= = Town 1 Town 2 Rural Area = The basic calculations are:

Transcription

1 Stratified Radom Samplig We ofte will ave groups i te populatio tat may ted to ave very differet values for te resposes of iterest I tese istaces we may wis to improve our estimates by samplig te groups separately A stratified radom sample is oe obtaied by separatig te populatio elemets ito ooverlappig groups, called strata, ad te selectig a simple radom sample from eac stratum Care must be take tat te selectio of samplig uits amog strata are idepedet Te objective of stratified radom samplig is to icrease te amout of iformatio obtaied for a give cost by reducig te variace of te populatio estimators Te idea is to produce strata tat are very omogeeous witi yet eterogeeous amog strata 3 Stratified radom samplig may also reduce survey costs if te populatio is divided ito groups tat provide for samplig coveiece (suc as stratificatio by area or regio 4 Stratified radom samplig also permits separate estimates of te populatio parameters for eac stratum 5 It may also serve to "spread out" te sample over te populatio to provide te "represetativeess" tat some ivestigators may require Tis is ofte doe for spatial or temporal samplig 6 I summary, te advatages of stratified radom samplig are (a icreased precisio over SRS, (b separate estimates for eac stratum or group, ad (c costs of samplig may be lower Te major disadvatage is tat a frame is eeded for eac stratum 7 et be te umber of strata i te survey ad let be te size of te simple radom sample take witi te t stratum et tere be elemets i te populatio i te t stratum 8 A estimator of te populatio mea µ from stratified radom samplig is y µˆ were is te stratum mea Its variace is give by --- y y ---- y i t i Var ( y were S Var ( y Var ( y is te variace of te mea i te stratum A estimator of tis variace is foud by substitutig for i te previous t s S formula to obtai var ( y s ( s 9 If i every stratum te sample estimator y is ubiased, te y usig te above formula is a ubiased estimator of te populatio mea µ EXST 70 8 Example : Average televisio viewig time is to be estimated i tows ad a rural area of a specific couty ad te a overall average viewig time is to be estimated A telepoe survey is used to determie te amout of time per week tat a family speds viewig te televisio Te followig summary statistics are provided: y s Tow Tow Rural Area var ( y Te basic calculatios are: y [ 55 ( ( ( 9000 ] var ( y ( 30 [ ( 55 ( 540 ( 6 ( 5303 ( 93 ( 636 ] 97 A approximate 95% cofidece iterval migt be costructed as y ± var( y wic is 7675 ± 97 Tis produces te iterval ( 4868, 3048 Tus, tis iterval icludes te true populatio average viewig time wit 95% cofidece ote, owever, tat tis iterval may ot be a good estimator We ave assumed tat eiter ( te idividual sample sizes are large or ( tat eac variace as a ci-square distributio Te sum of ci-squares is also ci-square, but te weigted sum of ci-squares wic permit differet expected values for eac variace, as doe ere, is geerally ot ci-square Tis is te same problem as occurs wit te two-sample t test wit uequal variaces Oe possible solutio is to compute a effective degrees of freedom usig Sattertwaite s (946 approximatio Defie g (, te te effective degrees of freedom to be used i te t table could be computed as g s 4 e g s mi were ( < e < ( EXST 70 83

2 For te telepoe survey of viewig time, g 55 ( , g 485, ad g g s 89, 708,, 698, 63, 057 ad e, wic would give a critical t-value wit degrees of freedom of 08 g s Repeat above aalysis wit SURVEYMEAS * STRATIFIEDSAS - Stratified Radom Samplig of Televisio *; * viewig time, i ours per week for tows ad a rural *; * area From Sceaffer et al 990:0 *; * *; * *; Optios S78 PS55 Pageo odate oceter FORMCHAR /\<>* ; Title "Stratified Radom Samplig of Televisio Viewig Time"; Data TV; egt Stratum $0; Iput Stratum & PopulatioSize SelectioProbSampleSize/PopulatioSize; SampligWeigt/SelectioProb; Do i To SampleSize; Iput Output; Drop i; Ed; abel Time"Viewig Time (Hrs/Wk" PopulatioSize"Stratum Size" SampleSize"Sample Size" SelectioProb"Selectio Probability" SampligWeigt"Samplig Weigt"; Datalies; Tow A Tow B Rural Area ; Proc Sort DataTV; By Stratum; Ru; Proc Prit DataTV Split" "; Sum SampligWeigt; Ru; EXST * Summarize te data wit MEAS for "ad" calculatios *; Proc Meas DataTV Mea Var StdErr; Class Stratum; Var Time; Ru; * Form a data set cotaiig stratum sizes for use wit *; * SURVEYMEAS Te stratum populatio sizes are i _total_ *; Data StratumSizes; Set TV; By Stratum; If FirstStratum; /* Keep oly te first observatio */ _total_populatiosize; Ru; * Aalysis wit SURVEYMEAS *; * First, produce estimates for eac stratum separately *; Proc SurveyMeas DataTV TotalStratumSizes Mea CM Sum CSum CV; By Stratum; Strata Stratum / ist; Var Time; Ru; * ow, estimate te total over all strata *; Proc SurveyMeas DataTV TotalStratumSizes Mea CM Sum CSum CV; Strata Stratum / ist; Var Time; Ru; EXST 70 85

3 Te executed program produced te followig results Stratified Radom Samplig of Televisio Viewig Time Viewig Stratum Sample Selectio Samplig Time Obs Stratum Size Size Probability Weigt (Hrs/Wk Rural Area Rural Area Rural Area Rural Area Rural Area Rural Area Rural Area Rural Area Rural Area Rural Area Rural Area Rural Area Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow A Tow B Tow B Tow B Tow B Tow B Tow B Tow B Tow B EXST Stratified Radom Samplig of Televisio Viewig Time Te MEAS Procedure Aalysis Variable : Time Viewig Time (Hrs/Wk Stratum Obs Mea Variace Std Error - Rural Area Tow A Tow B Stratified Radom Samplig of Televisio Viewig Time 3 StratumRural Area Te SURVEYMEAS Procedure Data Summary umber of Strata umber of Observatios StratumRural Area Stratum Iformatio Stratum Populatio Samplig Idex Stratum Total Rate Obs Variable Rural Area Time Std Error ower 95% Upper 95% Coeff of Variable Mea of Mea C for Mea C for Mea Variatio Time ower 95% Upper 95% Variable Sum Std Dev C for Sum C for Sum Time EXST 70 87

4 Stratified Radom Samplig of Televisio Viewig Time 4 StratumTow A Te SURVEYMEAS Procedure Data Summary umber of Strata umber of Observatios 0 StratumTow A Stratum Iformatio Stratum Populatio Samplig Idex Stratum Total Rate Obs Variable Tow A Time Std Error ower 95% Upper 95% Coeff of Variable Mea of Mea C for Mea C for Mea Variatio Time ower 95% Upper 95% Variable Sum Std Dev C for Sum C for Sum Time EXST Stratified Radom Samplig of Televisio Viewig Time 5 StratumTow B Te SURVEYMEAS Procedure Data Summary umber of Strata umber of Observatios 8 StratumTow B Stratum Iformatio Stratum Populatio Samplig Idex Stratum Total Rate Obs Variable Tow B Time Std Error ower 95% Upper 95% Coeff of Variable Mea of Mea C for Mea C for Mea Variatio Time ower 95% Upper 95% Variable Sum Std Dev C for Sum C for Sum Time EXST 70 89

5 Stratified Radom Samplig of Televisio Viewig Time 6 Te SURVEYMEAS Procedure Data Summary umber of Strata 3 umber of Observatios 40 Stratum Iformatio Stratum Populatio Samplig Idex Stratum Total Rate Obs Variable Rural Area Time Tow A Time 0 3 Tow B Time Std Error ower 95% Upper 95% Coeff of Variable Mea of Mea C for Mea C for Mea Variatio Time ower 95% Upper 95% Variable Sum Std Dev C for Sum C for Sum Time EXST Sample Size Allocatio i Stratified Radom Samplig For a give stratum, take a larger sample if Te stratum is larger Te stratum is more variable 3 Costs of samplig are ceaper i te stratum Two metods are available for fidig a optimal samplig size (goig back to te samplig objectives ad are Fid for a give fixed total cost Fid for a give precisio ( var ( y or boud o te error of estimatio Defie te cost fuctio to be C c 0 c, were c 0 is te fixed over-ead cost or te cost of "goig samplig", ad c is te cost of samplig a sigle elemet from stratum ote tat c 0 does ot cage as we vary te sample size allocatio Defie W to be te samplig weigt of te t stratum Te for a give fixed costs, te total sample size would be ( C c 0 ( S c ( S c If te variace of te sample mea is to be fixed (we wat to put a boud o te error of estimatio, te we would fid a optimal by W S c W S c --- var ( y --- W S EXST 70 9

6 or alteratively S c S c - var ( y S Sice we are costraiig te variace of te sample mea i te secod set of equatios so tat we are puttig a boud o te error of estimatio, te we ca substitute i te equatio for te variace te iformatio about te boud troug var ( y ( B Z were B is te boud we wat to put o te error of estimatio, ad Z is te quatile from te stadard ormal distributio correspodig to te cofidece tat we would like to ave tat te true populatio mea will be cotaied witi te boud If you are computig a sample size for estimatig a populatio total, te covert te boud for te total to a boud o te mea by dividig te boud o te total by te populatio size, B B total To allot te resultig sample size optimally amog te strata, use W S c W S c EXST 70 9 Example 3: Compute a optimal sample size ad allocatio for te telepoe survey of te televisio viewig times Assume te cost of samplig eac ouseold i te tows is $900,, ad te cost of samplig a rural ouseold is $600, Approximate te c c 9 c 3 6 strata stadard deviatios by S 5, S 5, ad S 3 0 Fid te optimal sample sizes to estimate te average viewig time wit a boud o te error of estimatio of ours wit 95% cofidece First, covert te boud ito te variace of te sample mea (our estimator Witout worryig about te t distributio ere, assume B var( y so tat var ( y ( B Here B so tat var ( y ( wic is used i te sample size formula ote: we are arguig tat te variace of te sample mea sould ot be larger ta so tat wit 95% cofidece, te boud o te error of estimatio will ot exceed ours ow, apply te sample size formulas for fixed variace Below are give some of te itermediate calculatios: S c 55 ( 5 9 ( 6 ( ( S c 55 ( 5 6 ( 5 93 ( S 55 ( 5 6 ( 5 93 ( 0 7, 5 ad 8835 ( ( 7, 5 ow allocate te 58 samplig uits amog te 3 strata, ( ad ( ( Tus, we would make 8 iterviews i tow, 3 i tow, ad 7 i te rural area to be 95% cofidet tat te boud o te error of estimatio of te populatio mea would ot exceed ours of televisio viewig time EXST 70 93,,

7 Example 4: Cotiue wit te previous example Wat was te cost of te first samplig allocatio if te cost of "goig samplig" is $00? Wat sample sizes sould be take if te total fuds available are $600? Te cost of te first samplig pla would be C 00 8 ( 9 3 ( 9 7 ( 6 $74 ( Usig te sample size formulas for fixed total costs, , 8835 wic yields a allocatio of 5, 8, ad 3 3 Te cost of tis samplig pla would be 00 5 ( 9 8 ( 9 3 ( 6 $605 Here, we eiter come up wit a additioal $5 or we reduce oe of te sample sizes, say te 8 to 7 i te secod stratum 50 Special Types of Sample Size Allocatio i Stratified Radom Samplig 5 Equal Allocatio amog strata Tis is typically used we te primary focus of te study is to test ypoteses about differeces amog te strata o te caracteristic(s measured Te t-test ad AOVA are most robust to violatios of assumptios we sample sizes are equal -- For te smallest variaces for testig ypoteses, te sample sizes are made proportioal to te stratum stadard deviatios, σ σ j j As we ofte will ot kow te stratum variaces before ad, te equal allocatio provides a type of robust samplig pla wit respect to ypotesis testig For may researc projects were testig ad model buildig are te primary focus, tis type of allocatio sceme sould be carefully cosidered However, if te focus is populatio parameter estimatio, te te followig sample size allocatio scemes ca yield muc more efficiet estimates ta equal allocatio EXST Proportioal Allocatio amog strata We wis to maitai te same samplig fractio (samplig effort for eac stratum Tis produces "self-weigtig" formulas wic ca simplify aalysis if eeded Te geeral use is we variaces amog strata are similar ad costs are also about equal ad S var ( y S 53 eyma Allocatio amog strata I tis istace we assume tat costs are te same i eac stratum, but tat te variaces may differ amog te strata S ad j S j j S var ( y S 54 Optimal Allocatio amog strata Here we wis to cosider te strata sizes, strata variaces, ad costs of samplig i eac stratum Tese formulas were give earlier ad are S c S c EXST 70 95

8 60 Commets o te use of Stratified Radom Samplig As i te oe-way aalysis of variace, we ca decompose te total variability of te data ito compoets of variace, a betwee or amog strata variace, ad a witi stratum variace: σ σ b σ w ( y y were σ b is te betwee strata variace, ad σ w is te pooled σ witi stratum variace Uder proportioal allocatio, var ( y, wic σ w for large is approximately var ( y Terefore, te ratio of te variace of te mea uder simple radom samplig (SRS to tat uder stratified radom samplig (Str is approximately, σ var ( y SRS var ( y Str σ b σ w σ w σ b σ w Tus, proportioal allocatio for stratified radom samplig will yield more efficiet estimates ta SRS weever te betwee strata variatio is large relative to te witi strata variatio Tis usually appes we te meas differ cosiderably amog te strata EXST Estimatio of Subgroup Caracteristics i Stratified Radom Samplig I tis situatio we will ave or more subgroups or subdomais tat were ot stratified o, but tat we wis to ave separate estimates for Sice te stratum weigts (usig te stratificatio variables are ulikely to represet te relative sizes of te subgroups i te strata, estimates usig our usual stratified radom samplig formulas will ot be ubiased (we will be usig te wrog weigts if we use te stratum weigts I order to costruct ubiased estimates, we eed te stratum sizes of eac subgroup, te we treat te estimatio problem as te usual stratified radom samplig were eac subgroup is treated as a separate populatio, recogizig tat te variaces will be coditioal upo te observed subdomai sample sizes, ad are, terefore, oly approximatios EXST 70 97

9 Post Stratificatio or Stratificatio After Samplig I tis samplig desig, we take a simple radom sample but we would like to get some of te gais i precisio as if we ad take a stratified radom sample Tis desig may be employed we it is ot possible to stratify te populatio prior to takig te sample Tus, we post-stratify te sample (stratificatio after te sample as bee selected wit te aim of icreased precisio of te estimators I order to use tis approac, we still eed te stratum sizes Agai, tis desig ofte is used we it is difficult, impractical, or impossible to stratify iitially If te populatio ca be stratified iitially, te a stratified radom sample is to be recommeded Usig te same otatio as for stratified samplig, our estimator of te populatio mea is µˆ --- y wic is te same estimator as uder (pre-stratificatio However, te variace of tis estimator must ow take ito accout te fact tat te stratum sample sizes are ow radom variables ad are ot fixed as i te usual stratified radom samplig estimator Var ( µˆ S S were S Te variace formula is basically te usual variace of te mea i ( y i y uder stratified radom samplig plus a compoet due to te radom stratum sample sizes A estimator of te post-stratified variace of te mea estimator is s s If is kow ad if 0 for eac stratum, te post stratificatio of te sample is early as precise as stratified radom samplig wit proportioal allocatio (Sceaffer et al 990:9 Te above variace approximatio oly works well we is large ad all terms are relatively large Tus, it is ot to our advatage to post stratify too fiely (ie, too may strata EXST Example 5: (From Example 58 of Sceaffer et al 990:3-3 A large firm kows tat 40% of its accouts receivable are wolesale ad 60% are retail To idetify a accout witout examiig te file, owever, is difficult A auditor takes a simple radom sample of size 00 ad te post stratifies te sample ito wolesale ad retail accouts Te summary statistics are: Wolesale Retail y 50 y 80 s 0 s 90 W 040 W 060 Estimate µ, te populatio mea, ad also give a 95% cofidece iterval estimate for te populatio mea otice tat i tis problem we are ot give te populatio size or te stratum sizes, owever, we are give te stratum weigts, W First write te post stratificatio formulas i terms of te W as µˆ W y To compute te variace we eed to be able to igore te fiite populatio correctio factor If te populatio is very large relative to te sample sizes te te approximatio below sould work, -- W s ---- ( W s To complete te calculatios we ave µˆ 04 ( ( ad [ 04 ( ( 90 ] [ 06 ( 0 04 ( 90 ] A approximate 95% cofidece iterval could be computed as 76 ± 797 wic yields te iterval ( 346, 406 EXST 70 99

10 Double Samplig or Two-Pase Samplig for Stratificatio For may problems it will be very difficult if ot impossible to stratify te populatio prior to samplig eve toug we migt kow tat it would be very appropriate to stratify Ufortuately we may ot be able to use te post stratificatio estimators as we will likely ot kow te stratum sizes eiter For example, we migt wis to estimate te average daily caloric itake for female graduate studets ad we would like to stratify o te basis of marital status (ot married, married ad a relative measure of obesity (ti, average, obese Te uiversity records may ot cotai te iformatio o marital status ad probably will ave o iformatio about a perso s eigt ad weigt ad so tis iformatio is ot available util we iterview te studet I additio, computig te caloric itake requires a detailed iterview o eatig abits over a oe-week time period ad so oly a limited umber of tese iterviews are possible Oe possible samplig desig tat could be used i tis situatio is called two-pase or double samplig Double samplig cosists of two pases of samplig I te first pase (pase sample a simple radom sample of size of te populatio is selected ad eac of tese idividuals is iterviewed to determie te caracteristics o wic stratificatio is to be based I our example, te graduate studets would be asked for teir marital status, eigt, ad weigt Te, te pase sample is stratified accordig to te stratifyig variables to give idividuals i te strata From tis pase "frame" a stratified radom sample (pase sample wit stratum sizes is selected otice tat We te make measuremets o all of te uits i te pase sample For our example, we would take simple radom samples witi eac of te 6 strata (married-ti, married-average,,umarried-obese from te origial sample ad would determie te caloric itake for eac of te people selected ito te pase sample otice tat if te pase sample is a simple radom sample, te te strata weigts W ca be estimated witout bias usig te estimator w For eac of te strata of te pase sample, compute te sample mea ad sample variace Te a estimator of te populatio y s mea is µˆ w y You sould otice tat bot w y ad are radom variables Tis must be cosidered we costructig a variace for te estimator If te samplig fractios i te secod pase sample, -----, are small ad te populatio size,, is large, te te variace ca be approximated by EXST w w s w ( y µˆ w If is very large relative to te suc tat we ca igore te correctio factors , te te variace ca be writte as w s w ( y µˆ Te secod term i te variace formula is tere because te w were estimated from te sample O a casual examiatio of tis formula it appears tat a desirable strategy would be to desig te sample so tat te sample meas are all early equal ad tus, elimiatig te secod term i te variace formula However, te beefits from stratificatio come about because te witi-stratum variaces, s, are very small Terefore, double samplig for stratificatio works by reducig te witi-stratum variaces but wit a pealty for estimatig te stratum sizes If stratificatio does result i small witi-stratum variaces, te te beefits of double samplig ca outweig te pealties imposed Double samplig ca also be very useful we te goals of te samplig desig iclude ypotesis testig ad modelig Tere may be several factors ad teir iteractios tat will be tested or modeled usig teciques suc as regressio, AOVA (aalysis of variace, or log-liear models I order for te iferece usig tese predictor variables to be strog, sufficietly large sample sizes eed to be obtaied for eac cell or cross-classificatio of te predictor variables We takig a simple radom sample from te populatio it may appe tat some table cells will ave very small or eve zero frequecy couts If te frequecies are zero, te desired ypoteses about iteractios or eve mai effects may be impossible to test or estimate To reduce or elimiate tis problem, double samplig may be used Here, te pase sample is agai used to cross-classify a large sample of subjects accordig to te predictor variables (stratificatio variables Te te pase sample is selected i suc a way as to isure tat eac cell of te cross-classificatio is sufficietly represeted i te sample Ofte times equal allocatio is used i pase as geerally, te purpose for te sample is testig or modelig, rater ta estimatio of populatio parameters per se EXST 70 0

11 Example 6: (Take from Sceaffer et al 990: 34 From a list of erollmets ad faculty sizes for America four-year colleges ad uiversities, it is desired to estimate te average erollmet (for te academic year Private istitutios ted to be smaller ta public oes, so stratificatio is i order However, te list is ot broke up tis way, eve toug te data are coded to idicate te type of college or uiversity Tus, te type of college (public or private ca be obtaied quickly, wile te erollmet data is more cumbersome to adle A oe-i-te systematic sample (select every tet elemet wile movig dow te list was take to obtai iformatio o te type of college Tis resulted i te followig: Private Public Total Subsamples of private ad public colleges gave te followig data o erollmets ad faculty size Estimate te average erollmet for America colleges ad uiversities i Private Public Erollmet Faculty Erollmet Faculty w w y 6809 y 5857 s 777 s EXST 70 0 Calculatios: µˆ ( w s ---- ( w s ---- [ w ( y µˆ w ( y µˆ ] ( [ ] [( ] [( 0596 ( ( 0404 ( ] ad terefore se ( µˆ otice tat te part of te variace formula tat deals wit te estimatio of te stratum sizes oly accouts for 5% of te total variace ad is, terefore, ot a great pealty If we compute a variace for te sample of size 3 as if it were a simple radom sample we would obtai a variace of wic is a stadard error of 906 wic is muc larger ta tat via double samplig Tus, it appears tat double samplig for stratificatio as improved te precisio of te estimates EXST 70 03