Combining Environmental Area Frame Surveys of a Finite Population

Abstract

New ways to combine data from multiple environmental area frame surveys of a finite population are being introduced. Environmental surveys often sample finite populations through area frames. However, to combine multiple surveys without risking bias, design components (inclusion probabilities, etc.) are needed at unit level of the finite population. We show how to derive the design components and exemplify this for three commonly used area frame sampling designs. We show how to produce an unbiased estimator using data from multiple surveys, and how to reduce the risk of introducing significant bias in linear combinations of estimators from multiple surveys. If separate estimators and variance estimators are used in linear combinations, there’s a risk of introducing negative bias. By using pooled variance estimators, the bias of a linear combination estimator can be reduced. National environmental surveys often provide good estimators at national level, while being too sparse to provide sufficiently good estimators for some domains. With the proposed methods, one can plan extra sampling efforts for such domains, without discarding readily available information from the aggregate/national survey. Through simulation, we show that the proposed methods are either unbiased, or yield low variance with small bias, compared to traditionally used methods.

Full text

Supplementary materials for this article are available at https:// doi.org/ 10.1007/ s13253-020-00425- z.
Combining Environmental Area Frame
Surveys of a Finite Population
Wilmer Prentius,XinZhao, and Anton Grafström
New ways to combine data from multiple environmental area frame surveys of a finite
population are being introduced. Environmental surveys often sample finite populations
through area frames. However, to combine multiple surveys without risking bias, design
components (inclusion probabilities, etc.) are needed at unit level of the finite population.
We show how to derive the design components and exemplify this for three commonly
used area frame sampling designs. We show how to produce an unbiased estimator using
data from multiple surveys, and how to reduce the risk of introducing significant bias
in linear combinations of estimators from multiple surveys. If separate estimators and
variance estimators are used in linear combinations, there’s a risk of introducing negative
bias. By using pooled variance estimators, the bias of a linear combination estimator can
be reduced. National environmental surveys often provide good estimators at national
level, while being too sparse to provide sufficiently good estimators for some domains.
With the proposed methods, one can plan extra sampling efforts for such domains,
without discarding readily available information from the aggregate/national survey.
Through simulation, we show that the proposed methods are either unbiased, or yield
low variance with small bias, compared to traditionally used methods.
Key Words: Combining data sources; Combining estimators; Environmental
monitoring; Linear combination estimator; Sample design properties.
1. INTRODUCTION
For a traditional finite population survey, one often think of some well-structured list
frame covering the population of interest, from which a statistician can draw a sample
according to some procedure, in order to produce an efficient and unbiased estimator of
some population parameter. When conducting environmental surveys, however, this is often
not the case.
Environmental surveys often lack well-structured, comprehensive list frames to sample
from. In such settings, it is common to use area frames covering the assumed spread of
W. Prentius (B)·X. Zhao ·A. Grafström, Department of Forest Resource Management, Swedish University of
Agricultural Sciences, 90183 Umeå, Sweden
(E-mail: wilmer.prentius@slu.se).
© 2020 The Author(s)
Journal of Agricultural, Biological, and Environmental Statistics
https://doi.org/10.1007/s13253-020-00425-z

W. Prentius et al.
the population of interest. Examples of environmental surveys using such area frames are
national forest inventories (Axelsson et al. 2010), agricultural inventories (Fecso et al. 1986),
landscape inventories (Allard 2017), among others. By using area frames, a sample unit
becomes a point from a continuous population—the area surface—why there is a need to
map the sample properties for the sampled points to the indirectly sampled units in the
population of interest.
Other desirable outcomes in environmental surveys are domain estimates, or their coun-
terparts, estimates created by aggregating domain estimates. In the first case, primary surveys
are seldom planned with domain estimates in mind, why complementary surveys are often
considered. The latter case may especially be considered when dealing with rare popula-
tions, or wanting to incorporate a previously conducted domain survey into an aggregate
survey (Benedetti et al. 2015).
Scenarios like these, or when dealing with two samples with different designs, connect
to the multiple-frame research area. When combining such samples, an optimal linearly
combined estimator should be weighted by the variance (Lohr and Rao 2006). Since true
variances are most likely not available, variance estimates are often used instead. However,
environmental surveys conducted using area frames often have target variables with highly
skewed distributions, since the units in the population of interest might be absent in large
parts of the area frame. Under such circumstances, the estimators and the variance estimators
are susceptible to correlation, which can introduce significant bias into linearly combined
estimates using variance estimates as weights (Grafström et al. 2019).
In order to reduce the bias of a combined estimate, we propose two methods: The first
approach is a generalization of the combining samples approach derived by Grafström et al.
(2019), which combines unit sample properties from an arbitrary number of designs into
design components for the combined design. The second approach uses a pooled variance
estimator to estimate the variance of each survey’s estimator by using all available informa-
tion from the surveys.
The targeted applications are primarily environmental surveys and monitoring, where it is
common to use area frames. Several countries have national landscape and forest monitoring
programs that may not be enough to produce regional or domain level estimates, and thus
need be complemented on some level to reach specific accuracy targets (Christensen and
Ringvall 2013).
With the methodology presented in this paper, there might be a need to link surveys
relating to different definitions of statistical units. Hence, this is something that should be
planned for from start. We need be able to detect if the same population unit is included in
more than one sample (or multiple times in the same sample). However, in most applications,
the size of the area being sampled is likely to be very large compared to the area covered
in the samples, which makes overlap not particularly common. In area-based surveys, we
are likely to have geographical coordinates for at least the statistical unit. These coordinates
can easily be used to detect possible overlap between different surveys. In the rare case of
possible overlap, it may be difficult identify exactly which population unit that is included
multiple times. If this is thought to be an issue, then it may be needed to use markings of
coordinates and/or population units in the field to make such identification easier.

Combining Environmental Area Frame Surveys of a Finite Population
In some cases, e.g., for unbiased variance estimation using a combined sample, we need
at least partial knowledge of the geographical coordinates of the sampled population units.
Such knowledge can be included by the use of accurate satellite-based positioning systems,
as is done, e.g., for permanent sample plots in the Swedish national forest inventory (Fridman
et al. 2014).
In Sect. 2, we provide a general procedure to produce unit sample properties for a discrete
population sampled using an area frame. Through Sect. 2.1, we show examples on unit
sample properties for a discrete population sampled through three different, commonly
used area frame designs. In Sect. 3, we recall the single and multiple count estimators that
are used to estimate population totals. Then, in Sect. 4, we present the theory for combining
samples, and for combining estimators using pooled variance estimators. In Sect. 5,weuse
a simulation to compare a naive linear combination with the combined sample and the linear
combination using pooled variance estimates. Finally, we discuss the results in Sect. 6.
2. UNIT SAMPLE PROPERTIES FOR GENERAL DESIGNS
Assume that there is a finite, but unknown population U, represented by fixed points on
an area of interest FU, that has some measurable properties of interest. If a sample point
X(k), with probability density function (pdf) f(k)(x), falls within the inclusion zone A(k)
iof
an unit i∈U, the unit is included in the sample.
Let Pbe the set of independent but not necessarily equally distributed sample points.
For any sample point X(k)∈P, and units {i,j}∈U, we make the following definitions:
S(k)
i:= IX(k)∈A(k)
i,(1)
π(k)
i:= Pr S(k)
i>0=A(k)
i
f(k)(x)dx,(2)
π(k)
ij := Pr S(k)
i>0,S(k)
j>0=A(k)
i∩A(k)
j
f(k)(x)dx,(3)
E(k)
i:= ES(k)
i=π(k)
i,(4)
E(k)
ij := ES(k)
iS(k)
j=π(k)
ij ,(5)
where I (·)denotes the indicator function, S(k)
iis the number of inclusions of unit iby sample
point X(k),π(k)
iis the first-order inclusion probability of unit iby sample point X(k), i.e.,
the probability of unit ibeing included into the sample by a sample point X(k),π(k)
ij is the
second-order inclusion probability for units i,jto be included in the sample simultaneously
by sample point X(k),E(k)
iis the (first-order) expected number of inclusions of unit iby
X(k), and E(k)
ij is the second-order expected number of inclusions of units i,jby X(k).
For the set of independent sample points P, we extend the definition in (1)to
S(P)
i:= 
X(k)∈P
S(k)
i.(6)

W. Prentius et al.
Expanding the definition of (4) to the first-order expected number of inclusions for unit i
by the set of sample points P,wehave
E(P)
i:= ES(P)
i=
X(k)∈P
E(k)
i,(7)
while it can be shown (see “Appendix” for further details), that the expected number of
inclusions of the second-order for units i,jby the set of sample points Pcan be extended
from (5)to
E(P)
ij :=ES(P)
iS(P)
j=E(P)
iE(P)
j+
X(k)∈P
E(k)
ij −E(k)
iE(k)
j.(8)
Moreover, the inclusion probabilities of the first and second-order of units i,jby the set
of sample points Pcan be expressed similarly to (2) and (3)as
π(P)
i:= Pr S(P)
i>0=1−
X(k)∈P
1−π(k)
i,(9)
π(P)
ij := Pr S(P)
i>0,S(P)
j>0=π(P)
i+π(P)
j
−⎛
⎝1−
X(k)∈P
1−π(k)
i−π(k)
j+π(k)
ij ⎞
⎠.(10)
For any set of sample points Pto be used to make an unbiased estimator of a parameter of
U, we require that all units in the population have positive inclusion probabilities, equivalent
to ensuring that a sampling design satisfies
∀i∈U∃X(k)∈P:π(k)
i>0.(11)
For an unbiased estimator of variance by any set of sample points P, we require that all
pairs of units {i,j}∈Uhave positive second-order inclusion probabilities, equivalent to
ensuring that a sampling design satisfies
∀{i,j}∈U∃{X(k),X(k)}∈P,k= k:π(k)
ij +π(k)
iπ(k)
j>0.(12)
While the requirements in (11) and (12) are necessary and sufficient for positive inclusion
probabilities of the first and second-order, they are in reality often not assessable if the units
in Uare unknown. Instead, sufficient counterparts with respect to FUcan be formulated as
∀x∈F∃X(k)∈P:f(k)(x)>0,(13)
∀{x,x}∈F∃{X(k),X(k)}∈ P,k= k:f(k)(x)f(k)(x)>0,(14)
where F, the sample frame, is connected to FUso that FU\Fdx=0, assuming reasonably
defined inclusion zones. It holds that (14) is sufficient for (13).

Combining Environmental Area Frame Surveys of a Finite Population
2.1. SAMPLE PROPERTIES FOR THREE COMMON DESIGNS
Provided the derived sample properties, it is easy to show the sample properties for three
common designs—i.i.d., one point per stratum stratified, and systematic—given uniform
sample point distributions. Assuming that unit i’s inclusion zones are identical for all sample
points within a specific design, i.e., A(k)
i=Aifor all X(k)
d, we define Fas the area enclosing
all possible inclusion zones, aFastheareaofF,aiastheareaofAi, and aij as the area of
Ai∩Aj.
An i.i.d. design defined by P1implies that f(k)
1(x)=f(k)
1(x)for every pair of sample
points X(k)
1,X(k)
1. The inclusion probabilities for units i,jby a single sample point X(k)
1
can thus be described as
π(k)
i=Ai
f(k)
1(x)dx=ai
aF
,
π(k)
ij =Ai∩Aj
f(k)
1(x)dx=aij
aF
.
From this, it follows that the first-order sample properties for unit iare
π(P1)
i=1−1−ai
aFn1,E(P1)
i=n1
ai
aF
,
with the second-order sample properties for units i,j
π(P1)
ij =π(P1)
i+π(P1)
j−1−1−ai+aj−aij
aFn1,
E(P1)
ij =n1(n1−1)
aFaF
aiaj+n1aij
aF
,
where n1denotes the cardinality of P1, i.e., the number of sample points in the design.
A systematic design with uniform pdf’s, and a repeating pattern in the inclusion zones
defined by the stratification (exemplified in Fig. 1), is a special case of the i.i.d. design where
only one point is sampled. Thus, for the systematic design, the sample properties for units
i,jare π(P2)
i=E(P2)
i=ai/aFand π(P2)
ij =E(P2)
ij =aij/aF.
The final example is the one point per stratum stratified design defined by P3, where one
point is sampled from each of a fixed number of disjoint strata. Let the stratum for sample
point X(k)
3be given as F(k)={x:f(k)
3(x)>0},a(k)
Fbe the area of F(k),a(k)
idenote the
area of Ai∩F(k), and let a(k)
ij denote the area of Ai∩Aj∩F(k). The inclusion probabilities
for units i,jby X(k)
3, given uniform pdf’s, can then be described as
π(k)
i=Ai
f(k)
3(x)dx=a(k)
i
a(k)
F
,
π(k)
ij =Ai∩Aj
f(k)
3(x)dx=a(k)
ij
a(k)
F
,

W. Prentius et al.
Figure 1. Examples of ai.i.d., bstratified, and csystematic frames and inclusion zones. The outer areas represent
the sample frames (F), the inner areas represents the areas of interest ( FU), and the circles represents the inclusion
zones (A) for units. In both aand b, the sample frame expands around the area of interest so that the largest of
the inclusion zones will always be fully within the area frame. In bfour disjoint strata of unequal sizes and shapes
are exemplified through the dashed lines. cshows inclusion zones for two units, where dashed circles and x’es
indicate the units’ positions. These types of inclusion zones would exemplify systematic plot sampling.
from which the results in (7), (8), (9), and (10) follows. In the case of equally sized and
disjoint strata, a(k)
F=aF/n3, where n3represent the number of strata/sample points.
3. SINGLE AND MULTIPLE COUNT ESTIMATORS
The sample properties derived in Sect. 2are needed for two common estimators used
when estimating the population total Y=i∈Uyiof a finite population U. The first of
these two estimators is the single-count (SC) Horvitz–Thompson estimator (Horvitz and
Thompson 1952), defined as
ˆ
YSC =
i∈U
yi
πi
I(Si>0),
where Sidenotes the number of inclusions of unit i,πi=Pr (Si>0)denotes the inclusion
probability for unit i, i.e., the probability for unit ito be included in the sample, and I (·)
denotes the indicator function. The variance of ˆ
YSC can be shown to be
Vˆ
YSC =
i∈U
j∈U
yi
πi
yj
πjπij −πiπj,
where πij =Pr Si>0,Sj>0denotes the second-order inclusion probability, i.e., the
probability for units i,jto be included in the sample simultaneously. Given that the second-
order inclusion probabilities are strictly positive for all pairs {i,j}∈U, an unbiased variance
estimator for ˆ
YSC is
ˆ
Vˆ
YSC =
i∈U
j∈U
yi
πi
yj
πjπij −πiπj
×I(Si>0)ISj>0
πij
.

Combining Environmental Area Frame Surveys of a Finite Population
The second estimator to be used in this paper is the multiple-count (MC), or Hansen–
Hurwitz, estimator (Hansen and Hurwitz 1943), defined as
ˆ
YMC =
i∈U
yi
Ei
Si,
where Ei=E[Si]denotes the expected number of inclusions for an unit i. The variance of
ˆ
YMC is
Vˆ
YMC=
i∈U
j∈U
yi
Ei
yj
EjEij −EiEj,
where Eij =ESiSjdenotes the second-order expected number of inclusions for two
units i,j. Given that the second-order expected number of inclusions are strictly positive
for all pairs {i,j}∈U, an unbiased variance estimator of ˆ
YMC is
ˆ
Vˆ
YMC=
i∈U
j∈U
yi
Ei
yj
EjEij −EiEjSiSj
Eij
.
As by the requirements in (13) and (14), the variance estimators presented here are not
applicable when using a one-per-stratum stratified or systematic sample design such as those
presented in Sect. 2.1. However, when combining two or more independent samples, these
criteria will be evaluated on the combined sample.
4. COMBINING SAMPLES
Let D={Pd}ddenote a combined sample, i.e., a set of independent sets of sample points
Pd. By extending the definition of (6) to the number of inclusions by the combined sample
as
S(D)
i:= 
Pd∈D
S(Pd)
i,(15)
the inclusion probability of unit iby a combined sample Dbecomes
π(D)
i=1−
Pd∈D
1−π(Pd)
i,(16)
similar to (9). Comparable to (7), (8), and (10), the rest of the necessary sample properties
for units i,jby a combined sample Dfollows as
π(D)
ij =π(D)
i+π(D)
j
−⎛
⎝1−
Pd∈D1−π(Pd)
i−π(Pd)
j+π(Pd)
ij ⎞
⎠,(17)
E(D)
i=
Pd∈D
E(Pd)
i,(18)

W. Prentius et al.
E(D)
ij =E(D)
iE(D)
j+
Pd∈D
E(Pd)
ij −E(Pd)
iE(Pd)
j.(19)
By using these combined sample properties, the estimators in Sect. 3can be applied directly.
When combining samples, for example in a multiple frame setting, the individual designs’
sample frames do not need to be identical, nor do they need to individually cover the area of
interest. The requirements in (11) and (12) needs to be fulfilled with respect to the sample
points in ∪dPd, i.e., the necessary condition for positive second-order inclusion probabilities
and positive expected number of inclusions for all pairs in the combined sample Dis
∀{i,j}∈U∃{X(k)
d,X(k)
d}∈∪
dPd,
(k,d)= (k,d):π(k)
ij +π(k)
iπ(k)
j>0,(20)
with sufficient counterpart
∀{x,x}∈F∃{X(k)
d,X(k)
d}∈∪
dPd,
(k,d)= (k,d):f(k)
d(x)f(k)
d(x)>0,(21)
both of which imply positive first-order inclusion probabilities and positive expected number
of inclusions for all units by the combined sample D.
If sample frames are extended in ways similar to those in Fig. 1, or if combining multiple
frames, there will be some oversampling. In such cases, it will be required to be able to
identify objects not part of the population of interest.
These results are not limited to area frames. As per an example in Lohr and Rao (2006),
it is possible to combine, for example, a sample taken from an area frame with full coverage
of the population of interest, and a list frame with unknown coverage of the population of
interest, as long as it is possible to identify units in the list frame that are not part of the
population of interest, and units sampled from the area frame that are also present in the list
frame.
4.1. COMBINING ESTIMATORS BY LINEAR COMBINATIONS
When combining a set of unbiased estimates formed of the samples in Dby linear
combinations, the form
ˆ
Y(D)
L=
Pd∈D
α(Pd)ˆ
Y(Pd)
is often considered, since it will yield an unbiased result. Often the inverse variance propor-
tion is used as the weight in order to increase accuracy. However, as described by Grafström
et al. (2019), if true variances are not available, using variance estimates may in certain
cases introduce bias to such a linear combination, especially when the variance estimator is
correlated with the estimator of the population parameter. We denote a linear combination

Combining Environmental Area Frame Surveys of a Finite Population
using variance estimates as
ˆ
Y(D)
L∗=
Pd∈D
ˆα(Pd)
∗ˆ
Y(Pd)
∗,ˆα(Pd)
∗=
ˆ
Vˆ
Y(Pd)
∗−1
Pd∈Dˆ
Vˆ
Y(Pd)
∗−1,
with ∗for either SC (single-count) or MC (multiple-count).
To overcome the issue with biased variance estimators, we propose a pooled variance
estimator, using all available information to estimate the separate variances. We denote the
linear combination estimator using such pooled variance estimates as
ˆ
Y(D)
LP∗=
Pd∈D
ˆα(Pd)
P∗ˆ
Y(Pd)
∗,ˆα(Pd)
P∗=
ˆ
VPˆ
Y(Pd)
∗−1
Pd∈Dˆ
VPˆ
Y(Pd)
∗−1,(22)
where
ˆ
VPˆ
Y(Pd)
SC =
i∈U
j∈U
yi
π(Pd)
i
yj
π(Pd)
jπ(Pd)
ij −π(Pd)
iπ(Pd)
j
×
IS(D)
i>0IS(D)
j>0
π(D)
ij
,
ˆ
VPˆ
Y(Pd)
MC =
i∈U
j∈U
yi
E(Pd)
i
yj
E(Pd)
jE(Pd)
ij −E(Pd)
iE(Pd)
j
×S(D)
iS(D)
j
E(D)
ij
,
are both unbiased estimators of the variances of the single and multiple count estimators,
given ∀{i,j}∈U,π(D)
ij >0 and ∀{i,j}∈U,E(D)
ij >0. Note that the final fractions for both
variance estimators for a design Pdassures that all available information are used through
S(D)
i,π(D)
ij and E(D)
ij , as defined in (15), (17) and (19). However, if many second-order
design properties are positive, but small, the variance estimators might produce negative
and unstable estimates, making them unsuitable for combinations.
5. SIMULATION
In order to evaluate the proposed combinations of samples and estimates, a simulation
study was performed. The simulation sampled 10,000 times from a simulated population
generated from the SLU (Swedish University of Agricultural Sciences) Forest Map (Reese
et al. 2003). The SLU Forest Map, previously known as kNN-Sweden, has extensive infor-
mation about Swedish forest land and is based on satellite and field data from the Swedish
national forest inventory (NFI). The map contains information about age, height, species

W. Prentius et al.
Figure 2. Location and the total biomass volume (m3/ha) for the area used as a boilerplate for simulating the
population. Darker colors indicate higher volumes (Color figure online).
Figure 3. Total biomass volume (m3/ha) per species for the simulated population. Darker colors indicate higher
volumes (Color figure online).
of wood and woodland for the country’s forest land. The basic format is raster data with a
resolution of 25 ×25 square meters.
From the SLU Forest map, an area of 1000 ×1000 square meters of southern Sweden
was cropped to represent the area of interest. Figure 2illustrates the location as well as the
total volume of the stand for the cropped area. Using individual tree data variables from the
Swedish NFI, the three dominating tree species—birch, pine, and spruce—were randomly
added to the population according to species-specific volume maps of the cropped area. In
the resulting population, the number of trees for each species is 7411 (13%), 24,428 (41%)
and 27,212 (46%), respectively. The resulting population is presented in Fig. 3, color-coded
by volume intensity.
For each of the 10,000 simulation runs, four samples were generated from the sample
frame using uniform densities—two i.i.d. samples, one systematic sample, and one stratified
sample. Each design used circular inclusion zones of common sizes per design, correspond-

Combining Environmental Area Frame Surveys of a Finite Population
Table 1. Sample designs used in the simulation study
Design n Radius (m) Sample frame (m2) Stratum size (m2) Sampled area (m2)
i.i.d. 1 10 10 1020 ×1020 3142
i.i.d. 2 40 5 1010 ×1010 3142
Systematic 16 8 1016 ×1016 254 ×254 3217
Stratified 16 8 1016 ×1016 254 ×254 3217
nSample size; Radius Radius of inclusion zones
ing to plot sampling. In order to have equal first-order expected number of inclusions for all
units, the sample frames were expanded around the area of interest in each direction by the
size of the inclusion zone radius, guaranteeing that all inclusion zones are fully within the
sample frames. In Table 1, the designs are described in further detail.
For each sample and combination, single (SC) and multiple count (MC) estimates were
calculated. To show the effect of different ways of combining data, we compared the esti-
mators using combined samples, with sample properties derived through (16), (17), (18)
and (19), with the estimators based on linear combinations of estimates using estimated
variances and pooled variance estimates as in (22).
As mentioned in Sect. 3, for variance estimators to be unbiased, we require positive
second-order sample properties for all pairs in the population. While the systematic and
stratified designs fulfills the requirements in (20) and (21) in combination with each other
or any of the i.i.d. designs, they do not fulfill (12) and (14) individually, while also being
prone to negative and unstable pooled variance estimates due to small second-order design
properties, making them unsuitable to use in a linear combination. In environmental surveys,
one often deal with this by using a more conservative variance estimator, for example by
using the i.i.d. variance estimator (Benedetti et al. 2015). However, using the i.i.d. variance
estimator might be too conservative, i.e., reducing the assumed efficiency of the stratified
and systematic designs.
For this simulation, second-order design properties were calculated as if they were sam-
pled using a i.i.d. design, when calculating the linear combination of estimates using pooled
variances. For the naive combination, plot variance estimates in the linear combination
ˆ
VPlot ˆ
Y(Pd)
MC =1
nd(nd−1)
X(k)
d∈Pd
y(k)
d−ˆyd2
,
ˆyd=1
nd
X(l)
d∈Pd
y(l)
d,
were used, where y(l)
dis the plot lestimate of the total. In order to reduce the efficiency
impact of the stratified and systematic designs, plot variances were calculated using a variant
of the local mean variance estimator proposed by Grafström and Schelin (2013)

W. Prentius et al.
Table 2. Results from 10,000 simulations for the i.i.d. 1 (i), systematic (sy), and stratified (st) designs showing
[empirical relative bias] and relative root-mean-squared error (RRMSE) for birches and all species in
percent
SC MC LPlot LPSC LPMC
Birches
i 50.22 50.14 – – [–] – [–] –
sy 42.79 42.79 [–] – [–] – [–] –
st 41.76 41.76 [–] – [–] – [–] –
i / sy 32.77 32.83 [-13.92] 36.79 [-0.70] 32.21 [-0.76] 32.19
i / st 32.49 32.55 [-13.92] 36.36 [-0.90] 31.90 [-0.96] 31.88
sy / st 30.01 30.05 [-12.32] 33.65 [-0.26] 30.05 [-0.27] 30.05
i / sy / st 25.95 26.01 [-18.98] 33.81 [-0.69] 25.64 [-0.73] 25.63
All species
i 28.53 28.49 [–] – [–] – [–] –
sy 21.62 21.62 [–] – [–] – [–] –
st 19.69 19.69 [–] – [–] – [–] –
i / sy 17.88 17.91 [-2.48] 18.83 [-0.83] 17.44 [-0.89] 17.44
i / st 17.23 17.25 [-2.40] 17.46 [-0.78] 16.55 [-0.84] 16.54
sy / st 14.71 14.69 [-2.44] 15.95 [-0.35] 14.69 [-0.35] 14.69
i / sy / st 13.63 13.65 [-3.32] 14.91 [-0.70] 13.25 [-0.74] 13.25
SC Single-count estimator; MC Multiple-count estimator; LPlot Linear combination weighted by plot variances;
LPSC Linear combination weighted by pooled SC-variances; LPMC Linear combination weighted by pooled
MC-variances
ˆ
VPlot ˆ
Y(Pd)
MC ,n∗=n∗
n∗−1
X(k)
d∈Pd
y(k)
d−ˆy∗
d(k,n∗)2
,
ˆy∗
d(k,n∗)=1
n∗
X(l)
d∈P∗
d(k)
y(l)
d,
where P∗
d(k)is the set of n∗sample points of design dclosest to X(k)
d. For this simulation,
the fixed number of neighbors was set to n∗=4.
The results, presented in Table 2, show that while any combination reduced the variance
in the estimator, the combination based on plot variance estimates introduced bias at least
three times of that generated by the pooled variance estimates. Because of the relatively
small probability of two sample points sampling the same tree, the SC and MC estimators
perform similarly.
In Table 3, bias, MSE, and variance estimates are presented for the i.i.d. 1 and 2 designs,
and the combinations of the two. Comparing the combined samples versus the combined
estimates, one can observe the trade-off between unbiased estimates and estimates with
reduced variances.
6. DISCUSSION
In Table 2, we showed that combined samples and linear combinations based on pooled
variances (pooled combination) will probably always be preferable to linear combinations

Combining Environmental Area Frame Surveys of a Finite Population
Table 3. Results from 10,000 simulations for the i.i.d. 1 and 2 designs showing [empirical relative bias] in percent,
mean variance estimates, and empirical mean-squared error (MSE) for birches and all species
Estimator Rel. bias Mean var. (104)MSE(10
4)
Birches
i.i.d. 1 SC [–] 26.08 26.02
MC [–] 26.16 25.95
i.i.d. 2 SC [–] 13.91 14.25
MC [–] 13.96 14.21
i.i.d. 1 / 2 SC [–] 9.93 10.07
MC [–] 9.99 10.12
LMC [-12.61] 6.63 12.15
LPSC [-3.83] 8.71 9.08
LPMC [-3.97] 8.74 9.07
All species
i.i.d. 1 SC [–] 1675.85 1716.50
MC [–] 1671.77 1711.94
i.i.d. 2 SC [–] 640.74 646.99
MC [–] 639.36 645.09
i.i.d. 1 / 2 SC [–] 573.51 589.58
MC [–] 573.24 591.06
LMC [-2.03] 437.48 538.30
LPSC [-2.03] 454.02 506.76
LPMC [-2.19] 453.07 507.65
SC Single count estimator; MC Multiple count estimator; LMC Linear combination weighted by estimated vari-
ances; LPSC Linear combination weighted by pooled SC-variances; LPMC Linear combination weighted by pooled
MC-variances
based on individual variances (naive combination), given that the target variable has a skewed
distribution. Even if no correlation exists between the estimator and its variance estimator,
the pooled combination should be more efficient than the naive combination, as more infor-
mation is used. The main drawback of the pooled combination is the need to compute
additional second-order design properties, which may be difficult if positional data is not
available or accurate enough to map the sample properties of the designs. Furthermore, for
some designs the pooled variance estimator might be unstable, which makes it an unsuitable
choice for such designs. However, the combined samples approach will function sufficiently
in most cases, as its estimate is not dependent on second-order design properties, why the
impact of absence of reliable positional data should be small, for most designs.
While the results from the simulation are conditional to the simulated population, we
expect the bias to be proportional to the heterogeneity of the population, why we may
draw some general conclusions. We believe both of these methods to be useful for domain
estimates. For the domain estimate of a primary survey, the target variable will have a
skewed distribution, even if the target variable over the domain is not. It is thus expected
that significant bias will be introduced by using the naive combination.
Another scenario where both presented methods might be useful are when combining
designs like those used in the simulation here, where it is not possible to get an unbiased
variance estimator for one or more of the individual designs. The pooled combination is
unbiased if the combined second-order sample properties are positive for all units in the

W. Prentius et al.
population, whereas the naive combination needs positive second-order sample properties
for all units and all designs. Furthermore, the combined samples approach has none of these
restrictions and is also more relaxed in terms of first-order sample properties.
Table 3provides results regarding MSE and variance estimates for i.i.d. designs. These
results highlight the bias–variance trade-off between the pooled combination and the com-
bined sample approaches. The combined samples approach produces unbiased estimators,
however, in the simulation, with larger empirical mean-squared errors than the pooled com-
binations. A statistician deciding between these two approaches should thus know to what
extent the end product needs to be accurate or reliable.
In Tables 2and 3, we see that the bias is, as expected, more apparent when dealing with
skewed target variables, as the volume of birch. It is not uncommon to reach acceptable
MSE’s for some dominant or aggregate target variable in a primary survey, here represented
by the total wood volume, while needing complementary surveys to study some target
variable with a more skewed distribution. The results of the simulation show that different
methods of combination will affect the reliability of the combined estimates.
Further research would study the effects of errors in the positioning of units, to see how
previously described mismatching would affect the estimates. For plot sampling procedures,
that are commonly used in forest inventories, one can assume two types of mismatching to
be common: One where there is a difference between the location of the studied plot and
the sampled location, and one where the positioning of units within a plot are inaccurate.
Depending on designs, these errors will have different effects.
ACKNOWLEDGEMENTS
We would like to thank the two anonymous reviewers and the associate editor for their helpful comments.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and
indicate if changes were made. The images or other third party material in this article are included in the article’s
Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds
the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this
licence, visit http://creativecommons.org/licenses/by/4.0/.
Funding Open access funding provided by Swedish University of Agricultural Sciences.
[Received February 2020. Accepted November 2020.]
APPENDIX: UNIT DESIGN PROPERTIES
Let Ube a finite, unknown population, representable by fixed points on an area of interest
FU. If a sample point X(k), with probability density function (pdf) f(k)(x), falls within the
inclusion zone A(k)
iof unit i∈U, the unit is included in the sample.

Combining Environmental Area Frame Surveys of a Finite Population
Let Pbe the set of independent sample points. For any sample point X(k)∈P, and units
{i,j}∈U, we make the following definitions:
S(k)
i:= IX(k)∈A(k)
i,(23)
π(k)
i:= Pr S(k)
i>0=A(k)
i
f(k)(x)dx,(24)
π(k)
ij := Pr S(k)
i>0,S(k)
j>0=A(k)
i∩A(k)
j
f(k)(x)dx,(25)
E(k)
i:= ES(k)
i=π(k)
i,(26)
E(k)
ij := ES(k)
iS(k)
j=π(k)
ij ,(27)
where I (·)denotes the indicator function, S(k)
iis the number of inclusions of unit iby
sample point X(k),π(k)
iis the first-order inclusion probability of unit iby sample point
X(k), i.e., the probability of unit ibeing included into the sample by a sample point X(k),
π(k)
ij is the second-order inclusion probability for units i,jby sample point X(k),E(k)
iis the
(first-order) expected number of inclusions of unit iby X(k), and E(k)
ij is the second-order
expected number of inclusions of units i,jby X(k).
For a set of independent but not necessarily equally distributed sample points P,we
extend the definitions to
S(P)
i:= 
X(k)∈P
S(k)
i,(28)
π(P)
i:= Pr S(P)
i>0,(29)
π(P)
ij := Pr S(P)
i>0,S(P)
j>0,(30)
E(P)
i:= ES(P)
i,(31)
E(P)
ij := ES(P)
iS(P)
j.(32)
It follows quite clearly from (31), (28), and (26) that
E(P)
i=
X(k)∈P
E(k)
i=
X(k)∈P
π(k)
i,
and by expanding (29), we can express it in terms of (24)
π(P)
i=1−Pr S(P)
i=0=1−Pr 
X(k)∈P
S(k)
i=0
=1−
X(k)∈P
1−π(k)
i.

W. Prentius et al.
Through some work, we can get the second-order expected number of inclusions for
units i,jby the set of sample points P
E(P)
ij =E
X(k)∈P
S(k)
i
X(k)∈P
S(k)
j=
X(k)∈P
ES(k)
iS(k)
j
+
X(k)∈P,X(k)∈P
k=k
ES(k)
iS(k)
j
=
X(k)∈P
E(k)
ij +
X(k)∈P,X(k)∈P
k=k
E(k)
iE(k)
j=E(P)
iE(P)
j
+
X(k)∈P
E(k)
ij −E(k)
iE(k)
j,
due to the independence of sample points in P. For the second-order inclusion probability
for units i,jby the set of sample points P, we start by showing that
π(P)
ij =Pr S(P)
i>0+Pr S(P)
j>0
−Pr S(P)
i>0∪S(P)
j>0
=π(P)
i+π(P)
j−1−Pr S(P)
i=0,S(P)
j=0.(33)
Through the independence between sample points in P, the following equality holds
Pr S(P)
i=0,S(P)
j=0=
X(k)∈P
Pr S(k)
i=0,S(k)
j=0,
and conversely, apparent from (33), we have
Pr S(k)
i=0,S(k)
j=0=1+π(k)
ij −π(k)
i−π(k)
j,
leading to
π(P)
ij =π(P)
i+π(P)
j−⎛
⎝1−
X(k)∈P
1−π(k)
i−π(k)
j+π(k)
ij ⎞
⎠.
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