View by Category

Loading...

PPT – Ch 4: Stratified Random Sampling STS PowerPoint presentation | free to download

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Ch 4 Stratified Random Sampling (STS)

- DEFN A stratified random sample is obtained by

separating the population units into

non-overlapping groups, called strata, and then

selecting a random sample from each stratum

Procedure

- Divide sampling frame into mutually exclusive and

exhaustive strata - Assign each SU to one and only one stratum
- Select a random sample from each stratum
- Select random sample from stratum 1
- Select random sample from stratum 2

Stratum H

Stratum 1

h1

h2

. . .

. . .

hH

Ag example

- Divide 3078 counties into 4 strata corresponding

to regions of the countries - Northeast (h 1)
- North central (h 2)
- South (h 3)
- West (h 4)
- Select a SRS from each stratum
- In this example, stratum sample size is

proportional to stratum population size - 300 is 9.75 of 3078
- Each stratum sample size is 9.75 of stratum

population

Ag example 2

Procedure 2

- Need to have a stratum value for each SU in the

frame - Minimum set of variables in sampling frame SU

id, stratum assignment

Ag example 3

Procedure 3

- Each stratum sample is selected independently of

others - New set of random numbers for each stratum
- Basis for deriving properties of estimators
- Design within a stratum
- For Ch 4, we will assume a SRS is selected within

each stratum - Can use any probability design within a stratum
- Sample designs do not need to be the same across

strata

Uses for STS

- To improve representativeness of sample
- In SRS, can get ANY combination of n elements in

the sample - In SYS, we severely restricted the set to k

possible samples - Can get bad samples
- Less likely to get unbalanced samples if frame is

sorted using a variable correlated with Y

Uses for STS 2

- To improve representativeness of sample - 2
- In STS, we also exclude samples
- Explicitly choose strata to restrict possible

samples - Improve chance of getting representative samples

if use strata to encourage spread across

variation in population

Uses for STS 3

- To improve precision of estimates for population

parameters - Achieved by creating strata so that
- variation WITHIN stratum is small
- variation AMONG strata is large
- Uses same principal as blocking in experimental

design - Improve precision of estimate for population

parameter by obtaining precise estimates within

each stratum

Uses for STS 4

- To study specific subpopulations
- Define strata to be subpopulations of interest
- Examples
- Male v. female
- Racial/ethnic minorities
- Geographic regions
- Population density (rural v. urban)
- College classification
- Can establish sample size within each stratum to

achieve desired precision level for estimates of

subpopulations

Uses for STS 5

- To assist in implementing operational aspects of

survey - May wish to apply different sampling and data

collection procedures for different groups - Agricultural surveys (sample designs)
- Large farms in one stratum are selected using a

list frame - Smaller farms belong to a second strata, and are

selected using an area sample - Survey of employers (data collection methods)
- Large firms use mail survey because information

is too voluminous to get over the phone - Small firms telephone survey

Estimation strategy

- Objective estimate population total
- Obtain estimates for each stratum
- Estimate stratum population total
- Use SRS estimator for stratum total
- Estimate variance of estimator in each stratum
- Use SRS estimator for variance of estimated

stratum total - Pool estimates across strata
- Sum stratum total estimates and variance

estimates across strata - Variance formula justified by independence of

samples across strata

Ag example 4

Ag example 5

- Estimated total farm acres in US

Ag example 6

Ag example 7

- Estimated variance for estimated total farm acres

in US

Ag example 8

- Compare with SRS estimates

Estimation strategy - 2

- Objective estimate population mean
- Divide estimated total by population size
- OR equivalently,
- Obtain estimates for each stratum
- Estimate stratum mean with stratum sample mean
- Pool estimates across strata
- Use weighted average of stratum sample means with

weights proportional to stratum sizes Nh

Ag example 9

- Estimated mean farm acres / county

Ag example 10

- Estimate variance of estimated mean farm acres /

county

Notation

- Index set for stratum h 1, 2, , H
- Uh 1, 2, , Nh
- Nh number of OUs in stratum h in the population
- Partition sample of size n across strata
- nh number of sample units from stratum h

(fixed) - Sh index set for sample belonging to stratum h

Stratum H

Notation 2

- Population sizes
- Nh number of OUs in stratum h in the population
- N N1 N2 NH
- Partition sample of size n across strata
- nh number of sample units from stratum h
- n n1 n2 nH
- The stratum sample sizes are fixed
- In domain estimation, they are random
- For now, we will assume that the sampling unit

(SU) is an observation unit (OU)

Notation 3

- Response variable
- Yhj characteristic of interest for OU j in

stratum h - Population and stratum totals

Notation 4

- Population and stratum means

Notation 5

- Population stratum variance

Notation 6

- SRS estimators for stratum parameters

STS estimators

- For population total

STS estimators 2

- For population mean

STS estimators 3

- For population proportion

Properties

- STS estimators are unbiased
- Each estimate of stratum population mean or total

is unbiased (from SRS)

Properties 2

- Inclusion probability for SU j in stratum h
- Definition in words
- Formula ?hj

Properties 3

- In general, for any stratification scheme, STS

will provide a more precise estimate of the

population parameters (mean, total, proportion)

than SRS - For example
- Confidence intervals
- Same form (using z?/2)
- Different CLT

Sampling weights

- Note that
- Sampling weight for SU j in stratum h
- A sampling weight is a measure of the number of

units in populations represented by SU j in

stratum h

Example

- Note weights for each OU within a stratum are

the same

Example 2

- Dataset from study

Sampling weights 2

- For STS estimators presented in Ch 4, sampling

weight is the inverse inclusion probability

Defining strata

- Depends on purpose of stratification
- Improved representativeness
- Improved precision
- Subpopulations estimates
- Implementing operational aspects
- If possible, use factors related to variation in

characteristic of interest, Y - Geography, political boundaries, population

density - Gender, ethnicity/race, ISU classification
- Size or type of business
- Remember
- Stratum variable must be available for all OUs

Allocation strategies

- Want to sample n units from the population
- An allocation rule defines how n will be spread

across the H strata and thus defines values for

nh - Overview for estimating population parameters

Special cases of optimal allocation

Allocation strategies 2

- Focus is on estimating parameter for entire

population - Well look at subpopulations later
- Factors affecting allocation rule
- Number of OUs in stratum
- Data collection costs within strata
- Within-stratum variance

Proportional allocation

- Stratum sample size allocated in proportion to

population size within stratum - Allocation rule

Ag example 11

Proportional allocation 2

- Proportional allocation rule implies
- Sampling fraction for stratum h is constant

across strata - Inclusion probability is constant for all SUs in

population - Sampling weight for each unit is constant

Proportional allocation 3

- STS with proportional allocation leads to a

self-weighting sample - What is a self-weighting sample?
- If whj has the same value for every OU in the

sample, a sample is said to be self-weighting - Since each weight is the same, each sample unit

represents the same number of units in the

population - For self-weighting samples, estimator for

population mean to sample mean - Estimator for variance does NOT necessarily

reduce to SRS estimator for variance of

Proportional allocation 4

- Check to see that a STS with proportional

allocation generates a self-weighting sample - Is the sample weight whj is same for each OU?
- Is estimator for population mean equal to

the sample mean ? - What happens to the variance of ?

Ag example 12

- Even though we have used proportional allocation,

rounding in setting sample sizes can lead to

unequal (but approximately equal) weights

Neyman allocation

- Suppose within-stratum variances vary

across strata - Stratum sample size allocated in proportion to
- Population size within stratum Nh
- Population standard deviation within stratum Sh
- Allocation rule

Caribou survey example

Optimal allocation

- Suppose data collection costs ch vary across

strata - Let C total budget
- c0 fixed costs (office rental, field

manager) - ch cost per SU in stratum h (interviewer

time, travel cost) - Express budget constraints asand determine nh

Optimal allocation 2

- Assume general case stratum population sizes,

stratum variances, and stratum data collection

costs vary across strata - Sample size is allocated to strata in proportion

to - Stratum population size Nh
- Stratum standard deviation Sh
- Inverse square root of stratum data collection

costs - Allocation rule

Optimal allocation 3

- Obtain this formula by finding nh such that

is minimized given cost constraints - The optimal stratum allocation will generate the

smallest variance of for a given

stratification and cost constraint - Sample size for stratum h (nh ) is larger in

strata where one or more of the following

conditions exist - Stratum size Nh is large
- Stratum variance is large
- Stratum per-unit data collection costs ch are

small

Welfare example

- Objective
- Estimate fraction of welfare participant

households in NE Iowa that have access to a

reliable vehicle for work - Sample design
- Frame welfare participant list
- Stratum 1 Phone
- N1 4500 households, p1 0.85, c1 100
- Stratum 2 No phone
- N2 500 households, p2 0.50, c2 300
- Sample size n 500

Welfare example 2

- Optimal allocation with phone strata

Optimal allocation 4

- Proportional and Neyman allocation are special

cases of optimal allocation - Neyman allocation
- Data collection costs per sample unit ch are

approximately constant across strata - Telephone survey of US residents with regional

strata - ch term cancels out of optimal allocation formula

Optimal allocation 5

- Proportional allocation
- Data collection costs per sample unit ch are

approximately constant across strata - Within stratum variances are approximately

constant across strata - Y number of persons per household is relatively

constant across regions - ch and Sh terms drop out of allocation formula

Subpopulation allocation

- Suppose main interest is in estimating stratum

parameters - Subpopulation (stratum) mean, total, proportion
- Define strata to be subpopulations
- Estimate stratum population parameters
- Allocation rules derived from independent SRS

within each stratum (subpopulation) - Equal allocation for equal stratum costs,

variances - Stratum variances change across strata

Subpopulation allocation 2

- Equal allocation
- Assume
- Desired precision levels for each subpopulation

(stratum) are constant across strata - Stratum costs, stratum variances equal across

strata - Stratum FPCs near 1
- Allocation rule is to divide n equally across

the H strata (subpopulations) - If Nh vary much, equal allocation will lead to

less precise estimates of parameters for full

population

Welfare example 3

- Suppose we wanted to estimate proportion of

welfare households that have access to a car for

households in each of three subpopulations in NE

Iowa - Metropolitan county
- Counties adjacent to metropolitan county
- Counties not adjacent to metro county

Welfare example 4

- Equal allocation with population density strata

Subpopulation allocation 3

- More complex settings If Sh vary across strata,

can use SRS formulas for determining stratum

sample sizes, e.g., for stratum mean - Result is
- May get sample sizes (nh) that are too large or

small relative to budget - Relax margin of error eh and/or confidence level

100(1-?) - Recalibrate stratum sample sizes to get desired

sample size

Welfare example 5

- 95 CI, e 0.10 for all pop density strata

Compromise allocations

Proportional Allocation

Equal Allocation

nh n /H

nh nNh /N

nh

nh

Nh

Nh

Nh

Square Root Allocation

Square root allocation

- More SUs to small strata than proportional

allocation - Fewer SUs to large strata than equal
- Variance for subpopulation estimates is smaller

than proportional - Variance for whole population estimates is

smaller than equal allocation

Nh

Square Root Allocation

Compromise allocations 2

- May want to set
- Minimum number of SUs in a stratum
- Cap on max number of SUs in a stratum
- Rule
- nh min for Nh lt A
- nh max for Nh gt B
- Apply rule in between A and B
- Square root
- Proportional

nh

max nh

min nh

A B Nh

nh

max nh

min nh

A B Nh

Welfare example 6

- Comparing equal, proportional and square root

allocation

Other allocations

- Certainty stratum is used to guarantee inclusion

in sample - Census (sample all) the units in a stratum
- For certainty stratum h
- Allocation nh Nh
- Inclusion probability ?hj 1
- Ad hoc allocations
- The sample allocation does not have to follow any

of the rules mentioned so far - However, you should determine the stratum

allocation in relation to analysis objectives and

operational constraints

Welfare example 7

- Ad hoc allocation

Determining sample size n

- Determine allocation using rule expressed in

terms of relative sample size nh /n - Rewrite variance of as a function of

relative sample sizes (ignoring stratum FPCs) - Sample size calculation based on margin of error

e for population total

Determining sample size n 2

- Rewrite variance of as a function of

relative sample sizes (ignoring stratum FPCs) - Samples size calculation based on margin of error

e for population mean

Welfare example 8

- Relative sample size for equal allocation
- Value of ?
- For 95 CI with e 0.1

STS Summary

- Choose stratification scheme
- Scheme depends on objectives, operational

constraints - Must know stratum identifier for each SU in the

frame - Set a design for each stratum
- Design for each stratum SRS, SYS,
- Determine n and nh
- Select sample independently within each stratum
- Pool stratum estimates to get estimates of

population parameters

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2014 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2014 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Ch 4: Stratified Random Sampling STS" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Unl University and other schools with their online training by sharing educational presentations for free