Geoscience Reference
In-Depth Information
where the random variables
e
i
are selected from a distribution with mean 0 and
variance
s
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
S
i
because of demographic variation. Consider flipping 10
coins. We know that the true probability of a head is 0.5, but we will not always
observe that value exactly. If you have 11 coins; the true value is not even in the
set of possible estimates. The same process operates in a population as demo-
graphic variation. Even though the true probability of survival is 0.5, we would
not necessarily see exactly half of the population survive on any given year.
Hence, what we actually observe are the quantities following:
Environmental Variation + Sampling Variation
I
Mean
Truth Year I
Observed Year I
ˆ
1
1
S
S
+
e
1
+
f
1
ˆ
2
2
S
S
+
e
2
+
f
2
ˆ
3
3
S
S
+
e
3
+
f
3
ˆ
4
4
S
S
+
e
4
+
f
4
ˆ
5
5
S
S
+
e
5
+
f
5
ˆ
6
6
S
S
+
e
6
+
f
6
ˆ
7
7
S
S
+
e
7
+
f
7
ˆ
8
8
S
S
+
e
8
+
f
8
ˆ
9
9
S
S
+
e
9
+
f
9
ˆ
10
10
S
S
+
e
10
+
f
10
-
w
Mean
S
S
where the
e
i
are as before, but we also have additional variation from sampling
variation, or demographic variation, or both, in the
f
i
.
The usual approach to estimating sampling variance separately from tem-
poral variance is to take replicate observations within each year so that within-
cell replicates can be used to estimate the sampling variance, whereas the be-
tween-cell variance is used to estimate the environmental variation. Years are
assumed to be a random effect, and mixed-model analysis of variance proce-
dures are used (e.g., Bennington and Thayne 1994). This approach assumes that
each cell has the same sampling variance. An example of the application of a
random effects model is Koenig et al. (1994). They considered year effects,
species effects, and individual tree effects on acorn production by oaks in cen-
tral California.
Classic analysis of variance methodology assumes that the variance within
