SC/64/NPM9

Distribution of correlation coefficients between θ and FIS in
mixed samples of two distinct stocks: Comments to Waples (2011)
Naohisa Kanda and Hiroshi Hatanaka
The Institute of Cetacean Research, 4-5, Toyomi-cho, Chuo-ku, Tokyo 104-0055, Japan

Contact email:kanda@cetacean.jp

ABSTRACTWaples (2011) presented very elegant analyses based on standard population genetics theory to test thetwo-stock hypothesis (i.e., O and J stocks in Hypotheses A and B) proposed for the North Pacific minke whales
around Japan. Logics of the analyses were that, when only two distinct stocks exist in a given area, the largest
departures from equilibrium (described as FIS) or higher linkage disequilibrium should be seen at the loci or pairsof loci that show the largest or strong allele frequency differences between the two distinct stocks (described asθ). Because the observed relationships of these genetic indices in the samples of the North Pacific minke
whales from SA7 bycatch, SA7-Kushiro, SA7-Sanriku, SA2, and SA11 were weaker than the expected
relationship estimated from artificial mixtures of only putative J and O individuals, it was suggested the samples
might have contained individuals from more than two stocks. However, it was also indicated that therobustness of the analyses should be evaluated because this approach was novel and the behaviors of these
indices were unclear under various situations such as different samples sizes. In this document, using computer
simulation, we examined the effect of different sample sizes to the distributions of the correlations of thesegenetic indices.
Among the results presented in Waples (2011), we only examined the correlation between θ and FIS in 1:1
mixture of individuals from the two distinct stocks that reflected the situation of the bycatch sample from SA7(J=93, O=90, unclassified=29) because we thought the argument to this single case should cover the other cases.
We generated 100 genetic data that contained genotypic data at 16 loci through computer simulations. We
examined distribution of the correlation coefficients under the three different cases in regard to the sample size:
5000 (same as simulated Ne, total sample size=10000), 400 (SA6 and SA9, 800), and 100 (SA7 bycatch, 200)
individuals from each of the two putatively distinct stocks that were differentiated each other with FST values
around 0.05. This level of genetic differentiation was comparable to that previously estimated between theputative J and O stocks (Kanda et al., 2009). After the data generation, we look at the distribution ofcorrelation coefficients between θ and FIS of the 16 loci among the 100 data in each of the three cases. The
correlation coefficients distributed as expected from Waples (2011) in the cases of 5000 and 400, but fluctuatedquite widely and some of the values were quite low in the case of 100. The results of our simulation exercises
indicated that lower level of the relationships observed in the SA2, SA7, and SA11 samples in Waples (2011)
could be due to the small sample size rather than due to mixture of the individuals from more than two stocks.
The conclusion of Waples (2011) therefore needs further evaluation to be used for the evidence against
Hypotheses A and B.
KEY WORDS: NORTH PACIFIC MINKE WHALE, SIMULATION, O STOCK, J STOCK,IMPLEMENTATION ASSESSMENT

INTRODUCTION
We have presented the evidence of statistically significant departures from the expected Hardy Weinberg
genotypic proportions with homozygous excess in the samples from SA7 (including bycatches and scientificcatches) as one of the supportive results for that minke whales in the area are the mixture of the two stocks, J andO, rather than an additional intermediate stock (Kanda et al., 2010). Waples (2011) indicated based on standard
population genetics theory that if there are only two stocks, loci with the largest allele frequency differences
between them (described as θ) should show the largest departures from equilibrium (described as FIS), andlinkage disequilibrium should be higher for pairs of loci that show strong allele frequency differences between
the two distinct stocks. In order to test the two-stock hypothesis, then, Waples (2011) used the samples from
SA6 and SA9 to characterize putative pure J and O stocks, artificially mixed the two samples to look at theexpected relationships of the above genetic indices, and compared these genetic indices to the ones observed
from the natural samples obtained from SA2, SA7 and SA11 where it was believed that whales from only the1

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two stocks existed according to Hypotheses A and B. Because the observed values were lower than theexpected ones, Waples (2011) concluded that the samples from SA2, SA7 and SA11, respectively, might havecontained individuals from more than two stocks. However, it was also raised that the robustness of theanalyses should be evaluated because this approach was novel and the behaviors of these values were unclear.
In this document, in order to verify Waples (2011), we describe the expected distribution of the correlations
between θ and FIS using computer simulation and examine the effect of sample size on the distributions.
Among the results presented in Waples (2011), we only examined the case of 1:1 mixture of individuals from thetwo distinct stocks that reflected the situation of the bycatch sample from SA7 (J=93, O=90, unclassified=29).
METHODS
The software EASYPOP (Balloux, 2001) was used to generate 100 sets of genotypic data for each of thescenarios and the software FSTAT 2.9.3 (Goudet, 1995) was to calculate θ (Weir and Cockerham, 1984) perlocus between the stocks and FIS per locus for the data two stocks combined as one.
For the generation of the genotypic data, two stocks were assumed, each of which consists of diploid
individuals with a constant size and equal sex ratio with random mating. Effective population size (Ne) of eachstock was set to be 5000. Census population size (N) of Ne=5000 can be approximately 20000 when ratio ofNe to N to be 1/3 to 1/4 (Roman and Palumbi, 2003), which was comparable to the IWC’s accepted populationabundance of the minke whales in the North Pacific. The simulation produces genotype data set for 16
independent nuclear gene loci for each individual. The number of the loci simulated and maximum number of
the allelic states (13) was set based on the observed minke whale data. Bidirectional migration was assumed
with an equal migration rate of 0.0002. Mutation rate of 5x10-4 was chosen to represent microsatellite loci.
For each simulation parameter set, we made 100 replicates. Number of generation was 5000 for each replicate
before collecting data. In order to look at the effect of sample size, we conducted three different cases of thesimulations: at the final generation of each replicate, all of the 5000 individuals each was sampled, 2) 400
individuals each were sampled, 3) 100 individuals each were sampled from two simulated stocks for geneticanalysis. The sample size of 400 (total sample size=800) approximately reflects the case of artificial mix of thesamples from SA6 (411) and SA9 (466) and the sample size of 100 (total sample size=200) does the case of theSA7 bycatch sample (J=93, O=90, unclassified=29) in Waples (2011).
RESULTS AND DISCUSSION
Values of θ between the two generated stocks ranged from 0.038 to 0.065 (avg=0.051, sd=0.006) among the100 data set in the case of sample size 5000, 0.037 to 0.066 (0.050, 0.006) among the 100 data set in the case ofsample size 400, and 0.033 to 0.070 (0.050, 0.007) among the 100 data set in the case of sample size 100.
Because the FST value between the J and O stocks we observed was 0.049 (Kanda et al., 2009), these generateddata should be reasonable to examine the distribution of correlations between the FIS and θ for our purpose.
Fig.1 shows the distributions of the correlation coefficient values among the 100 data in each of the threedifferent sample size scenarios. When all the individuals in the simulated stocks were used, very highcorrelations were observed in all of the 100 data set. This result demonstrated that, in theory, the proposed
hypothesis in Waples (2011) based on population genetics was correct. Although the observed correlation
coefficients from the case of the sample size 400 fluctuated, most of the values were high and the results
supported the high correlation (0.83) observed from the artificial mixture of the individuals from SA6 and SA9
in Waples (2011). Contrary to these two cases, the correlation coefficients among the 100 data set in the case ofsample size 100 fluctuated widely with some negative values, and about 40% of the values were around 0.3 orsmaller. The reliability of the results can be affected strongly by the sample sizes and thus the low correlation
coefficient observed from the SA7 bycatch sample (0.3) in Waples (2011) could be due to the small sample size.
In this paper, we only examined the case of SA7 bycatch sample. Because the sample sizes for the othercases (SA7-Kushiro, SA7-Sanriku, SA2, and SA11) were similar to or lower than the SA7 bycatch sample, we
think that the same pattern of fluctuated distributions will likely be observed when we simulate the other samples.
Similarly, because both FIS and linkage disequilibrium describe the departure from equilibrium, the sample size
effect will likely be appeared for the relationship between θ and linkage disequilibrium, too.
In regard to the stock structure issue for North Pacific minke whales, the conclusion of Waples (2011)
therefore needs further evaluation to be used for the evidence against Hypotheses A and B that proposed only
two stocks in the western North Pacific.
ACKNOWLEDGEMENTS
We thank Toshihiro Mogoe for technical assistance and Luis A. Pastene for valuable comments on the paper.
REFERENCES
Balloux, F. 2004. EASYPOP (version 1.7): a computer program for the simulation of population genetics.
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SC/64/NPM9Goudet, J. 1995. FSTAT, version 1.2: a computer program to calculate F-statistics. J. Hered. 86:485-486.
Kanda, N., Goto, M., Kishiro, T., Yoshida, H., Kato, H., and Pastene, L.A. 2009. Microsatellite analysis ofminke whales in the western North Pacific. Paper SC/J09/JR30 presented to the JARPN II Review
Workshop, Tokyo, January 2009.
Kanda, N., Goto, M., Nagatsuka, S., Kato, H., Pastene, L.A., and Hatanaka, H. 2010. Analyses of genetic andnon-genetic data do not support the hypothesis of an intermediate stock in sub-area 7. Paper
SC/S10/NPM9 presented to the First Intersessional Workshop for western North Pacific common minke
whales, Tokyo, September 2010.
Roman, J, and Palumbi, S.R. 2003. Whales before whaling in the North Atlantic. Science 301:508-510.
Waples, R.S. 2011. Can evidence for spatial and/or temporal genetic heterogeneity of North Pacific minke
whales be explained by different mixture fractions of the same two core stocks, or is it necessaru topostulate an additional stock (s)? Paper SC/63/RMP7 presented to the IWC Scientific Committee, May
2011, Tromso, Norway
Weir, B. S. and Cockerham, C.C. 1984. Estimating F-statistics for the analysis of population structure.
Evolution 38: 1358–1370.

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<1.0Correlation coefficients

Fig.1. Distribution of correlation coefficients between θ and FIS among the 100 computer generated genotypic
data set for different sample sizes (5000, 400, and 100) from each of the two putatively distinct stocks.

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