Is it appropriate to compare the coefficient of variation between discrete (e.g., clutch size) and continuous data (e.g., egg width)?
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1$\begingroup$ Appropriate for what purpose? And how do you propose to make the comparison? $\endgroup$– whuber ♦Commented Jul 17, 2015 at 13:32
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$\begingroup$ A simple comparison to say, "clutch size was more variable than egg width in our population." Does this clarify my question? $\endgroup$– user82580Commented Jul 17, 2015 at 13:40
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$\begingroup$ Theoretically, a species should optimize their egg size and vary clutch size; therefore, there is a tradeoff between egg and clutch size. Some people essentially look at the coefficient of variation to determine if clutch size is more variable than egg size within a population to confirm the optimal egg size theory. My question is related to the appropriateness of making a comparison between a discrete and continuous variables'coefficient of variation. $\endgroup$– user82580Commented Jul 17, 2015 at 13:48
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There is a great variety of the different discrete variables: nominal, ordinal, and count. Gender is nominal: you cannot attach a meaningful number to gender. Strongly disagree/disagree/agree/strongly agree is ordinal: they represent discretization of an underlying spectrum. You can code them as 0/1/2/3, or 1/2/3/4, or -2/-1/1/2, with an understanding that higher values of the coded variable represent the greater strength of agreement, although the difference between 1 and 2 may not be the same as the difference between 2 and 3. Clutch size is a count variable: the value of 1 is meaningful, and the differences between 2 and 3 is the same difference as between 8 and 9. For most purposes, that works pretty much the same way as measuring egg width: the difference of 1cm between 1.5 cm and 2.5 cm is the same as between 3.2 cm and 4.2 cm. For both clutch size and egg width, the value of zero (nothing) is meaningful. So I would not see any objections to at least computing the CV for either variable.
Now, whether the substantive interpretation you offer would hold water should be an argument. For one thing, you have not convinced me! You would need to find sufficiently different nesting conditions to argue that a species reacts to the change in the environment by increasing CV of one or the other variable -- or, if you at the liberty of doing so, set up an experiment with whatever conditions are appropriate (dry vs. wet, warm vs. cold, plenty fish vs. no fish -- you know better.)
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1$\begingroup$ I wonder about the equations you are making in this answer. The total energy expended by a bird to make a clutch would be pretty close to the total mass. In this sense, a difference of egg widths from $1.5$ to $2.5$ cm corresponds to a mass change of a factor $(2.5/1.5)^3 = 4.6$ whereas a change from $3.2$ to $4.2$ is a factor of $(4.2/3.2)^3 =2.3$: it's only half as big. This suggests abandoning CV as having any kind of meaningful relationship to the underlying biology and trying to exploit biological understanding to express the variables appropriately and make useful comparisons. $\endgroup$– whuber ♦Commented Jul 17, 2015 at 16:02
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$\begingroup$ Back to my original question, can you compare CV between clutch size (count data) and egg width (continuous data) as a technique to test if a population is varying clutch size rather than egg size? There have been several papers that use this as a method to test the optimal egg size theory, but it seems inappropriate to me to compare dispersion this way. $\endgroup$ Commented Jul 17, 2015 at 18:07
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1$\begingroup$ @whuber, great comment. I also wondered whether the "natural" CV of the egg size need to be comparable to the "natural" variation in clutch size, but could not find a good way to express my concern (other than the second paragraph). Now, your simple example shows that the CV is not even scale independent (as I am used to in my economics work with figures like employment or revenue). $\endgroup$– StasKCommented Jul 19, 2015 at 18:18