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A Weighting Class Adjustment Estimator for the Total under a Stratified Sampling Design in a Continuous Domain

Breda Munoz

Virginia Lesser*

Oregon State University

R82-9096-01

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This presentation was supported under STAR Research Assistance Agreement No. CR82-9096-01 awarded by the U.S. Environmental Protection Agency to Oregon State University. It has not been formally reviewed by EPA. The views expressed in this document are solely those of authors and EPA does not endorse any products or commercial services mentioned in this presentation.

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Overview

• Introduction– Assumptions

• Estimator for the Total in a continuous domain

• Effect of missing data in estimator for the total

• Adjustment estimator

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R

T y d s s

Assumptions

• probability sample

• Estimate of the population total of a variable Y

• Missing at random

John Day stream network

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Estimating the Total

• Horvitz-Thompson Estimator for the total in a continuous domain (Cordy, 1993):

- Unbiased

• Estimator for the Variance of the total

• Other: Total and variance estimators (Yates and Grundy, 1953)Local Variance Estimator (Stevens and Olsen, 2003)

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observed

missing

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HT-total estimator under missing data

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8000 10000 12000 14000

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6000 8000 10000 12000 14000

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8000 12000 16000 20000

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15% missing 30% missing 50% missing

92% 89% 70%

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Accounting for missing data

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Variance of the Adjustment Estimator

• Observe that:

22* * *ˆ ˆ ˆHT HT HTVar Y E Y E Y

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HT hi hi

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Variance of the Adjustment Estimator

, stratathh s s

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Variance of the Adjustment Estimator

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Var Y w y d y y d dw w

s ss s s s s s s

s s

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Population: John Day Middle Fork stream reaches

• Area of 785 mi2

• 143 stream reaches divided in survey segments (~1 mile)– 6536 survey

segments

• We simulate a continuous multivariate normal spatial random process

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Population: John Day Middle Fork stream reaches

• The population of stream reaches was stratified in 6 strata based on the number of survey segments:

“<10 ” “10-20” “20-30”

“30-50” “50-100” “>100”

• 1,000 samples of size 100

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Data

Site Outcome Strata U Prob. response If U<1-P then R=0 S1 Y1 1 U1 .70 missing S2 Y2 1 U2 .70 observed S3 Y3 2 U3 .85 observed S4 Y4 2 U4 .85 missing S5 Y5 3 U5 .90 observed S6 Y6 3 U6 .90 missing S7 Y7 4 U7 .75 observed S8 Y8 4 U8 .75 observed S9 Y9 5 U9 .80 observed S10 Y10 5 U10 .80 missing

178000 10000 12000 14000 16000

00

00

00

8000 10000 12000 14000 16000

00

00

00

8000 12000 16000 200000

00

00

15% Missing Rate 30% Missing Rate 50% Missing Rate

94.8% 94.1% 77.4%

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