SC/65a/EM04

Further Investigation of Whether Correlations amongst
Data are Invalidating the Conclusion of a StatisticallySignificant Trend in Antarctic Minke Body Condition overTime
KENJI KONISHI1 AND DOUG S BUTTERWORTH 2

1Institute of Cetacean Research, 4-5, Toyomi-cho, Chuo-ku, Tokyo 104-0055, Japan

2Marine Resource Assessment and Management Group (MARAM), Department of Mathematics andApplied Mathematics, University of Cape Town, Rondebosch 7701, South Africa

ABSTRACTA statistically significant decline in body condition in the Antarctic minke whale over theJARPA period was reported in Konishi et al. (2008). Subsequently, however, questions wereraised in the IWC Scientific Committee as to whether the model used had adequately
accounted for the data structure in JARPA (de la Mare, 2011), i.e. whether neglect ofcorrelations in the data had led to negative bias in the confidence interval estimated for thedecline. Earlier a jack-knife approach with year as the sampling unit was used to account forwithin-year correlations, and showed that while the confidence interval estimates reported
originally had been negatively biased, the estimated decline remained statisticallysignificant at the 5% level. This approach did not, however, take account of possible betweenyear correlations. This paper investigates the possibility that such correlations could havebiased these interval estimates appreciably. The results show no sign of appreciable interannual auto-correlation, and indicate that the decline reported by Konishi et al. (2008)
remains statistically significant at the 5% level. We suggest that his constitutes sufficient
statistical evidence to confirm this significance, so that this matter might now be regarded as
resolved by the Scientific Committee.
INTRODUCTION
Konishi et al. (2008) reported a decline of body condition in Antarctic minke whales over the18 years of the JARPA period, having taken account of comments made on an earlier analysis
at the JARPA review meeting (IWC, 2008). Subsequently, an SC member questioned whether
adequate account had been taken in the regression analysis used of structure of JARPA data,suggesting that doing so could result in a different result with respect to the statisticalsignificance of the trend reported (de la Mare, 2011). Walløe, (IWC, 2012) responded byreporting results for the jack-knife approach. This is a non-parametric approach which isapplied with an enlarged sampling unit to account for the concerns about non-independenceof data when treating an individual whale as the sampling unit, as had had been voiced byde la Mare. Naturally if this sampling unit is chosen to be larger than necessary to accountfor the lack of independence in the data, the resultant standard error estimate becomes
positively biased. These jack-knife results confirmed that the estimated decline in blubber
thickness remained statistically significant at the 5% level (IWC, 2012). Skaug (2012) also
examined the blubber thickness trend using mixed-effect models, including an additionalvariance structure in accord with specific requests made by the EM Sub-Committee. The
variance of the estimated trend became somewhat higher; however the point estimate of thetrend changed little and remained significant at the 5% level.
de la Mare (2012) again suggested that spatial pseudo-replication in JARPA data needed tobe considered. Some members considered, however, that his concern about non-independence
amongst the data had already been addressed by Walløe’s response above. However the jackknife approach applied had used year (i.e. season) as the sampling unit. Therefore, while it
subsumes auto-correlation at a within-year temporal scale, it would not have taken account1

of possible auto-correlation over periods longer than one year. To address this, we haveconducted an additional test to assess whether there is any such auto-correlation effectwhich might negatively bias the earlier jack-knife estimate of the variance of the decline inblubber thickness to any meaningful extent.
MATERIALS AND METHODS
Jack-knife method
Further to a jack-knife analysis of the form carried out by Walløe (IWC, 2012), we repeatedsuch jack-knife computations with sampling units of two and three years to check for any
appreciable increase in the standard error of the trend estimate that would suggest thepresence of positive inter-year correlation effects sufficient to change conclusions concerning
the statistical significance of the trend estimate.
The model used here is the best model (in terms of AIC) selected by Skaug (2012). It iswritten below in standard R notation where BLm is body length, ‘DateNum’ is the datewithin a year and ‘YearNum’ is the year number, with the trend parameter of interest being
the slope associated with ‘YearNum’. LongCat is a categorical variable coding for seven areas,
the slope associated with ‘DateNum’ being variable over years and treated as a random effect,a random intercept is associated with ‘Year’, and a random intercept and slope (of‘DateNum’) is associated with ‘LongCat’:

Lmer(BT11 ~ (DateNum|Year) + (DateNum|LongCat) + Diatom + LongDegE + YearNum +Latitude + BLm + Sex, data=blubber,REML=reml))
RESULTS
Jack-knife method
The annual trend estimates (in cm/yr) and associated jack-knife standard error estimates foreach length of sampling unit (note that the 95% confidence intervals are based on thestandard error estimates coupled to the assumption of distribution normality) are;1 year: -0.0172 (-0.0211 to -0.0133 95%CI) SE=0.0020 No. of sampling units: 18
2 year: -0.0175 (-0.0209 to -0.0140 95%CI) SE=0.0018 No. of sampling units: 9
3 year: -0.0173 (-0.0244 to -0.0103 95%CI) SE=0.0036 No. of sampling units: 6
All these trend estimates are very close to that from the original model (Skaug 2012) and the
95% confidence intervals do not overlap 0 even with increasing length of the sampling unit.
Note that some increase in the standard error would be expected as the length of thesampling unit is increased, as the effective extent of the period over which the regression
slope is estimated is thereby reduced.
These results indicate that autocorrelation at the inter-year level is not introducing any biasthat is of a level that would be of concern.
DISCUSSION
IWC report SC/59/Rep1 states.: “Finally, the Workshop agreed that the JARPA dataset
provides a valuable resource to allow investigation of some aspects of the role of whales
within the marine ecosystem. With appropriate analyses, this has the potential to make animportant contribution to the Scientific Committee’s work in this regard, as well as the workof other relevant bodies such as CCAMLR (IWC 2008).”2

This study has addressed concerns that the earlier jack-knife results (based on a one-yearsampling unit) put forward by Walløe (IWC, 2012) did not constitute a sufficientdemonstration that the precision with which the trend in blubber thickness had been
estimated was such as to confirm statistical significance at the 5% level. The test carried out
above show no evidence of auto-correlation amongst years of sufficient size to negate thisresult,We therefore suggest that this issue might be considered by the Scientific Committee to havebeen resolved. These JARPA data have demonstrated an important trend to have occurred in
the Antarctic ecosystem over the past two decades.
REFERENCES
de la Mare, W. K. 2011. Are reported trends in Antarctic minke whale body condition
reliable? Paper SC/63/O16 presented to the IWC Scientific Committee, June 2011, Tromsø,
Norway. 25pp.
de la Mare, W. K. 2012. Lurking variables and the interpretation of statistical analyses ofdata collected under JARPA Paper SC/63/O16 presented to the IWC Scientific Committee,
June 2012, Panama City, Panama. 65pp.
International Whaling Commission 2008. Report of the Intersessional Workshop to Review
Data and Results from Special Permit Research on Minke Whales in the Antarctic, Tokyo, 48 December 2006. Journal of Cetacean Research and Management 10 (Supplement): 411-45
International Whaling Commission 2012. Report of the Sub-Committee. Annex K1. Report ofthe Working Group in Ecosystem Modelling. Journal of Cetacean Research and Management.
13 (Supplement): 256-62..
Konishi K, Tamura T, Zenitani R, Bando T, Kato H, Walløe L. 2008. Decline in energy storage
in the Antarctic minke whale (Balaenoptera bonaerensis) in the Southern Ocean. Polar Biol.
31(12):1509-20
Skaug, H. J. 2012. Results of mixed-effects regression analysis of blubber thickness inAntarctic minke whales from data collected under JARPA. Report of the Scientific Committee.
Appendix 2 of Annex K1. Journal of Cetacean Research and Management 13 (Supplement):
262.

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