Ch 4: Stratified Random Sampling STS

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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

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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
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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

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Ag example 2
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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

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Ag example 3
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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

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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

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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

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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

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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

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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

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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

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Ag example 4
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Ag example 5

• Estimated total farm acres in US

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Ag example 6
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Ag example 7

• Estimated variance for estimated total farm acres
  in US

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Ag example 8

• Compare with SRS estimates

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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

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Ag example 9

• Estimated mean farm acres / county

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Ag example 10

• Estimate variance of estimated mean farm acres /
  county

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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
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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)

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Notation 3

• Response variable
• Yhj characteristic of interest for OU j in
  stratum h
• Population and stratum totals

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Notation 4

• Population and stratum means

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Notation 5

• Population stratum variance

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Notation 6

• SRS estimators for stratum parameters

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STS estimators

• For population total

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STS estimators 2

• For population mean

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STS estimators 3

• For population proportion

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Properties

• STS estimators are unbiased
• Each estimate of stratum population mean or total
  is unbiased (from SRS)

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Properties 2

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

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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

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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

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Example

• Note weights for each OU within a stratum are
  the same

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Example 2

• Dataset from study

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Sampling weights 2

• For STS estimators presented in Ch 4, sampling
  weight is the inverse inclusion probability

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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

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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
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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

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Proportional allocation

• Stratum sample size allocated in proportion to
  population size within stratum
• Allocation rule

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Ag example 11
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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

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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

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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 ?

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Ag example 12

• Even though we have used proportional allocation,
  rounding in setting sample sizes can lead to
  unequal (but approximately equal) weights

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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

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Caribou survey example
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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

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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

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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

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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

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Welfare example 2

• Optimal allocation with phone strata

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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

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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

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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

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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

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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

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Welfare example 4

• Equal allocation with population density strata

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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

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Welfare example 5

• 95 CI, e 0.10 for all pop density strata

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Compromise allocations
Proportional Allocation
Equal Allocation
nh n /H
nh nNh /N
nh
nh
Nh
Nh
Nh
Square Root Allocation
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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
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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
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Welfare example 6

• Comparing equal, proportional and square root
  allocation

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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

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Welfare example 7

• Ad hoc allocation

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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

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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

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Welfare example 8

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

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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