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A Test for Comparing Diversities Based on the Shannon Formula

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J. theor. Biol. (1970)129,
151-154
A Test for Comparing Diversities Based on the Shannon Formula
Several indices of dispersion have been suggested by ecologists, the most commonly used being the measure corresponding to the entropy concept defined by
s
H = - C pi
ln
pi
i=l
where
pi
2 ,
Cpi
= 1. Thus the statistic is
h = -i h/n) ln Cniln)
and this can be regarded as an estimate of
H
being given a random sample of n observations from data falling into the s categories c,(l, 2,. . . , s) with probabilities
~~(1, 2,. . . ,
s). It is well known that h and
H
lie between zero and Ins, the latter being achieved when the observations (or category probabilities) are uniformly distributed (since x In x -+ 0 as x + 0 it is customary to take ni In ni as zero when 12i = 0). The distributional properties of h must be known before one can compare the diversities of populations. Much has been done recently toward establish- ing the distribution of h as well as other measures of diversity by Bowman, Hutcheson, Odum & Shenton (1969). Beforelooking at the distribution of h, we record the exact values of the first two moments of h. From Hutcheson (1969) and Bowman et al. (1969),
Eh
= In n --(n - l)P,(ln 2 - In 1) -
(::_:)P(lnn (“i’)
ln(n-l)+...+(-l)“-‘In1 >
(1)
where
P, = i p:.
j=l
151
152
K. HUTCHESON
For example, 11 = 2 Eh = (l-P,)In2 11 = 3
Eh
= -2(P,-P,)-ln2+(1-P,)In3. where B = integer part of (n-b-2)/2. These expressions can be used as long as n and s are not excessively large (say, S, n, up to about 200). Even with large computers the time to compute Var h can be significant. In order for one to make use of the moments of h in a practical situation one must find good estimates of Eh and Var h. From Bowman et al. (1969) the series expansions are, Eh=-Cpilnpi-(s-1)/2n+(l-Cp,~‘)/ 12n2+C(p;‘-p,y2)/12n3+. . . (3) Var h = [C pi ln2 pi - (C pj In pJ’]/n + (S l)/2n2 + +(-1+~p~~‘-~p,~11np~+~p,~‘~p~Inpi)/6n3+.... (4) It is shown, by Bowman et al. (1969), that the distribution of h is asymp- totically normal. If the categories are equiprobable the distribution is x2. It is well known that the sampling distribution of the statistic
z
G1-~2)-(P1 -P2>
(of/n, + a /n,)“’ is normal with mean 0 and variance 1. This is approximately true for sufficiently large values of nl and n2 even if the populations are not normal. If the values of c1 and g2 are not known and if it is known that each population is normally (or nearly normal) distributed, then the statistic obtained by substituting the estimates Var h, and Var h, for CJ~ and e2 yields a statistic which is approximately t-distribution with degrees of freedom d.f. = [Var h1 +Var h,]’ (Var h,)’ (Var h,)’ ____- + nl n2 As noted in section 1 exact calculations are very unwieldy for n, s, 2200. However, the equation (4) above gives a good approximation to the sample
LETTERS TO THE EDITOR
153 variance and asymptotically approaches the sample variance. Thus, h--h,-(C11P2) t = (Var h, + VBr h2) “’ is asymptotically t with d.f. given above. Using data from Stoddard & Norris (1967) we will compare the diversity of the bird casualties of January 1964 to that of January 1965. The number of individuals killed in January 1964 was 57 with 16 species represented. The number of individuals killed in .January 1965 was 23 with 11 species represented. For 1964, II, = 2.44155 Vdr 11, = 0.0148724 for 1965, II, = 1.80785 Var 11~ = 0.0620399 with 35 degrees of freedom. We would reject the hypothesis of equal diversities at the 5 “J level. The true variance for 1965 was found to be Var 11~ = 0.0547087, thus the above test is conservative. The data from Table 1 of Pielou (I 966) gives for plot 6 earlier, species = 6, individuals == 496. I1 = 0.97931 plot 6 later, species = 6, individuals := 443, /I = 1.02472. To test for a difference in diversity we have 1.02472 - 0.9793 1 ’ = (0~0012556+0~0013055)“2 = 0.8973 with degrees of freedom large. Thus, we cannot reject the hypothesis of equal diversity. I would like to acknowledge the help of Professors L. R. Shenton, J. B. Douglass and M. C. Carter.
Department of Statistics and Institrrte of Ecology, University of Georgia,
KERMIT HUTCHESON
Athens, Georgia, U.S.A. (Received 27 February 1970)
154
K. HUTCHESON
REFERENCES
BOWMAN, K. O., HUTCHESON, K., ODUM, E. P. & SHENTON, L. R. (1969). Znrernational Symposium
on
Stutisticd Ecology.
Vol. 3. University Park: Pennsylvania State University Press.
HUTCHESON,
K. (1969). Ph.D. Dissertation.
PIELOU, E. C. (1966). J. theor. Biol. 10, 370. STODDARD, H. L. & NORRIS, R. A. (1967). Bull. Tall Timbers Res. Stn 8, 1.

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