Lecture 6Forestry 3218

Forest Mensuration II Forest Mensuration II

Lecture 6

Double SamplingCluster SamplingSampling for Discrete Variables

Avery and Burkhart,Chapter 3

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Double Sampling (two-phase sampling) Double Sampling (two-phase sampling)

Double sampling with regression and ratio estimator

Double sampling for stratification

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Double Sampling with Regression and Double Sampling with Regression and Ratio Estimators Ratio Estimators

Remember: regression and ratio estimators require known

Take a large sample in which x alone is measured – allow a good estimate of

Establish a regression or ratio relationship between paired x and y

x

x

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Double Sampling with RegressionDouble Sampling with Regression

Estimate of the population mean of y

)( 212 xxbyyRd

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Double sampling with regression vs. Double sampling with regression vs. regression estimationregression estimation

Complete enumeration of x vs. a large sample of it

Both gain precision from using regression estimators

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Double Sampling With RatioDouble Sampling With Ratio

)( 1xRyMRd

2n

r

x

yR

where

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Double Sampling for StratificationDouble Sampling for Stratification

Recall: stratified random sampling requires that the strata size (Nh) be known in advance of sampling

Double sampling for stratification applies when– Nh is not known, but can be estimated by sampling

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Double Sampling for StratificationDouble Sampling for Stratification

1. Estimate Nh using a large sample

)(ˆ1

1

n

nNN h

h

N

yNy

L

hhh

std

1

ˆ

2. Estimate overall population mean

How is this different from that in stratified random sampling?

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Cluster SamplingCluster Sampling

A practical problem– A forester needs to estimate average seedling

heights or root collar of a nursery. Seedlings are grown on benches, blocks, or clusters of styrofoam

How are you going to sample?

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Cluster SamplingCluster Sampling

A cluster sample is a sample in which each sampling unit is a collection, or cluster, of elements

Reasons1. A list of elements is

not available, but a list of clusters is

2. Even when a list of elements is available, it is more economical to randomly select clusters than individual elements

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Cluster SamplingCluster Sampling

We need to know:– How many clusters in the population (N)– How many clusters selected (n), often by simple

random sampling– How many elements in a cluster (m)

– Measured value for sampled elements (yij), e.g., seedling height

Estimation of population mean

n

ii

n

i

m

jij

c

m

y

y

1

1 1

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Two-stage SamplingTwo-stage Sampling

What if there are too many elements in a cluster? For examples, – You want to know seedling dry weight of the previous

example

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Sampling for Discrete VariablesSampling for Discrete Variables

For qualitative attributes such as dead or alive, deciduous or evergreen – binomial distribution

Species composition – multinomial distribution

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Sampling for Discrete VariablesSampling for Discrete Variables

Estimate proportion

)N

n(S n

)P(PP

SS

S

111 Estimate standard error of the

proportion

total

alivePS #

#

ntSP

SPS 2

1 Estimate confidence interval

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Sampling for Discrete VariablesSampling for Discrete Variables

Use Cluster Sampling for Attributes – recall how we calculate mean, variance, and standard error of the mean for simple random sampling

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Relative Efficiencies of Sampling PlansRelative Efficiencies of Sampling Plans

Measure by cost or time with the same level of accuracy (not precision, why?)

When samples are unbiased, standard error of mean can serve as a measure of accuracy

Most efficient plan is:

min { (standard error)2×cost (time) }

Remember: The objective of sampling design is to obtain a specified amount of information about a population parameter at minimum cost