Download Estimating Variance Components in Population Dynamics: A Statistical Approach and more Study notes Biology in PDF only on Docsity!
Lecture 14. Estimating Variance Components.
Reading: Burnham, K. P., D. R. Anderson, G. C. White, C. Brownie, and K. H. Pollock. 1987. Design and Analysis Experiments for Fish Survival Experiments Based on Capture-Recapture. Am. Fish. Monograph No. 5, Pages 260-278.
We have discussed various sources of variance that impact the dynamics of a population: demographic, environmental and spatial (process), individual heterogeneity, and genetic variances. In addition, the concept of sampling variance, or the uncertainty of our estimates of population parameters, has been frequently mentioned. This chapter covers the statistical methodology to estimate the different variance components from data.
Consider the example situation of estimating survival rates each year for 10 years from a deer population. Each year, the survival rate is different from the overall mean, because of snow depth, cold weather, etc. Let the true, but unknown, overall mean be S. Then the survival rate for each year can be considered to be S + some deviation:
Environmental Variation
i Mean Year i Year i
1 S S + e 1 S 1
2 S S + e 2 S 2 3 S S + e 3 S 3
4 S S + e 4 S 4 5 S S + e 5 S 5 6 S S + e 6 S 6
7 S S + e 7 S 7 8 S S + e 8 S 8 9 S S + e 9 S 9
10 S S + e 10 S 10 Mean S S ¯ S ¯
The estimator of S is S ¯:
S^ ¯ '
j
10
i ' 1
Si
10
Fˆ^2 '
j
10
i ' 1
( Si & S ¯)^2
10
with the variances of the Si :
where the random variables ei are selected from a distribution with mean 0 and variance F 2. In reality, we are never able to observe the annual rates because of sampling variation or demographic variation. For example, even if we observed all the members of a population, we would still not be able to say the observed survival rate was Si because of demographic variation. Consider flipping 10 pennies. We know that the true probability of a head is 0.5, but we will not always observe that value exactly. The same process operates in a population as demographic variation. Even though the true probability of survival is 0.5, we would not necessarily see exactly ½ of the population survive on any given year.
Hence, what we actually observe are the quantities:
Environmental Variation + Sampling Variation i Mean Year i Year i
1 S S + e 1 + f 1 S ˆ 1 2 S S + e 2 + f 2 S ˆ 2
3 S S + e 3 + f 3 S ˆ 3 4 S S + e 4 + f 4 S ˆ 4
5 S S + e 5 + f 5 S ˆ 5 6 S S + e 6 + f 6 S ˆ 6 7 S S + e 7 + f 7 S ˆ 7
8 S S + e 8 + f 8 S ˆ 8 9 S S + e 9 + f 9 S ˆ 9 10 S S + e 10 + f 10 S ˆ 10
Mean S S ¯ S ˆ
where the ei are as before, but we also have additional variation from sampling (or demographic
var( ˆ S ) ' F^2 % E [var( ˆ S * S )] 10
var( ˆˆ S ) '
j
10
i ' 1
( ˆ Si & S ˆ)^2
10(10 & 1)
E ˆ(var( ˆ S | S )] '
j
10
i ' 1
var( ˆ ˆ Si * Si )
10
Fˆ^2 '
j
10
i ' 1
( ˆ Si & S ˆ)^2
(10 & 1)
j
10
i ' 1
var( ˆ ˆ Si * Si )
10
wi '
F^2 % var( ˆ Si * Si )
S ˆ '
j
10
i ' 1
wi S ˆ i
j
10
i ' 1
wi
var( ˆ S ) '
j
10
i ' 1
wi
i.e., the total variance is the sum of the environmental variance plus the expected sampling variance. This total variance can be estimated as
We can estimate the expected sampling variance as the mean of the sampling variances
so that the estimate of the environmental variance obtained by solving for F 2
However, normally, the sampling variances are not all equal, so that we have to weight them to obtain an unbiased estimate of F^2. The general theory says to use a weight, wi
so that the estimator of the weighted mean is
with theoretical variance (see Box 1 for a derivation of this result)
var( ˆ ˆ S ) '
j
10
i ' 1
wi ( ˆ Si & S ˆ)^2
j
10
i ' 1
wi (10 & 1)
j
10
i ' 1
wi
j
10
i ' 1
wi ( ˆ Si & S ˆ)^2
j
10
i ' 1
wi (10 & 1)
j
10
i ' 1
wi ( ˆ Si & S ˆ)^2
(10 & 1)
j
10
i ' 1
wi ( ˆ Si & S ˆ)^2
(10 & 1)
P^210 & 1," L
j
10
i ' 1
wi ( ˆ Si & S ˆ)^2
(10 & 1)
P^210 & 1,"
U 10 & 1
and empirical variance estimator
When the wi are the true (but unknown) weights, we have
giving the following
Hence, all we have to do is manipulate this equation with a value of F^2 (that is imbedded in the wi term) to obtain an estimator of F^2.
To obtain a confidence on the estimator of F^2 , we can substitute the appropriate chi-square values in the above relationship. To find the upper confidence interval value, Fˆ , solve the 2 U equation
and for the lower confidence interval value, Fˆ^2 L , solve the equation
As an example, consider the following fawn survival data from Little Hills.
because the process variance of 8 is a function of the process variance of the parameter
var( 8 ) ' (^) j , i , j
2 var( aij )
assuming no covariances between the aij elements. In order to persist, a population must have a limited amount of variation. Thus, natural selection will select against high process variance in a parameter that 8 is very sensitive too.
Literature Cited
Bennington, C. C. and W. V. Thayne. 1994. Use and misuse of mixed model analysis of variance in ecological studies. Ecology 75:717-722.
Burnham, K. P., D. R. Anderson, G. C. White, C. Brownie, and K. H. Pollock. 1987. Design and Analysis Experiments for Fish Survival Experiments Based on Capture-Recapture. Am. Fish. Monograph No. 5, Pages 260-278.
Burnham, K. P. and G. C. White. 2002. Evaluation of some random effects methodology applicable to bird ringing data. Journal of Applied Statistics 29(1-4):245-264.
Koenig, W. D., R. L. Mumme, W. J. Carmen, and M. T. Stanback. 1994. Acorn production by oaks in central California: variation within and among years. Ecology 75:99-109.
Pfister, C. A. 1998. Patterns of variance in stage-structured populations: evolutionary predictions and ecological implications. Proceedings of National Academy of Science 95:213-218.
var( ˆ S ) '
j
10
i ' 1
wi
wi '
F^2 % var( ˆ Si * Si )
var( ˆ Si )
var( ˆ S ) ' var
j
10
i ' 1
wi S ˆ i
j
10
i ' 1
wi
j
10
i ' 1
wi
2 var (^) j
10
i ' 1
wi S ˆ i
j
10
i ' 1
wi
2 j
10
i ' 1
var wi S ˆ i
j
10
i ' 1
wi
2 j
10
i ' 1
wi^2 var( ˆ Si )
j
10
i ' 1
wi
2 j
10
i ' 1
wi
j
10
i ' 1
wi
Box 1. Derivation of the result that
Starting with the result that
the derivation is as follows.
