Regression Estimation - Survey Sampling Techniques - Lecture Slides, Slides of Survey Sampling Techniques

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Regression

estimation

What

if

relationship

between

y

and

x

is

linear,

but

does

NOT

pass

through

the

origin

Better

model

in

this

case

is

x

B

B

y

1

0





y

x

B

0

B

1

slope

Regression

estimation

New

estimator

is

a

regression

estimator

To

estimate

is

predicted

value

from

regression

of

y

on

x

at

U

x

x

reg

y

U

y



 x x B y x B B y

U

U

reg











1

1

0

ˆ

ˆ

ˆ

ˆ

This is theadjustment tobasic estimator(sample mean)

Estimating

pop

mean

Sample

variances,

correlation,

covariance











 



  

  







 













n i

i

i

xy

y

x

xy

n i

i

x

n i

i

y

y y x x n s

s

s

s

r

x

x

n

s

y

y

n

s

1 1

2

2

1

2

2

1

1

1

1

1

1

Estimating

variance

Note:

Regression

estimator

residual

is

different

from

ratio

estimation

residual

Different

models

different

predicted

values









i

i

i

i

i

n i

i

e

e

reg

y y x B B y e

e

n

s

n s

N n

y

V

where

1

0

1

2

2

2





i

y

Tree

example

Goal:

obtain

a

precise

estimate

of

number

of

dead

trees

in

an

area

Sample

Select

n

out

of

N

plots

Make

field

determination

of

number

of

dead

trees

per

plot,

y

i

number

of

dead

trees

from

field

plot

i

Population

For

all

N

plots,

have

photo

determination

on

number

of

dead

trees

per

plot,

x

i

number

of

dead

trees

from

photo

plot

i

Calculate

dead

trees

per

plot

U

x

Tree

example

Tree

example

Regression

analysis

output

Source

DF

Sum of Squares

Mean Square

F Value

P-value

Model

1

Error

23

5.78957 = MSE

= 133.16018 / 23

Total

24

MSE

n n

s

e

)

1

(

)

2

(

2

^ 



Tree

example

Estimated

mean

number

of

dead

trees/plot









41 .

0

~

4080 .

0

25 54834 .

5

100

25

1 1

ˆ

ˆ

t

trees/plo

dead

99 .

11

3 .

11

613274 .

0

059292 .

5

ˆ

ˆ

ˆ

2

0

0





  

  





  

  















n s

n N

y

E

S

x

B

B

y

e

reg

U

reg

Tree

example

Lohr

uses

t

distribution,

but

we’ll

stick

to

normal

approximation

Approx

95%

CI

for

t

y

is









]

[

2 /

yreg

yreg

t

E

S

z

t



Domain

estimation

under

SRS

Often,

we

want

to

study

the

characteristics

of

one

or

more

subsets

of

the

population

Goal

is

to

estimate

and

compare

subpopulation

(i.e.,

domain)

population

parameters

If

we

have

a

SRSWOR

design

(Ch

then

we

have

not

designed

the

sample

specifically

to

estimate

parameters

for

the

domain

Use

domain

estimation

in

this

case

If

we

have

used

stratified

sampling

(Ch

with

strata

defined

by

the

subpopulations

of

interest,

this

is

very

simple

Use

stratum

estimators

(more

later)

Domain

estimation

framework

Divide

population

into

D

mutually

exclusive

and

exhaustive

domains

Index

set

for

domains:

d

D

May

not

have

information

on

domain

membership

prior

to

survey

(unless

it

is

in

the

sampling

frame)



Domai

n D



d



d







d

D



Domai

n

Domain

estimation

framework

Population

sizes

(may

be

unknown)

N

d

number

of

SUs

in

domain

d

in

the

population

N

N

1

N

2

N

D

After

collecting

data,

can

calculate

the

number

of

SUs

in

each

domain

n

d

number

of

sample

units

in

domain

d

n

n

1

n

2

n

D

Notation

U

d

=

index

set

for

population

domain

A

d

=

index

set

for

sample

domain

Sum

over

all

i

belonging

to

domain

d

Population

Sample



d

U

i



d

A

i

Boat

owner

example

Population

N

boat

owners

(currently

licensed)

Sample

n

owners

selected

using

SRS

Population

has

domains

(size

N

d

unknown)

d

own

open

motor

boat

ft.

(large

boat)

d

do

not

own

this

type

of

boat

Of

the

n

sample

owners:

n

1

owners

of

open

motor

boat

ft.

n

2

owners

do

not

own

this

kind

of

boat