untitled

or by different methods. A quick and dirty comparison is provided by the normal

approximation, which is adequate if each survey covered more than 30 sampling units.

The two estimates are significantly different when:

(est 1 −est 2 )√[Var(est 1 ) +Var(est 2 )] >1.96

If sample sizes are too low, or if more than two surveys are being compared, the

determination of significance should be made by one-factor analysis of variance.

If the surveys are not independent, as when the same transects are run each year,

a comparison may still be made by analysis of variance but with TRANSECTSnow declared

a factor in a two-factor analysis. Chapter 16 goes further into this and other uses of

analysis of variance.

If a comparison shows that two or more independent estimates of the same popula-

tion are not significantly different, we may wish to merge them to provide an

estimate more precise than the individual estimates. This is a procedure quite dis-

tinct from stratification where estimates from different populations are combined to

give an overall estimate. Merging is restricted to the same population estimated more

than once. We must make sure that environmental (e.g. different seasons) and

biological (e.g. significant mortality or emigration) conditions do not differ between

censuses. Merging is particularly powerful in obtaining a reduced confidence inter-

val from a series of individual censuses each with very wide confidence intervals. If

one obtains a single estimate with a wide confidence interval (say because too few

samples were counted) then it will often pay to repeat the census as soon as possible

and merge the two results.

There are two methods. First, there is a quick and dirty method, to be used only

when the individual estimates were each made with about the same sampling inten-

sity. The merged estimate Yˆcan then be calculated as:

Yˆ=(Y 1 +Y 2 +Y 3 +...+YN)/N

where there are Nsurveys. It has a variance of:

Var(Yˆ) =[Var(Y 1 ) +Var(Y 2 ) +Var(Y 3 ) +...+Var(YN)]/N^2

Thus the merged estimate is simply the mean of the individual estimates, and its

variance is the mean of the individual-estimate variances divided by their number.

SE(Yˆ) is the square root of Var(Yˆ). From these the merged density estimate is D=

Yˆ/Awhich has a standard error of SE(D) =SE(Yˆ)/A.

Second, a more appropriate method, particularly for surveys utilizing markedly dif-

ferent intensities of sampling, is provided by Cochran (1954), who also considers

more complex merging. Here the contribution of an individual estimate to the

merged estimate is weighted according to its precision. Letting w=1/ Var(Yˆ):

Yˆ=(w 1 Y 1 +w 2 Y 2 +w 3 Y 3 +...+wNYN)/(w 1 +w 2 +w 3 +...+wN)

with a variance of:

Var(Yˆ) =1/(w 1 +w 2 +w 3 +...+wN)

234 Chapter 13

13.5.5Merging
estimates