Comparison of Variance Estimators for Two-dimensional, Spatially-structured

Sample Designs.

Don L. Stevens, Jr.

Susan F. Hornsby*

Department of Statistics

Oregon State University

The research described in this presentation has been funded by the U.S. Environmental Protection

Agency through the STAR Cooperative Agreement CR82-9096-01 Program on Designs and Models

for Aquatic Resource Surveys at Oregon State University. It has not been subjected to the Agency's

review and therefore does not necessarily reflect the views of the Agency, and no official endorsement

should be inferred

R82-9096-01

Preview

• Widely accepted that a more or less regular pattern of points (e.g., systematic sampling) is more efficient than SRS

• A variety of variance estimators for estimated mean are available for 1-dimensional systematic sampling

• We will examine the behavior of some variance estimators for 2-dimensional systematic and spatially-balanced (not necessarily regular) designs

Variance Estimators

• Wolter (1985) identified eight 1-dimensional variance estimators for 1-dimensional systematic sampling

• D’Orazio (2003) extended three of these to 2-dimensional systematic sampling

• Stevens & Olsen (2003) developed an estimator for 2-dimensional spatially-balanced samples

Simulation Study

• D’Orazio used simulation to compare estimators on a lattice generated from a Gaussian random field using several covariance functions– 32 x 32 lattice– Calculated variance estimator for all 16

possible 8 x 8 samples– Generated the random field using the Gaussian

Random Field package in R

0 5 10 15 20 25 30

05

10

15

20

25

30

xg

yg

0 5 10 15 20 25 30

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10

15

20

25

30

xg

yg

Simulation Study

• Replicate D’Orazio’s study for the exponential covariance model, with the addition of the NBH estimator

• Check the behavior of the estimators on a spatially-patterned surface that is not stationary.

Variance Estimators

• Simplest approach: assume SRS:

n

sfpcVSRS

2

ˆ

N

nNfpc

Variance Estimators

• Horizontal stratification

0 5 10 15 20 25 30

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15

20

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30

2/

1

21222)(

1)(ˆ

n

jjjhs yy

nfpcV

Variance Estimators

• Vertical stratification

0 5 10 15 20 25 30

05

10

15

20

25

30

2/

1

21222)(

1)(ˆ

n

jjjvs yy

nfpcV

Variance Estimators

• 1st Order autocorrelation correction– 1-dimension

, the Durbin-

Watson

statistic

SRSw

ac VD

V ˆ2

ˆ

n

jj

n

jjj

w

yy

yy

D

1

2

1

1

21

)(

)(

1

1

21 )(

)1(2

1)(ˆ

n

jjjac yy

nnfpcV

Variance Estimators• 1st Order autocorrelation correction

– 2-dimension

, Geary’s c index of

spatial autocorrelation

1 if and are neighbors

0, otherwise

j k

jk

y y

2

1

1

2

2

)(

s

yy

cn

j

n

jkjk

n

j

n

jkjkkj

g

SRSgac VcV ˆˆ

Variance Estimators

• Cochran’s Autocorrelation Correction– 1-dimension

ˆ ˆcac SRSV wV

otherwise

ρw

,1

0ˆ ),1ˆ/1/(2)ˆln(/21

1 lagat ation autocorrel estimates

Variance Estimators

• Cochran’s Autocorrelation Correction– 2-dimension

– Use Moran’s I in place of in formula for w

ˆ ˆcac SRSV wV

1

1

( )( )

ˆ( 1)

n n

j k jkj k j

n n

jk SRSj k j

n y y y y

I

n V

Stevens & Olsen NeighborhoodEstimator

• General form for variable probability, continuous population

2

2

1

1ˆ1 i i

j kNBH ij ik

n i j D k Dj k

i

y yV w w

Di is the set of neighbors for point i

Stevens & Olsen NeighborhoodEstimator

Weights are chosen so that

Weights are a decreasing function of distance, and vanish outside of local neighborhood

1 1

1n n

ij iji j

w w

and wij =0 for jDi

Stevens & Olsen NeighborhoodEstimator

• For constant probability, finite population

2

2

1ˆ ( )i i

NBH ij j ik ki j D k D

V fpc w y w yn

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xg

yg

grf.sim[, , i] Y

Z

grf.sim[, , i] Y

Z

grf.sim[, , i] Y

Z

grf.sim[, , i] Y

ZGaussian Random Fields

X

YZ

X + Y

X

YZ

N(0.25,0.075) * N(0.25,0.1)

X

Y

xy[,1]^2 + xy[,2]^2 -xy[,1] -xy[,2] +.1* cos(20*xy[,1]) + 0.1*sin(15*xy[,2]) +1.5

z.ptch32.ary[, , i]

Y

Z

z.ptch32.ary[, , i]

Y

Z

z.ptch32.ary[, , i]

Y

Z

z.ptch32.ary[, , i]

Y

ZPatchy Surfaces

Result GRF cv=1-exp(-2x)

Mean Bias 95% Coverage

VSRS 0.011834 0.00801 0.99943

Vsh 0.005722 0.00190 0.97706

Vsv 0.005728 0.00190 0.978625

Vac 0.005709 0.00189 0.982000

Vcac 0.001436 -0.0023 0.788187

VNBH 0.004949 0.00113 0.973062

Result GRF cv=1-exp(-0.5x)

Mean Bias 95% Coverage

VSRS 0.01448 0.00693 0.99356

Vsh 0.01275 0.00521 0.98719

Vsv 0.01270 0.00515 0.98731

Vac 0.01270 0.00515 0.98881

Vcac 0.00626 -0.0013 0.89950

VNBH 0.00992 0.00238 0.97681

0.0 0.5 1.0 1.5 2.0

0.85

0.90

0.95

1.00

CV Parameter

Cov

erag

e

srs

shsv

ac

cac

nbh

Results Patchy Surface

Mean Bias 95% Coverage

VSRS 0.00087 0.00058 0.99950

Vsh 0.00040 0.00011 0.95796

Vsv 0.00032 0.00002 0.93487

Vac 0.00036 0.00006 0.96290

Vcac 0.00009 -0.0002 0.74199

VNBH 0.00033 0.00003 0.95408

Conclusions

• The hs, vs, ac, and nbh estimators all seem to work reasonably well for both the GRF and patchy surfaces

• The nbh estimator seems to give coverages that are a bit closer to nominal than the hs, vs, or ac estimators

• The nbh works for variable probability, spatially constrained designs for which the other estimators do not.