Design considerations in
forest inventories with dual
estimation objectives
1
STEEN MAGNUSSEN, AND JOHANNES BREIDENBACH
Dual estimation objectives
2
 Population- or stratum-level objectives for a study variable Y.

The mean (total), estimates of error, confidence intervals.
 Stand-level objectives for the same study variable Y.

The mean (total), estimate of errors, confidence intervals.
 Auxiliary variables X correlated with Y are used to improve precision
of population-level estimates. X is known for all elements in
population.
 Model-assisted inference for population parameters
 Traditional designs have sparse within-stand sampling

most one sampled element in n selected stands.
 The model that links X to Y cannot accommodate stand effects.
 Stand effects cannot be quantified.
Do we have stand effects?
3
 No categorical ‘yes’ or ‘no’.
 Evidence is scarce.
 Reasonable to anticipate stand effects.
 Expression of stand effects:
Two observations of Y in a single stand are more
similar than two observations picked at random from
all stands. Extensions to residual errors.
 Estimation via a partitioning of variance
(among- and within-stands).
A common scenario
4
 Population with M stands of size mi.
 Sparse sampling of stands.
 Assume a linear model for element k in stand i.
 A working element-level model.
pooled
yk |i  βt X k |i  eIi  ek |i ,
i  1,..., M , k|i  1,...., mi
A common scenario
5
 Synthetic estimator of a stand mean.
mi
yˆ iSYN  mi1  yˆ k |i
k |i 1
 Estimator of variance in a synthetic mean.


ˆ X  ˆ 2 m 1
Vˆ yˆ iSYN  X it Σ
 i
eIi  ek |i i
A common scenario
6
 Synthetic estimate of a stand mean is possibly biased
 can we ignore the term eIi?
 Model-based estimator of variance in a synthetic
estimate of a stand man is possibly anti-conservative.
A better estimator would be:


SYN
t ˆ
2
2
1
ˆ
ˆ
ˆ
ˆ
V yi
 X i Σ  X i   eIi   ek|i mi
An alternative estimator of variance
7
 Default design: One observation of Y in each of n
stands.
 Estimate stand effects in yˆ k |i .
 Adjust design-effects in estimates of variance in Y.
 
ˆ x   2   2 m 1
V AH yˆ i  xit Σ
 i
eIi
ek |i i
 e2  r  ˆ yˆ  ˆ y2
Ii
r =ˆ y2ˆ  ˆ yˆ 2  N 
Magnussen, S. 2016 A new mean squared error estimator for a synthetic
domain mean. For. Sci., in press.
Do stand effects matter?
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 Stand effects lead to a covariance among
all elements in a stand.
 Strong impact on the among-stand
variance.
 Strong impact on the within-stand
variance.
 The impact on variance components
depends on stand size.
Do stand effects matter?
9
 Examples from simulations of tri-variate (y,x1,x2)
populations with marginal gamma distributions.
 300 stands, 225 elements per stand.
 Sample size n of 30, 50, 100, and 200;
 a maximum of one element per stand.
 Correlation between x1 and y of 0.3, 0.6, and 0.9,
correlation between x2 and y, and between x1 and
x2 is weaker.
Do stand effects matter?
10
 Stand-effect in intercept of model.
 Relative stand effect (fraction of total variance)
varied from 0.10 to 0.35.
 Performance indicators for stand-level inference:


R = Squared root of ratio of average estimated variance
to 
average actual 
root mean squared error;
coverage of nominal 95% confidence intervals.
Do stand effects matter?
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The performance ratio R declines as stand effects increase.
The apparent banding reflects effects of sample size and the strength
of the correlation between x1 and y.
Do stand effects matter?
12
Coverage declines to unacceptable levels as stand effect increase
Design considerations
13
 Given dual estimation objectives;
 population/strata, and stands.
 Given performance standards ;
 R and coverage of nominal 95% confidence intervals.
 Given a default design with sampling of n stands;
 one element per stand
 design meets population-level inference objectives.
 Anticipating some level of stand effects;
 design may compromise stand-level inference.
A suitable design alternative?
14
 A two-stage probability sampling design with n1




stands selected in the first stage, and n2 elements
selected within each stand selected in the first
stage.
Constraint: fixed total sample size, n = n1 × n2.
Desired: maximize n1 for given n.
Alternative design should meet stand-level
performance criteria.
Alternative design will be less efficient for
population-level inference.
In search of a design alternative
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 Set a minimum acceptable stand average of R;
 here min(R)= 0.80.
 Find the lowest level of stand effects for which this
criterion can still be met with the default design with
sparse stand sampling.
 Do you anticipate greater stand effects?


If yes, begin search for an alternative design;
find design with a notional high probability of a positive
estimate of the stand effect (variance).
A search example
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 With Monte-Carlo simulations find trends in stand
average of R as a function of stand effects,
correlation structure, and sample size n.
 As an example: simulations with standard Gaussian
tri-variate populations,



three levels of correlation between y and x1;
1200 stands, stand sizes of 9, 16, 25,…, 144 elements;
sparse sampling of stands.
 Find the critical level of stand effects above which
the average R-value drops below 0.8 with a default
design with sparse sampling of stands.
A search example
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For each population, set of design parameters ψ, level of
stand effects, and sample size compute:

M


R  ,  e2Ii , n 
SYN
ˆ
ˆ
V
y
a i
i 1
mi
m  y
M
i 1
1
i
k 1
k |i

 yˆ k |i 
2
Fit trend surface (here for all average ratios less than
0.95).
0.16
Rˆ  0.94   eI2 i   y2   x2.31

1.74
n
1, y
A search example
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 Fix ratio at 0.80 and solve for
 y    .
2
eI i
2
y
The solution is the critical level of stand effect for a given design and
population. An greater value leads to an average R-value below 0.8.
Anticipated stand effects
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 After obtaining a critical lower level of stand effects
‘tolerable’ under a default design (with sparse
sampling of stands) and sample size n, the analyst
must now decide if anticipated stand effects are
larger than the critical value.
 Use information from older inventories, special
purpose studies, or cost plus loss in support of a
decision.
Meeting stand-level objectives
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 Search for combinations Ω of n1 and n2 so that:
  n1 , n2 | n  n1  n2 , n2  1 




n2 e2Ii  crit   
   0.20 
  n1 , n2  : P  Fn1 ( n2 1),n1 1   1  2
2



  y   eIi  crit   

Choose - among possible designs - the design with the largest value of
n1 (i.e. number of first-stage stands).
Estimators for two-stage designs
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 Population-level estimators,
 Model-assisted regression estimators
(e.g. Särndal et al. 1993).
 Stand-level estimator of mean:
n2
 1 mi
mi  yˆ k |i   eˆk |i sample size in ith stand is n2
k |i 1
k |is

yˆ i  
mi
 m 1 yˆ sample size in ith stand is 0
k |i
 i k
|i 1
Estimators for two-stage designs
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 Variance estimator for a stand mean:
Vˆ
 yˆ 
i
 2 n2 n2
ˆ i   kl  g k ek  g l el 
m
k 1 k 1


 if sample size in ith stand is >1


ˆ x  ˆ 2  ˆ 2 m 1
 xit Σ
 i
eIi
ek |i
i


if sample size in ith stand is  1
Assessment of a proposed design
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 Monte-Carlo simulations.
 Default sampling design (n= n1 and n2=1).
 Alternative design (n= n1 × n2 and n2>1).
 Compare stand-level and population level
performance w.r.t. R and coverage.
2
2
 Vary level of  y   eIi   y to assess
robustness of design.
Twelve Monte-Carlo examples
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 Populations with 1600 stands.
 Equal probable stand sizes {32,42,….,122}.
 Four sample sizes (120, 240, …,480).
 Three variables (y, x1,x2).
 Standard Gaussian distributions.
 Correlation ρ(y, x1) = {0.5,0.7,0.9} 0.5.
 Correlation ρ(y, x2) = 0.25 ρ(y, x1).
 Correlation ρ(x1, x2) = 0.5.
 Stand effects in y: ρy= {0.05,0.1,0.2,0.3}.
 Stand effects in x: ρx = 0.7 ρy.
Twelve Monte-Carlo examples
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Design
n
ρ2(y, x1)
ρcrit
n1
n2
P(F ≤ 1)
1
120
0.5
0.083
30
4
0.17
2
120
0.7
0.056
15
8
0.21
3
120
0.9
0.042
15
8
0.27
4
240
0.5
0.074
80
3
0.14
5
240
0.7
0.050
60
4
0.20
6
240
0.9
0.037
30
8
0.19
7
360
0.5
0.069
180
2
0.18
8
360
0.7
0.047
120
3
0.20
9
360
0.9
0.035
60
6
0.18
10
480
0.5
0.066
240
2
0.15
11
480
0.7
0.045
160
3
0.18
12
480
0.9
0.033
96
5
0.17
Stand-level performance of designs
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Stand-level performance of designs
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Estimates of among-stand variance
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Population-level performance of designs
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Conclusions
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 A pairing of: a sampling design with at most one
sampled element from each of n stands; and dual
(pop. and stand) estimation objectives can lead to a
failure to meet stand-level objectives when:
- there are (non-trivial) stand effects in the target
population (stratum);
- and a model-based estimator of variance is used
for a synthetic estimate of a stand mean.
Outlook
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 Explore locally linear regressions?
 local in X-space.
 An X-based clustering of stands?
 Reduction of stand effects within X-clusters;
 modelling with indicators of cluster membership.
 Stand structure indicators (in X-space).
 Two-phase sampling with n2 random (1,2,3)?
 What are sample inclusion probabilities?
Thank you
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