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values are consistent in stepwise, logistic or exponential population growth and are not
especially affected in diverse mutation models (Kimmel et al., 1998). Two methods were
employed to determine the statistical significance of the  (ln ) obtained in diverse pink river
dolphin populations. The first was to apply a jackknife procedure (Efron & Tibshirani, 1993)
to obtain the variance of  and, with this variance, a Student's t test and 95 and 99 %
confidence intervals were estimated. A second procedure was throughout empirical
distributions of ln  from 500 coalescence simulations with a  = 5. In this case, a 95 %
confidence interval was determined (- 0.23, 0.25).
A second test employed was that from Zivothovsky et al. (2000), which calculates an
expansion index:
S k = 1 - ((K - (R k V/2)/5V 2 ),
where K and V are the unnormalized kurtosis (fourth central moment) and the allele size
variance is estimated from a sample and corrected for sampling bias, respectively, whereas R k
= k m / 2 m (they are the kurtosis and the variance in the repeat number mutational changes).
The expressions used to estimate V and K are:
V =  i = 1…n p i (X i - X) 2 and K =  i = 1…n p i (X i - X) 4 , where X =  i = 1…k p i k, and k
represents the alleles in a locus given and p i , the allele frequencies. All the other terms were
defined in the previous analysis. The value of R k employed was 6.3 because this value was
obtained for dinucleotide microsatellites by Dib et al. (1996), and because dinucleotide
microsatellites were used in the current study. Feldman et al. (1999), used the same data and a
geometrical distribution of mutational events, and obtained an estimated  2 m of 2.5, which is
basically the same as what was obtained by using a truncated Poisson distribution
(Zhivothovsky et al., 2000) and by myself in the present work ( 2 m = 2.45) The value of S k is
expected to be 0 in a general symmetric stepwise mutation model for a population in
equilibrium and of constant size (this was derived by Zhivotovsky & Feldman, 1995). The S k
is positive if an expansion affected the population and is contrarily negative if a bottleneck
affected the population. To obtain demographic conclusions of this analysis, the within-
population variance and the expansion index are averaged for all the microsatellites studied
within each population and their dynamics are compared. Zhivotovsky et al. (2000) showed
that a significant correlation existed between V and S k (r = 0.58) for a human data set, but this
correlation was moderate and, in fact, both statistics could react differently to the changes in
population size and have different patterns in different populations. Noteworthy consequences
could be extracted from the differential behavior of both of these statistics in a given
population. To measure the statistical significance of the S k values, two methods were carried
out. First, a bootstrap over loci (10,000 runs) was completed. Secondly, such as in the first
test, a jackknife procedure was performed to obtain the variance of S k and, with this variance,
a Student's t test and a 95 % confidence interval were estimated.
Another procedure used to detect any possible reduction in population size was that
created by Garza & Williamson (2001). This procedure is based on the ratio M = k/r, where k
is the total number of alleles detected in a locus given and r is the spatial diversity; that is, the
distance between alleles in number of repeats and the overall range in allele size. When a
population is reduced in size, this ratio will be smaller than in equilibrium populations. To
calculate this M value, the program will simulate an equilibrium distribution of M and give
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