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时间：2021-03-31

STA304H1S/1003HS Winter 2021: Week 8

We should learn...

What is systematic sampling? It’s advantages and disadvantages...

How does systematic sampling compare with SRS?

What is repeated systematic sampling?

Shivon Sue-Chee Systematic Sampling 1

Systematic Sampling (Ch.7)

choose a random starting point and choose each 10th, or 15th, or

20th observation

Eg, every 10th person leaving a shopping centre

Eg, every 15th item in an ordered list of accounts

Eg, every half hour/fifteen minutes/ ... take an item from an

assembly line for inspection

Eg, our class heights (for n=10)

Shivon Sue-Chee Systematic Sampling 2

Class example

example: our heights (for n=10)

> N=length(height); N

[1] 217

> n=10

> #Systematic random sampling

> k=floor(N/n);k

[1] 21

> #random starting point

> set.seed(1234)

> start=sample(1:k,1); start

[1] 19

> sys<-seq(start,N,k);sys

[1] 19 40 61 82 103 124 145 166 187 208

> sample_sys<-height[sys];

> sample_sys

[1] 170.0 160.0 170.5 180.0 181.0 170.0 175.0 175.0 172.0 180.0

Shivon Sue-Chee Systematic Sampling 3

Advantages and disadvantages of systematic

sampling

easier, especially with no sampling frame

easier to organize, especially with untrained interviewers

more precise if the y ’s in the population are ordered

equivalent to a SRS if the y ’s in the population are in random order

hard to estimate the variance of y¯

so we usually use the SRS variance estimate

very biased if the y ’s in the population are cyclical, and the sampling

interval coincides with the cycle

see Figures 7.1 – 7.3

Shivon Sue-Chee Systematic Sampling 4

Population types

Shivon Sue-Chee Systematic Sampling 5

What are Random, Ordered and Periodic

populations? Examples...

Random population (ρ ≈ 0)

The elements of the population are in random order.

Ordered population (ρ < 0)

The elements of the population have values that trend upward or

downward when listed.

Periodic population (ρ ≈ 1)

The elements of the population tend to cycle upward and downward in a

regular pattern when listed.

Shivon Sue-Chee Systematic Sampling 6

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 7

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 8

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 9

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 10

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 11

Inference from systematic samples: §7.3, 7.4

When are the SRS formulas okay?

if the sample is ‘similar’ to a simple random sample

i.e. population is unordered, with respect to the variable of interest

SRS variance will be an over-estimate if the population is ordered

SRS estimate will just be wrong if the population is periodic and the

sampling interval coincides with the cycle

e.g. data on store sales taken every 7th day; data on rainfall taken

every 12 months; ...

Shivon Sue-Chee Systematic Sampling 12

Inference from systematic samples: §7.3, 7.4

When are the SRS formulas okay?

if the sample is ‘similar’ to a simple random sample

i.e. population is unordered, with respect to the variable of interest

SRS variance will be an over-estimate if the population is ordered

SRS estimate will just be wrong if the population is periodic and the

sampling interval coincides with the cycle

e.g. data on store sales taken every 7th day; data on rainfall taken

every 12 months; ...

Shivon Sue-Chee Systematic Sampling 13

Inference from systematic samples: §7.3, 7.4

When are the SRS formulas okay?

if the sample is ‘similar’ to a simple random sample

i.e. population is unordered, with respect to the variable of interest

SRS variance will be an over-estimate if the population is ordered

SRS estimate will just be wrong if the population is periodic and the

sampling interval coincides with the cycle

e.g. data on store sales taken every 7th day; data on rainfall taken

every 12 months; ...

Shivon Sue-Chee Systematic Sampling 14

Why is SRS variance estimate too big if the

population is ordered?

By a systematic sample, we estimate a population mean µ by:

µˆ = y¯sy =

∑n

i=1 yi

n

(7.1)

and an estimate of the variance of y¯sy is:

V̂(y¯sy ) =

(

1− n

N

)s2

n

= V̂(y¯)

where s2 = 1n−1

∑n

i=1(yi − µˆ)2.

equivalent to SRS

Shivon Sue-Chee Systematic Sampling 15

Why is SRS variance estimate too big if the

population is ordered?

BUT, true variances are different!

V (y¯sy ) =

=

=

σ2

n

[1 + (n − 1)ρ]

(

1− n

N

) N

N − 1

compared to SRS, where

V (y¯) =

σ2

n

(

1− n

N

) N

N − 1

if ρ < 0,V (y¯sy ) < V (y¯), so V̂ (y¯sy ) is too big!

Shivon Sue-Chee Systematic Sampling 16

... comparing variances

V (y¯sy ) =

σ2

n

[1 + (n − 1)ρ]

(

1− n

N

) N

N − 1

− 1

n − 1 < ρ < 1

ρ is the average of the correlation coefficients between all possible

pairs of observations in the systematic sample of size n

a. Random, ρ ≈ 0 V (y¯sy ) V (y¯) V̂ (y¯)

b. Ordered, ρ < 0 V (y¯sy ) V (y¯) V̂ (y¯)

c. Periodic, ρ ≈ 1 V (y¯sy ) V (y¯) V̂ (y¯)

Shivon Sue-Chee Systematic Sampling 17

Comparing systematic samples: random vs ordered

population

Estimating average height

Refer to class R codes and output

Type Estimate Variance Estimate

a. Random, ρ ≈ 0 V̂ (y¯sy )

b. Ordered, ρ < 0 V̂ (y¯sy )

Shivon Sue-Chee Systematic Sampling 18

Example (Lohr, 5.12)

sampling of dumps and landfills

to see if toxic waste is leaking

from containers

choose a random point in the

landfill area; construct a grid

containing that point

take soil samples from each grid

point

gives good coverage if there is little prior knowledge about where the

toxic materials might be

but could fail if the material is regularly placed

Shivon Sue-Chee Systematic Sampling 19

Example (Lohr, 5.12)

sampling of dumps and landfills

to see if toxic waste is leaking

from containers

choose a random point in the

landfill area; construct a grid

containing that point

take soil samples from each grid

point

gives good coverage if there is little prior knowledge about where the

toxic materials might be

but could fail if the material is regularly placed

Shivon Sue-Chee Systematic Sampling 20

What is Repeated systematic sampling? §7.6

example (Table 7.2): population of N = 960 elements, numbered

consecutively

choose ns = 10 random starting points, take systematic samples of

size 6

Random Second Third Sixth

Sample Starting element element element

Number Point in sample in sample . . . in sample

1 6 166 326 . . . 806

2 17 177 337 . . . 817

3 21 181 341 . . . 821

4 42 202 362 . . . 842

5 73 233 393 . . . 873

6 81 241 401 . . . 881

7 86 246 406 . . . 886

8 102 262 422 . . . 902

9 112 272 432 . . . 912

10 145 305 465 . . . 945

Shivon Sue-Chee Systematic Sampling 21

... repeated systematic sampling

estimate population mean by averaging the row averages

µˆ =

ns∑

i=1

y¯i

ns

(7.12)

where ns is the number of systematic samples

estimate σ2 by the variance across the rows

s2y¯ =

∑ns

i=1(y¯i − µˆ)2

ns − 1 , V̂ (µˆ) =

(

1− n

N

) s2y¯

ns

(7.13)

Why repeated SYS?

No need to make any assumption about the order of the population.

Shivon Sue-Chee Systematic Sampling 22

Homework

Readings: §7.1 – §7.6

Skip §7.7

HW: 7.2, 7.4, 7.5, 7.11, 7.15, 7.19, 7.24

Shivon Sue-Chee Systematic Sampling 23

学霸联盟

We should learn...

What is systematic sampling? It’s advantages and disadvantages...

How does systematic sampling compare with SRS?

What is repeated systematic sampling?

Shivon Sue-Chee Systematic Sampling 1

Systematic Sampling (Ch.7)

choose a random starting point and choose each 10th, or 15th, or

20th observation

Eg, every 10th person leaving a shopping centre

Eg, every 15th item in an ordered list of accounts

Eg, every half hour/fifteen minutes/ ... take an item from an

assembly line for inspection

Eg, our class heights (for n=10)

Shivon Sue-Chee Systematic Sampling 2

Class example

example: our heights (for n=10)

> N=length(height); N

[1] 217

> n=10

> #Systematic random sampling

> k=floor(N/n);k

[1] 21

> #random starting point

> set.seed(1234)

> start=sample(1:k,1); start

[1] 19

> sys<-seq(start,N,k);sys

[1] 19 40 61 82 103 124 145 166 187 208

> sample_sys<-height[sys];

> sample_sys

[1] 170.0 160.0 170.5 180.0 181.0 170.0 175.0 175.0 172.0 180.0

Shivon Sue-Chee Systematic Sampling 3

Advantages and disadvantages of systematic

sampling

easier, especially with no sampling frame

easier to organize, especially with untrained interviewers

more precise if the y ’s in the population are ordered

equivalent to a SRS if the y ’s in the population are in random order

hard to estimate the variance of y¯

so we usually use the SRS variance estimate

very biased if the y ’s in the population are cyclical, and the sampling

interval coincides with the cycle

see Figures 7.1 – 7.3

Shivon Sue-Chee Systematic Sampling 4

Population types

Shivon Sue-Chee Systematic Sampling 5

What are Random, Ordered and Periodic

populations? Examples...

Random population (ρ ≈ 0)

The elements of the population are in random order.

Ordered population (ρ < 0)

The elements of the population have values that trend upward or

downward when listed.

Periodic population (ρ ≈ 1)

The elements of the population tend to cycle upward and downward in a

regular pattern when listed.

Shivon Sue-Chee Systematic Sampling 6

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 7

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 8

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 9

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 10

Inference from systematic samples: §7.3, 7.4

comparable to SRS

When systematic sampling is nearly equivalent to SRS, we can

estimate V (y¯sy ) by V̂ (y¯).

see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)

HW: there’s a (small) mistake in formula (7.3) in the 7th edition:

what is it?

usual technique for estimating sample size n (§7.5)

Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).

Shivon Sue-Chee Systematic Sampling 11

Inference from systematic samples: §7.3, 7.4

When are the SRS formulas okay?

if the sample is ‘similar’ to a simple random sample

i.e. population is unordered, with respect to the variable of interest

SRS variance will be an over-estimate if the population is ordered

SRS estimate will just be wrong if the population is periodic and the

sampling interval coincides with the cycle

e.g. data on store sales taken every 7th day; data on rainfall taken

every 12 months; ...

Shivon Sue-Chee Systematic Sampling 12

Inference from systematic samples: §7.3, 7.4

When are the SRS formulas okay?

if the sample is ‘similar’ to a simple random sample

i.e. population is unordered, with respect to the variable of interest

SRS variance will be an over-estimate if the population is ordered

SRS estimate will just be wrong if the population is periodic and the

sampling interval coincides with the cycle

e.g. data on store sales taken every 7th day; data on rainfall taken

every 12 months; ...

Shivon Sue-Chee Systematic Sampling 13

Inference from systematic samples: §7.3, 7.4

When are the SRS formulas okay?

if the sample is ‘similar’ to a simple random sample

i.e. population is unordered, with respect to the variable of interest

SRS variance will be an over-estimate if the population is ordered

SRS estimate will just be wrong if the population is periodic and the

sampling interval coincides with the cycle

e.g. data on store sales taken every 7th day; data on rainfall taken

every 12 months; ...

Shivon Sue-Chee Systematic Sampling 14

Why is SRS variance estimate too big if the

population is ordered?

By a systematic sample, we estimate a population mean µ by:

µˆ = y¯sy =

∑n

i=1 yi

n

(7.1)

and an estimate of the variance of y¯sy is:

V̂(y¯sy ) =

(

1− n

N

)s2

n

= V̂(y¯)

where s2 = 1n−1

∑n

i=1(yi − µˆ)2.

equivalent to SRS

Shivon Sue-Chee Systematic Sampling 15

Why is SRS variance estimate too big if the

population is ordered?

BUT, true variances are different!

V (y¯sy ) =

=

=

σ2

n

[1 + (n − 1)ρ]

(

1− n

N

) N

N − 1

compared to SRS, where

V (y¯) =

σ2

n

(

1− n

N

) N

N − 1

if ρ < 0,V (y¯sy ) < V (y¯), so V̂ (y¯sy ) is too big!

Shivon Sue-Chee Systematic Sampling 16

... comparing variances

V (y¯sy ) =

σ2

n

[1 + (n − 1)ρ]

(

1− n

N

) N

N − 1

− 1

n − 1 < ρ < 1

ρ is the average of the correlation coefficients between all possible

pairs of observations in the systematic sample of size n

a. Random, ρ ≈ 0 V (y¯sy ) V (y¯) V̂ (y¯)

b. Ordered, ρ < 0 V (y¯sy ) V (y¯) V̂ (y¯)

c. Periodic, ρ ≈ 1 V (y¯sy ) V (y¯) V̂ (y¯)

Shivon Sue-Chee Systematic Sampling 17

Comparing systematic samples: random vs ordered

population

Estimating average height

Refer to class R codes and output

Type Estimate Variance Estimate

a. Random, ρ ≈ 0 V̂ (y¯sy )

b. Ordered, ρ < 0 V̂ (y¯sy )

Shivon Sue-Chee Systematic Sampling 18

Example (Lohr, 5.12)

sampling of dumps and landfills

to see if toxic waste is leaking

from containers

choose a random point in the

landfill area; construct a grid

containing that point

take soil samples from each grid

point

gives good coverage if there is little prior knowledge about where the

toxic materials might be

but could fail if the material is regularly placed

Shivon Sue-Chee Systematic Sampling 19

Example (Lohr, 5.12)

sampling of dumps and landfills

to see if toxic waste is leaking

from containers

choose a random point in the

landfill area; construct a grid

containing that point

take soil samples from each grid

point

gives good coverage if there is little prior knowledge about where the

toxic materials might be

but could fail if the material is regularly placed

Shivon Sue-Chee Systematic Sampling 20

What is Repeated systematic sampling? §7.6

example (Table 7.2): population of N = 960 elements, numbered

consecutively

choose ns = 10 random starting points, take systematic samples of

size 6

Random Second Third Sixth

Sample Starting element element element

Number Point in sample in sample . . . in sample

1 6 166 326 . . . 806

2 17 177 337 . . . 817

3 21 181 341 . . . 821

4 42 202 362 . . . 842

5 73 233 393 . . . 873

6 81 241 401 . . . 881

7 86 246 406 . . . 886

8 102 262 422 . . . 902

9 112 272 432 . . . 912

10 145 305 465 . . . 945

Shivon Sue-Chee Systematic Sampling 21

... repeated systematic sampling

estimate population mean by averaging the row averages

µˆ =

ns∑

i=1

y¯i

ns

(7.12)

where ns is the number of systematic samples

estimate σ2 by the variance across the rows

s2y¯ =

∑ns

i=1(y¯i − µˆ)2

ns − 1 , V̂ (µˆ) =

(

1− n

N

) s2y¯

ns

(7.13)

Why repeated SYS?

No need to make any assumption about the order of the population.

Shivon Sue-Chee Systematic Sampling 22

Homework

Readings: §7.1 – §7.6

Skip §7.7

HW: 7.2, 7.4, 7.5, 7.11, 7.15, 7.19, 7.24

Shivon Sue-Chee Systematic Sampling 23

学霸联盟