Statistical inference on measures of niche overlap

. Estimates of measures of niche overlap are often reported without any indication of sampling variance or an accompanying confidence interval. We have investigated the delta, Jackknife, and bootstrap methods for making statistical inferences on four measures of niche overlap: the coef- ficient of community, Morisita's index, Horn's index, and the Euclidian distance. Our qualitative conclusions are: (1) The bias of these estimators was usually < 10% of the mean unless the sample size was small and the number of resource categories large. The jackknife and bootstrap can significantly reduce this bias. (2) The variance of the bootstrap and jackknife estimators was usually greater than that of the "standard" estimator. (3) Under a variety of circumstances, the population sampled may actually represent several unrecognized subpopulations. In such cases confidence intervals generated by the jackknife and delta techniques can be quite inaccurate, while the nonparametric confidence intervals derived from the bootstrap are highly accurate.


INTRODUCTION
Until recently little attention has been paid to the statistical properties of niche overlap measures (Petraitis 1979, Ricklefs and Lau 1980, Maurer 1982, Smith and Zaret 1982. Many of these overlap measures are complicated functions of the underlying observations. It is therefore not surprising that most ecologists have not attempted to estimate the variance or construct confidence intervals for these measures of overlap (Orians and Horn 1969, Pulliam and Enders 1971, Jaksic et al. 1981, Paine et al. 1981. Ricklefs and Lau (1980) have examined the bias and variance of four measures of niche overlap by sampling theoretical populations on a computer. They suggest that empirical workers conduct computer simulations using their own data to gain insights concerning the sampling variance and distribution of these overlap measures. In fact, in recent years the use of computer simulations has allowed the development of a growing body of "computer intensive" statistics (Efron 1979b), which sidestep the mathematical difficulties encountered in analyzing many statistics having non-normal distributions. We shall be examining one example of these techniques, the "bootstrap" estimator (Efron 1979a).
In addition we will derive analytic approximations for the variance of the overlap indices studied by Ricklefs and Lau. Using this variance estimate we will also describe a method for constructing confidence intervals. The methods involved in these analytic approximations come under the general heading of the "delta method." This method is based on Taylor series approximations. Smith and Zaret (1982)  in general, bias will be greater when many resource categories are used rather than a few. Smith (1982) has also used the delta method to derive the approximate sampling variance of several measures of niche breadth and has shown these approximations to be quite accurate.
We will describe a third method, the jackknife, for both point and interval estimation. We examine statistical properties of all three methods by conducting extensive computer simulations. These computer simulations allow us to evaluate the accuracy of these methods, since each involves approximations. We also use the results from the computer simulations as a means of discerning if one method is superior to the other.

THE DELTA METHOD
We are interested in calculating the niche overlap of two species (populations) which utilize up to n different resources. Let the probabilities of resource utilization for the first species be given by p, P2, . . ., Pn and for the second species q, q2, . . , qn. With these population quantities we can calculate measures of niche overlap in a variety of ways (Schoener 1970, Pianka 1974, Hurlbert 1978, Petraitis 1979, Ricklefs and Lau 1980, Maurer 1982). We will restrict our attention here to the four measures considered by Ricklefs and Lau (1980): coefficient of community (S.), Morisita's index (S2), Horn's index (S3) and the Euclidian distance (S4).
In addition to obtaining an estimate of Sj we would like to be able to estimate the sampling variance of each Sj. It is possible to derive the approximate sampling variance of the Sj using the delta method (see Bishop et al. 1975:486-488). This method yields the Although the method of estimating the Sj just described is straightforward it may not be the best available. Before considering alternative methods of estimation we will make some analytic statements about the S estimators. In particular we would like to know if E(S,) = $j, i.e., is the estimator unbi Horn's index is a concave function that has 025/O 02S/Oq 2 + 2a2S/ap0aq <0 for all i. Consequently we can invoke Jensen's inequality to prove that E(S3) ' S3 (see Karlin and Taylor 1975:249). This result does not tell us the magnitude of the bias but it does say that it will always be negative when it is not 0. This fact explains the persistent negative bias in the simulations of Ricklefs and Lau (1 980: Table 1). It is possible to estimate the magnitude of the bias for S2 and S3 by expanding these functions in a Taylor series about the points (pl, . p. , pn) and (ql, . .., qj). This technique has been used by Smith and Zaret (1982) who give approximations for the bias of 52 and S3. An exact expression for the bias of S9 is also given in Smith and Zaret (1982). van Belle and Ahmad (1974) have previously derived the delta estimate of variance for Morisita's index.

THE JACKKNIFE
A second method of estimating the indices (Eq. 1) is the jackknife statistic (see Miller 1974 for a review).
The jackknife has been applied to an ecological diversity index (Zahl 1977), population size estimation (Burnham and Overton 1979), and genetic distance estimates (Mueller 1979). The jackknife provides an easy method for estimating the overlap indices and their variance. The jackknife can also be useful in reducing bias. When the bias takes the form a/N + O(N2), then the jackknife will remove the first term of the bias.
Let S_-i be the same as before, except that the ith observation has been deleted before the calculation.
Since we assume we have made N. observations for variable are known exactly, there is less to be gained from the bootstrap. However, we expect that the ben-efits of the bootstrap will increase as the differences between the assumed distribution and the real one increase. We will illustrate this technique with the overlap measures already discussed.
Suppose we have the estimates j and q from independent samples of size Ni (i = 1, 2). We produce one bootstrap replicate as follows. Using a random number generator we take two independent multinomial samples of size Ni where the underlying probabilities for each sample are f and q, respectively. We designate the sample so obtained p1*, ql*. We then substitute these values in Eq. 1 and obtain a bootstrap estimate S*i , In the present study this process was repeated 1000 times. We obtain bootstrap estimates of the variance and bias of S, as,  6) In addition to estimating the bias of various overlap measures, we will use the 1000 replicates of each S, to approximate the cumulative distribution function. A Bayesian justification for this approach is given in Efron (1981). Once the cumulative distribution function has been estimated, confidence intervals can be derived. To describe the procedure for estimating confidence intervals, we first introduce some notation which closely follows Efron (1981). Let our estimate of the cumulative distribution function for Si, CDFi(t), be If there is substantial bias in Si, th interval generated by Eq. 7 will be biased in a similar fashion. Efron (1981) has described a method for correcting this bias which we now describe. Let cI( ) be the cumulative distribution function of a standard normal random variable. We can now define z, = Ib-CDF (Si) and t(zx) = a. A bias-corrected 1-2a confidence interval is then given by [CDF-1('(2z0 + zD), CDF-1('(2z0 + z1_))]. (8) Obviously if the estimated bias is 0, then from Eq. 6, Si* = Si and the confidence intervals produced from Eqs. 7 and 8 will be the same.
It is highly unlikely that the estimators, S, and S, and S/*, are equivalent. A preference for one of these estimators over the other should be based on statistical criteria. The criteria we have chosen to examine are the bias, variance, and mean-squared error (MSE), which equals the variance plus the squared bias, of each estimator in addition to the associated confidence intervals. We would prefer the estimator with the smallest value of all these quantities. In practice the variance of $~ and Sj will be used to construct a confidence interval. The confidence interval on Sj* is determined nonparametrically. The accuracy of the confidence intervals on S3 and $j will depend on the accuracy of the was taken for each species to estimate p, and q1.
2) Sj, $j, var(Sj), and var(S) were calculated and a 95% confidence interval was constructed for each estimator, i.e., S$j ? t2N-2,0.025 (var(S.))''2. These examples are identical to ones used by Smith and Zaret (1982) and the underlying pi's and q,'s are given in Table 2. For each example we let N = 20, 60, and 200. Only the results for N = 20 are shown in Table   2 because the results for N = 60 and 200 are qualitatively similar.
Perhaps the most restrictive assumption used in both the jackknife and delta methods is that the count vector has a multinomial distribution. To examine how our estimators behave when the underlying distribution is not multinomial we conducted several additional simulations. In particular, we considered a population that was "contaminated" by a second subpopulation which samples the environment differently. For instance, suppose we were determining the overlap in diets of two species of lizards. Individuals were sampled at random and their gut contents examined. We might then classify the insects into various genera. Usually the results from each lizard would be pooled to get our final estimate of p and q. It is possible, however, that some fraction of the individuals we sampled were actually choosing items with different probabilities from the rest of the population (see Mosteller and Tukey 1977: 17, for a brief discussion of contaminated distributions). We have studied this problem with the following sort of numerical simulations. We restricted ourselves to two categories. We assumed a base population for each species composed of individuals which sample the environment with probabilities pI and qP. We called these individuals type I. We also assumed a fraction C of all individuals sampled the environment with probabilities A, and q1. We called these individuals type II. Random samples from this population were generated as follows. Ten individuals of each species were sampled and the probability that type II individuals were included in the sample was C. Each individual was then given its own sample of 20 resource items; the probabilities used to generate these samples depended on whether a given individual was type I or type II. Since there were only two resource categories, each individual in the sample could be characterized by the frequency of category one items it contained in its sample of 20 resource items. For species one, let the frequencies of resource item one for each individual be x,, x2, . . , x,0 and for species two, Y., Y2, . * *, Y1o We conducted the bootstrap sampling on this set of data while keeping in mind that we can no longer distinguish type I from type II individuals. Each bootstrap replicate sampled 10 individuals, with replacement, for each species. From each individual a sample of 20 resource items was chosen: the probability of the item being from category one was determined by the par- * Percent of all cases in which the jackknife had a smaller bias, variance, and mean squared error than the delta estimator, and percent of all cases in which the confidence level of the jackknife measure was closer than the delta confidence level to 95%. t Percent of all cases in which the jackknife estimate of variance was closer than the delta estimate to the observed value.
t Percent of all cases in which the bias was significantly greater than 0 is listed for both the jackknife and delta method.

RESULTS
A summary of the results for the 25 two-category simulations which examined the jackknife and delta estimators only is given in Table 1. From examination of Table 1 it is clear that the jackknife is effective at reducing the bias for S2 and S3. This result is consistent with the form of the bias for S2 and S3 and the known properties of the jackknife. These same two overlap statistics also seem to have significant bias in a greater Although the jackknife has a decided advantage in bias reduction, its variance and mean-squared error are larger than the delta method's. The data are not shown in Table 2, but results from additional computer simulation show that the delta estimator has more accurate confidence intervals at sample sizes of 20 and 60 but loses this advantage at the highest sample size (200).
In Table 3 we present the results of the simulations involving various degrees of contamination (0, 0.10, or 0.25). Since the results for the Euclidian distance were identical to the coefficient of community, only the latter results are presented. The most striking result in Table 3 is that even low levels of contamination can severely affect the accuracy of confidence intervals produced by either the jackknife or delta methods despite the large sample sizes. In marked contrast are the confidence intervals of the bootstrap, which are exceedingly accurate. There are also a few instances where the jackknife or delta estimators are biased while the bootstrap is unbiased (within the accuracy of our simulations). The bootstrap invariably has a higher variance and MSE: however, it is larger by only 3-11%. Ricklefs and Lau (1980) suggested that computer simulations be used to estimate confidence intervals there is a cost to reducing bias in the form of increased variance of the estimator.

DISCUSSION
A basic assumption in this and other work (Ricklefs and Lau 1980, Smith 1982, Smith and Zaret 1982 is that the count vector of niche categories has a multinomial distribution. We have considered deviations from this assumption by examining populations in which one multinomial distribution is contaminated with a second to various degrees. There are a variety of circumstances where such distributions are likely to be found. If there is genetically based variation for resource preference, then the resulting population might be a mixture of several multinomial populations. Contaminated distributions might also arise if individual resource use is dependent on some form of preconditioning or experience. In any case, the bootstrap has a tremendous advantage when there is a contaminated distribution. This advantage is due to the nonparametric confidence intervals that the bootstrap generates. Consequently, if it seems unwise to make the multinomial assumption for a given data set, the bootstrap would be the best method for estimating niche overlap. Implementation of the bootstrap requires access to computing facilities, while the delta and jackknife techniques can be carried out with calculators.