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Bias Corr. of Bootstrapings on Lambda

Reno Lindberg reno at pop.bio.aau.dk
Sat Feb 1 14:53:43 EST 1997

In estimating Lambda and its Confidence intervals for a Lefkovitch
projection matrix, I found out that there is a couple of ways to estimate
Lambda and the confidence intervals based on Bootstrapings. In building my
Bootstrap program I resample the entire population on the individual level
with information on individual growth, leaf production, survived/dead, and
individual size. 
The simplest is the percentile method where you just take the 2.5 and the
97.5 percentiles of the bootstrapped lambda distribution. 
If there is a large difference from the observed lambda and the mean
bootstrapped lambda it is suggestedthat a bias correction is made on the
Lambda. The confidence intervals should also be bias corrected then. The
question is now how to report these values?
A; Observed Lambda= 0.9629
B; Bootstrapped mean lambda= 0.9484
C; BIAS= -0.0145 according to  (McPeek and Kalish, 1993, pp. 247) and
(Caswell, 1989, pp. 190) 
D; Bias corrected Bootstrapped mean lambda=0.9774 =(2*Observed
Lambda-Bootstrapped mean lambda)according to  (McPeek and Kalish, 1993,
pp. 247) and (Caswell, 1989, pp. 191) 
E; Confidence intervals= (0.89685,0.98974), according to (Dixon, 1993, pp 299)
F; Bias corrected confidence intervals= (0.91871, 1.00318) (Dixon, 1993,
pp. 301) 

I find it most logical to report (A); (B, E); and (D,F).
I can understand that it could be necessary to correct for bias in Lambda
and Confidence Interval estimation. 
I do on the other hand NOT understand why it is better simply to subtract
the difference from the observed value (See C and D) in the BIAS
correction of Lambda. I would say that the bias correction then skews the
estimation as much to the opposite site of the observed value (See D.)! 
The BIAS  correction of the confidence, isn¼t that a normal distribution
approximation. When I look at my graph of Lambda frequencies, it looks
skewed to the right. The BIAS  correction also moves the confidence
interval upwards. Is it the right thing to do a BIAS correction, since the
lambda distribution should not necessarily be normal distributed after a
bunch of Bootstrapings. Shouldn¹t the percentile method not be as good as
the BIAS corrected method?

What does the bias tell about my population? Is it because that very few
individuals have a large impact on Lambda and they are seldomly all
sampled. I have asked a little around, but no one seems to have a good

reno at pop.bio.aau.dk

Caswell, H. (1989). Population matrix models. Eds. Vol. Sinauer Associates
Inc., Sunderland, Maryland

Dixon, P. M. (1993). The Bootstrap and the Jack-knife: Describing the
Precision of Ecological Indices. In: Design and Analysis of Ecological
Experiments (Eds. S. M. Scheiner and J. Gurevitch), Vol. Chapman & Hall,
New York,  pp. 290-318

Efron, B. and R. J. Tibshirani (1993). An introduction to the bootstrap.
Eds. Vol. 57, Chapman & Hall, New York, pp. 436

McPeek, M. A. and S. Kalish (1993). Population Sampling and Bootstraping
in Complex Designs: Demographic Analysis. In: Design and Analysis of
Ecological Experiments (Eds. S. M. Scheiner and J. Gurevitch), Vol.
Chapman & Hall, New York,  pp. 232-252

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