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Post-stratification. Sometimes there is an obvious stratification variable Don’t know stratum assignment for each SU  can’t stratify Take a SRS, e.g. Know stratum totals, N h , which can be used to improve estimation relative to SRS estimators

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post stratification
Post-stratification
  • Sometimes there is an obvious stratification variable
  • Don’t know stratum assignment for each SU  can’t stratify
    • Take a SRS, e.g.
  • Know stratum totals, Nh, which can be used to improve estimation relative to SRS estimators
    • Very common for household and population surveys
    • Census data provide number of persons, households per area, by age, …
food spending example
Food spending example
  • Objective: estimate the average amount spent on food per week in NC
  • Possible stratification variable: household composition
    • Family households might be expected to have higher food bills than non-family households
  • Sampling frame
    • List of all households in NC
    • No information on household composition
  • From U.S. census data, the distribution of household composition is known
food spending example 2
Food spending example – 2
  • 2000 Census data on household composition in NC
post stratification 2
Post-stratification – 2
  • Design phase
    • SRS of n OUs (could be another design)
    • Identify poststrata
  • Sample selection phase
    • SRS of n Ous
    • After sampling, get n1 , n2 , …, nH - BUT can’t determine at this point
  • Data collection phase
    • Include a question that gathers information on stratum assignment
    • OU i belongs to poststratum h
    • Can determine values for n1 , n2 , …, nH
    • Note that values for nh are random – differ for each sample
food spending example 3
Food spending example – 3
  • Select SRS of n = 1000 households
  • Collect data on household composition
    • List each household member and relationship to respondent
  • Tabulate number of households for different size categories (nh)
  • Use Census 2000 population information on number of households for composition categories (Nh)
post stratification 3
Post-stratification – 3
  • Note that sample composition across post-strata is different from population composition
    • Consider percentage distribution across post-strata for population (column 3) and for sample (column 5)
    • Could improve estimates by “calibrating” to the post-stratum population totals – this is the basis for post-stratification estimator
  • Another way to look at the sample composition is to compare the expected sample size for post-strata with the observed sample size for post-strata obtained from the SRS
    • Expected sample size for post-stratum h
post stratification 4
Post-stratification – 4
  • Estimating a population mean
    • Domain estimation for means, then pool stratum estimates
    • Variance approximation (nh > 30, n large)
food spending example 6
Food spending example – 6
  • Food expenditures last week
food spending example 7
Food spending example – 7
  • Estimate population mean
  • Estimate SE of estimated mean
post stratification 5
Post-stratification – 5
  • Formulas involve weighted averages of stratum sample means and variances
    • Mean estimator looks like stratification estimator
    • Variance estimator is not the stratification variance estimator
  • Estimating a population total?
  • Estimating a population proportion?
post stratification 6
Post-stratification – 6
  • Estimator for population total
  • Weight under post-stratified estimator
    • whj = Nh /nh
post stratification and nonresponse
Post-stratification and nonresponse
  • May get disproportionate allocation across poststrata because of differential stratum nonresponse rates
  • Same approach can be used to improve estimation by using ratio of post-stratum population size to total population in averaging estimates across post-strata
implicit assumption
Implicit assumption
  • Sample post-stratum mean from responding units is an unbiased estimate of the population post-stratum mean
  • Distribution of Y for responding part of post-stratum population is (approximately)
    • Same as distribution for whole poststratum population
    • Same for the nonresponding poststratum population
  • Often a poor assumption