By Thomas C. Grubb, Jr. , Department of Zoology, Ohio State University
The Boxwood Press; Pacific Grove; 1986
(Used with Permission)
The Chisquare 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 _{X}^{2} 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 givingup 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 givingup 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 givingup times, as shown in the two lefthand columns of Table A2.2. Notice that some tied values occur in Table A2.1; there are two 30second catch times and three 90second givingup 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.
Some possible average catch times and givingup times of Great Blue Herons fishing in 15 different habitats and/or on15 different ways.
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 righthand columns of Table A2.2. Find S Difference between ranks and S Difference^{2} 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.
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 Difference^{2}, and to subtract the result from 1. Substituting our heron S Difference^{2} 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 givingup 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 givingup 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 givingup time is caused by catch time.
Those knowledgeable in statistics might claim that we could have used a "onetailed" rather than a "twotailed" 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 twotailed correlation will also be significant in a onetailed 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 Difference^{2} would give r = 1, 0, and +1? How would catch times and givingup times have to be ranked relative to each other to produce these three values of Difference^{2}?
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 Difference^{2}.
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.





















































