Beyond Birding - Field Projects for Inquisitive Birders

By Thomas C. Grubb, Jr. , Department of Zoology, Ohio State University

The Boxwood Press; Pacific Grove; 1986

(Used with Permission)

Birds in the Classroom
The Bird Groups
Back to Home
Glossary
Themes
Bibliography


Preface

Chapter One - Ornithology as a Science

Chapter Two - Analytical Ornithology

Chapter Seven - When Do Great Blue Herons Give Up?

Chapter Fifteen - What Determines Individual Distance?

Chapter Twenty Three - What To do When You Know this Book

Appendix 2 - The Spearman Rank Correlation Test

The Chi-square Test and the Median Test described in Appendix 1 can tell us when one set of numbers differs from what we might expect by chance, or when two sets of numbers are more different from each other than expected by chance. Suppose, though, that we have a hypothesis which states that two kinds of quantities vary together in some predictable way. For example, the longer the wing length of any subspecies within a given species, the greater the distance it migrates, or the older the male Ruffed Grouse, the more females he mates with per breeding season. These are called positive correlations. The two quantities vary together such that as one gets larger, so does the other. Negative correlations can also be hypothesized where as one quantity gets larger, the other gets smaller. For example, as the average daily temperature during winter increases, the length of time that Northern Cardinals spend looking for food decreases, or the more turbid the water, the fewer fish Belted Kingfishers catch per hour.

We need some way of assessing whether the two quantities we measure actually are varying together as predicted by our hypothesis, or whether any apparent correlation produced is just due to the amount of variation in results we could expect from chance happenstance. Several analytical statistics are available which test the strength of a correlation. One of these, the Spearman Rank Correlation Test, will be used for this book's projects.

When applied to any two sets of results, the Spearman Test produces a Spearman Correlation Coefficient, r. This r can take values between -1 and + 1. When r = -1, we have two sets of numbers that have a perfect negative correlation. That is, without exception, as the value of one quantity in our sample becomes larger, the value of the second quantity gets smaller. Similarly an r = +1 indicates a perfect positive correlation. Without exception, every larger value of one quantity is accompanied by a larger value of the second quantity. If r varies between -1 and + 1, what does r = 0 mean? It means that there is no correlation between the two quantities. They are completely independent of one another. If we tested the correlation between the number of songs per minute in Scarlet Tanagers and the number of bagel stands in the nearest city, we would probably calculate an r close to 0. In most cases, r will not equal -1, 0, or +1; it will be somewhere in the middle. Say we analyzed the correlation between the rate at which Cattle Egrets walked through a pasture looking for grasshoppers and the height of the grass, and we found that t = -0.37. We might suspect that the higher the grass, the slower the birds walked, but was the negative correlation strong enough, close enough to -1, to rule out the possibility that - 0.37 was due to chance? We want to know whether r= -0.37 is statistically different from r = 0. As part of the Spearman analysis we can compare calculated r with r in a table of values expected solely due to chance, just as we did with X2 values in Appendix 1.

Let's work through a Spearman Rank Correlation Test of some imaginary results from the fishing behavior of Great Blue Herons described in Chapter 7. Our hypothesis was that herons used average catch time to tell them what their giving-up time should be. We predicted that if our hypothesis were correct, then over a large range of habitats and seasons of the year, we should see a positive correlation between catch time and giving-up time. Now we can add that we predict r to be significantly greater than 0.

Let's assume that we do the study and determine the 15 pairs of average values shown in Table A2.1. You might want to construct a scattergram of these made up results, as describe in Chapter 7. The first step in the Spearman Analysis is to rank the values within each category from smallest to largest. Here, we assign ranks 1 through 15 to catch times, and do the same for giving-up times, as shown in the two left-hand columns of Table A2.2. Notice that some tied values occur in Table A2.1; there are two 30-second catch times and three 90--second giving-up times. In these cases, we assign the mean rank to each of the tied scores, and the next higher score in the list receives the rank normally assigned to it. Thus, to values 30, 45, 60, 60, 60, and 90 seconds, we would give the ranks, 1, 2, 4, 4, 4, and 6.

Table A2.1.

Some possible average catch times and giving-up times of Great Blue Herons fishing in 15 different habitats and/or on15 different ways.

Habitat and/or day
Average catch time (in seconds)
Average giving-up time (in seconds)
1
122
107
2
66
32
3
79
61
4
94
111
5
43
48
6
64
72
7
30
34
8
91
76
9
110
119
10
76
90
11
61
90
12
104
59
13
67
64
14
30
46
15
92
90

The next step in the Spearman Test procedure is to find the difference between rank for each pair of numbers and then square that difference, as shown in the two right-hand columns of Table A2.2. Find S Difference between ranks and S Difference2 between ranks as shown at the bottom of Table A2.2. Although we will have no further use for S Difference, it is a good idea to calculate it anyway. Since S Difference must equal 0, it can give you a check on your arithmetic. If your S Difference does not sum to 0, you will want to go back and discover your error.

Table A2.2. computational procedures for calculating Spearman's r

Habitat and/or day
Rank of average catch time
Rank of average giving-up time
Difference between ranks
Difference between ranks squared
1
15
13
2
4
2
6
1
5
25
3
9
6
3
9
4
12
14
-2
4
5
3
4
-1
1
6
5
8
-3
9
7
1.5
2
-0.5
0.25
8
10
9
1
1
9
14
15
-1
1
10
8
11
-3
9
11
4
11
-7
49
12
13
5
8
64
13
7
7
0
0
14
1.5
3
-1.5
2.25
15
11
11
0
0

S Diff. = 0

S Diff. = 178.5

To find Spearman's r we will use the equation:

In words, this equation directs us to multiply the number of pairs (n) times the square of the number of pairs minus 1, to divide this quantity into the number obtained by multiplying 6 times S Difference2, and to subtract the result from 1.  Substituting our heron S Difference2 from Table A2.2 in the equation gives us:

 

So r = 0.682. We now wish to test the null hypothesis that there is no relation between catch time and giving-up time, that our calculated r of 0.682 is just due to a chance outcome. As we did in Appendix 1, we will use the 5% level of rejection in comparing our calculated r with that expected solely due to chance. Table A2.3 lists the critical values of r at the 5% level of significance for various numbers of pairs. The r value in Table A2.3 corresponding to a sample size of 15 pairs is 0.62. Our calculated r of 0.682 is larger, so we reject the null hypothesis. Since the positive correlation between catch time and giving-up time in Great Blue Herons has less than a 5% probability of being due to chance alone, we have proven true our prediction, and we have failed to reject the alternative hypothesis that giving-up time is caused by catch time.

Those knowledgeable in statistics might claim that we could have used a "one-tailed" rather than a "two-tailed" test, since we predicted the direction of the correlation. For the sake of clarity and simplicity, I have chosen to ignore the distinction. In any case, a significant two-tailed correlation will also be significant in a one-tailed analysis.

Doing a little exercise will help you gain insight into the way r is related to the ranking procedure. In our heron example, what values of Difference2 would give r = -1, 0, and +1? How would catch times and giving-up times have to be ranked relative to each other to produce these three values of Difference2?

In summary, to carry out the Spearman Rank Correlation Test:

1. Rank the values of each variable from smallest to largest.

2. Find the difference between ranks for each pairs of numbers. Square these differences and sum them to obtain Difference2.

3. Use the equation to calculate Spearman's r.

4. Compare your calculated r with the r value in Table A2.3 corresponding to your numbers of pairs. If calculated r is equal to or greater than tabulated r, reject the null hypothesis at the 5% level of confidence. If calculated r is less than tabulated r, reject the alternative hypothesis at the 5% level of confidence.

Table A2.3
Number of pairs
Values of r
5
1.00
6
0.94
7
0.89
8
0.83
9
0.78
10
0.75
11
0.73
12
0.71
13
0.67
14
0.64
15
0.62
16
0.60
17
0.58
18
0.56
19
0.55
20
0.53
21
0.52
22
0.51
23
0.50
24
0.48
25
0.47
26
0.46
27
0.46
28
0.45
29
0.44
30
0.43