate the Spearman Rank Correlation Coefficient (rs). Enter the rs value in the box above.
5
1.00 0.89
6∑D 2
7
0.79
n -n
8
0.74
9
0.68
Spearman rank correlation coefficient
10
0.65
Step six: Compare the rs value to the table of critical values (right) for the appropriate number of pairs. If the rs value (ignoring sign) is greater than or equal to the critical value then there is a significant correlation. If rs is positive then there is a positive correlation. If rs is negative then there is a negative value correlation.
12
0.59
15
0.521
20
0.45
25
0.398
30
0.362
rs = 1 –
2 (a) Identify the critical value for the plant-precipitation data:
(b) State if the correlation is positive or negative:
(c) State whether the correlation is significant:
CIE2.indb 215
Critical value
6
1. State the null hypothesis (Ho) for the data set:
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Number of pairs of measurements
3
No Cla t fo ssr r o Us om e
Step one: Rank the data for each variable. For each variable, the numbers are ranked in descending order, e.g. for the variable, volume, the highest value 570 species is given the rank of 1 while its corresponding frequency value is given the rank of 7.5. Fill in the rank columns in the table above in the same way. If two numbers have the same rank value, then use the mean rank of the two values (e.g. 1+2 = 3. 3/2= 1.5).
KNOW 4/01/16 4:42 pm