Ch 4: Stratified Random Sampling (STS) 4/16/2017 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 Stat804

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 h=1 h=2 . . . . . . h=H

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 Stratum (h) Stratum size (Nh) Sample size (nh) 1 (NE) 220 21 2 (NC) 1054 103 3 (S) 1382 135 4 (W) 422 41 Total 3078 300

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 Stratum (h) SU (j) 1 2 3 …

Ag example – 3 Stratum (h) SU (j) 1 2 3 … 220 4 421 422

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 Stratum (h) Stratum size (Nh) Sample size (nh) Sample mean ( ) Estimated stratum total ( ) 1 (NE) 220 21 97,630 21,478,558 2 (NC) 1054 103 300,504 316,731,379 3 (S) 1382 135 211,315 292,037,391 4 (W) 422 41 662,295 279,488,706 Total 3078 300 Acres devoted to farms / co Total farms acres for stratum

Ag example – 5 Estimated total farm acres in US

Ag example – 6 Stratum (h) Stratum size (Nh) Sample size (nh) Sample variance ( ) 1 (NE) 220 21 7,647,472, 708 2 (NC) 1054 103 29,618,183,543 3 (S) 1382 135 53,587,487,856 4 (W) 422 41 396,185,950,266 Total 3078 300

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 h=1 h=2 . . . h=H Stratum 1 Stratum H

Notation – 2 Population sizes Partition sample of size n across strata 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 Population and stratum totals 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 We’ll 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 as and 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 Sample design 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 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 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 < A nh = max for Nh > 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 Set a design for each stratum 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