Stratified Random Sampling Summary Notes in Progress

Transcription

1 Stratified Radom Samplig Summar otes i Progress ecture 3- Basic Estimatio Metods witi strata ad overall, Examples, Samplig Allocatio Rules. ecture 4- Samplig Allocatio Rules, Optimal Allocatio Proof Outlie, Example of Calculatios, Post Stratificatio

2 Probabilit Samplig We kow properties of samples Geeral Teor 6 Hase-Hurwitz wr Horvitz-Tompso a desig Simple Radom Samplig Witout Replacemet -5 simple samplig uits ested Samplig Uits Cluster Samplig Multi-Stage Samplig 3 Improvig Samplig Desig wit Auxiliar Iformatio Regressio Metods Ratio Estimator 7 Regressio Estimator 8 Stratified Radom Samplig Double Samplig 4 Practical Advatages

3 Improvig Samplig Desigs wit Auxiliar Iformatio Regressio Metods Use relatiosip betwee ad x i liear regressio models. Stratified Radom Samplig Use auxiliar variable s to set up rougl omogeeous groups or strata Double or Two-Pase Samplig Practical Advatages as sometimes ot eoug iformatio o Frame to use eiter directl witout Two-pase samplig

4 samplig uits i populatio, i te sample, sample witout replacemet. Te ke result is te fiite populatio correctio factor o te variace ad stadard error of te sample mea or proportio due to te samplig beig witout replacemet. Te ke parameters are mea, proportio ad populatio total. Applicatios to surves of uma pops, area surves to estimate aimal or plat populatio parameters Modif to Sstematic Radom Samplig sometimes. Good samplig desig if te populatio is omogeeous. ow we move oto cosiderig ow to sample eterogeeous populatios usig stratified radom samplig to improve te precisio over simple radom samplig.

5 Stratified Radom Samplig Ver Widel used optio. Useful we te populatio is eterogeeous ad it is possible to establis strata wic are reasoabl omogeeous witi eac oe.

6 Break up a regio ito 3 omogeeous abitat strata uits uits 3 uits uits uits 3 uits

7 Te populatio is divided up ito omogeeous strata. Te stratum sizes are, Kow o Frame Witi eac stratum a simple radom sample of size, is take. + It is importat to realize tat te samplig is idepedet i te differet strata. ote- Similar to subpopulatios but ow eac subpopulatio is sampled separatel ote: Aalogous to a radomised complete block desig i experimetal desig

8 Experimetal desig- Completel radom desig for omogeeous exptl uits Radomised complete block desig for eterogeeous exptl uits Samplig desig- Simple radom samplig for omogeeous samplig uits Stratified radom samplig for eterogeeous samplig uits

9 Stratified Radom Samplig-W Stratif? Te stratum parameters are of iterest i teir ow rigt Icrease efficiec of estimators of overall populatio parameters b coice of strata tat are omogeeous over te samplig uits witi eac. To make te surve easier to admiister operatioall.

10 Stratified Radom Samplig-W Stratif? Te stratum parameters are of iterest i teir ow rigt Habitat meas of iterest i a wildlife surve Regioal meas of iterest i a statewide political surve

11 Stratified Radom Samplig-W Stratif? Icrease efficiec of estimators of overall populatio parameters b coice of strata tat are omogeeous over te samplig uits witi eac. Te overall populatio total, for example, ca be estimated more efficietl.

12 Stratified Radom Samplig-W Stratif? To make te surve easier to ru admiistrativel. For example, a ealt surve migt ave special subpopulatios strata like studets livig i dorms or prisoers i jails or ospital patiets tat could be sampled differetl to te rest of te public.

13 Stratified Radom Samplig How to Stratif ad How Ma Strata? Pick Homogeeous Strata to icrease efficiec Usuall Use 5-0 Strata. If too ma te te sample size witi strata is too low

14 Stratified Radom Samplig Estimatio Metods Idividual stratum meas, totals ad teir stadard errors. Over all populatio meas ad totals ad teir stadard errors.

15 Stratum Estimatio Metods:Poit Estimates / / Sample Populatio j j j j j j s,..,.,,.., j respose of uit j i stratum µ σ µ τ µ τ µ µ

16 Stratum Estimatio Metods:Variaces ad SE s variaces te tat SEs are te square roots of Recall j j Were s s Var Var s Var Var µ τ µ

17 Overall Populatio Estimatio Metods Populatio Mea µ Populatio Total τ

18 Overall Populatio Mea Estimatio Metods / Sample Estimates / Populatio st st j j W W µ µ µ µ µ µ

19 Estimatio of Overall Populatio Mea µ. µ st Tis is a weigted average of te idividual stratum meas. Here te weigts are te relative stratum sizes. W st W /,,..,

20 Estimatio of Overall Populatio Mea µ. st st s Var W Var W E W E Idepede ce Uses Variace. Ubiased. Properties µ µ

21 Estimatio of Overall Populatio Total τ. variaces of roots square are SEs Recall Recall Variace Ubiased Properties / st st st st st s Var Var Var Var E E µ τ τ τ τ τ τ τ τ µ τ

22 Estimatio of Overall Populatio Proportio p. Estimate Variace / Estimate Poit st st p p p Var W p Var p W p p

23 Cofidece Iterval Estimatio Mea ad Total µ st ± z α / SE µ st p st ± z α / SE p st τ st ± z α / SE τ st Tere are better approximatios usig te t dist See P i te Text.Tis is referred to as Sattertwaite s approximatio.

24 Motivatig Stratificatio: Example. A artificial example cotrasts: Simple radom samplig Stratified radom samplig were a perfect variable as bee foud to stratif o. ote- Te stratified estimate of te mea as 0 variace ad SE. Urealistic but makes te poit of te value of stratificatio!

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26 Stratificatio Examples Text P0-. Ver simple example of calculatios 4 Strata example. Ver simple example of calculatios Mail Surve Example of Estimatio of Proportios Mule Deer Aerial Surve Example of Estimatio of Populatio Totals wit Cofidece Itervals.

27 Estimatio of Overall Populatio Mea µ: Calculatio Example 4 Strata. W Tis is a weigtedaverageof te stratum meas. Here te weigtsare te relativestratumsizes. W st µ st 90,, 0.0,W 0.50, st / 00,, 3, 0.,W 5.00, W 3,,..,4 400, 3 3 0, ,W 30.00, 30, W i

28 Estimatio of Overall Populatio Mea µ : Example. ou like if 0.Ceck ourself , 9.0, 6.0, , 0., 0.0, 83 0,,,, 90 30, 400, 00, 90, st st Var s s s s W W W W s Var Var W Var

29 ecture 4. Brief Review of Stratificatio Estimatio Cocepts usig te Examples Surve Plaig- How to coose te overall sample size ad ow to allocate to te umber of samplig uits i eac idividual stratum.

30 Recurrig Teme: Improve Precisio of Estimates for Fixed Costs Oe could argue tat moe is cetral te fuctioig of our societ. We use stratified radom samplig because we are trig to improve te precisio of estimates wile keepig costs costat. If we fid a good stratificatio based o auxiliar iformatio we will do muc better ta if we use simple radom samplig. ote tis is te same idea we used we we developed ratio ad regressio estimators. M opiio is tat stratificatio is a bit more practical i ma cases ta usig te regressio estimators but bot approaces are ver useful.

31 Improvig Samplig Desigs wit Auxiliar Iformatio Regressio Metods Use relatiosip betwee ad x i liear regressio models. Stratified Radom Samplig Use auxiliar variable s to set up rougl omogeeous groups or strata Double or Two-Pase Samplig Practical Advatages as sometimes ot eoug iformatio o Frame to use eiter directl witout Two-pase samplig

32 Recurrig Teme: Improve Precisio of Estimates for Fixed Costs Bot use of regressio models ad stratificatio are ver useful for gaiig efficiec i estimatio i fiite populatio samplig. M opiio is tat stratificatio is a bit more practical i ma cases ta usig te regressio estimators but Q of te Week? W do ou tik tis migt be true or do ou disagree?

33 Q of te Week: Stratificatio vs Regressio Models

34 Estimatio of Overall Populatio Proportio p: Example

35 Estimatio of Overall Populatio Proportio p: Example Tis example illustrates ma ke poits:. Te stratum estimates are of value i teir ow rigt. Te overall populatio parameter estimate -i tis case te proportio is also importat. 3. Tere are importat practical issues i ruig a surve ad i tis case te orespose could be a issue causig bias.

36 Stratificatio Example: Helicopter Surve to Cout Mule Deer Kufeld et al 980 Joural of Wildlife Maagemet, 44, strata of differet sizes based o differet regios i differet abitats. Samplig Uit is a plot were a complete cout of mule deer is made. Ver good example of use of te stadard samplig metodolog. Idividual stratum estimates ad te overall populatio estimates sum of te stratum estimates. Stud desig is crucial i suc a expesive surve Reasoable precisio of estimates

37 Stratificatio Example: Helicopter Surve to Cout Mule Deer: Results Table.

38 Stratificatio Example: Helicopter Surve o Mule Deer: Summar. Tis example illustrates ma ke poits:. Te stratum estimates ma be of value i teir ow rigt. Te overall populatio parameter estimate -i tis case te proportio is also importat. 3. Tere are importat practical issues i ruig suc a complex ad costl surve. 4. Tikig carefull about te desig i advace is a o braier to use a colloquialism. 5. I tis case te odetectio of some aimals is a issue possibl causig bias. Tere are metods to adjust for odetectio wic are covered i m oter class ST 506.

39 Stratificatio Example: Agler Surve o Small ake: Aget preset all da ad iterviews aglers as te leave. Use two strata WD, WE. M T W T F S S If Exted to mo. surve te 4 strata x WD-Weekdas -sample 8 WE-Weekeds 8 -sample 4 Higer Rate

40 Stratified Radom Samplig:Allocatio Surve Plaig- I simple radom samplig we just ad to figure out ow large te sample eeded to be to a acieve a precisio goal. ow i stratified radom samplig it is a little more complex: - How to coose te overall sample size. - How to allocate to te umber of samplig uits i eac idividual stratum.

41 Stratified Radom Samplig: Sample Allocatio Rules to Obtai Better Precisio -Equal Allocatio allocate te sample size equall i all te strata. -Proportioal Allocatio allocate proportioal to te size of te strata-ver widel used -Optimal allocate proportioal to size ad stratum variaces ad iversel proportioal to costs i te differet strata.

42 Stratified Radom Samplig: Sample Allocatio Rules to Obtai Better Precisio Equal Allocatio allocate te sample size equall i all te strata. ot usuall sesible uless all strata are equal size i terms of overall estimates precisio. However, mabe good if ou wat to compare stratum meas as our primar focus

43 Stratified Radom Samplig: Sample Allocatio Rules: Equal Allocatio /

44 Stratified Radom Samplig: Sample Allocatio Rules: Equal Allocatio / Earlier 4 Stratum Example 4 / 0.5 Use 83x0.5, 0.75,,, 83x0.5, 0.75, 3, x0.5, 0.75, x0.5

45 Stratified Radom Samplig: Sample Allocatio Rules: Proportioal Allocatio Allocate proportioal to te size of te strata-ver widel used / W

46 Stratified Radom Samplig: Sample Allocatio Rules: Proportioal Allocatio / W 83x0.0, 83x0., 36.5, ote weigts give earlier. Use rouded values 8.3, 8, 3 9., 9, 3 37, x0.44, for total x0.35

47 Stratified Radom Samplig: Sample Allocatio Rules to Obtai Better Precisio Optimal Allocatio Takes accout of stratum sizes, differet variaces, ad differet costs of samplig i differet strata Optimal Allocatio is ot used as muc as proportioal allocatio but it ca result i a gai i precisio if te costs ad variaces are kow or well estimated from a prior stud or a pilot surve.

48 Geeral Optimal Allocatio Metodolog σ Var st We wat to miimise te variace for fixed Total Cost were te Cost Fuctio is : C c c 0 is is c 0 + te overead cost te per samplig uit cost. We cosider te costraied fuctio i terms of ad use te agrage multiplier approac. Tis will be outlied o te witeboard ad is discussed i simpler form i text o P6. c te

49 Geeral Optimal Allocatio Results l c c c C c c 0 / OverallSample Size / / Relative Sample Sizes σ σ σ σ

50 Fiseries Agler Surve Book Example of te Allocatio Calculatios 000, Total Cost C $00 000, 3 500, W 0.57, W 0.86, W Prior St Deviatio Values σ 0, σ 0, σ 3 0, Cost Values Overead Cost co $00 Per Uit Cost c $.00, c $.00, c 3 $0.64,

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53 Self Weigted Stratified Samples allocatio i.e We are usig proportioal / / ad ol if if / / W st j j st

54 Post Stratificatio.6 P4 Used sometimes o a simple radom sample we te stratum sizes are ot kow. Post Stratified Variace Sowig Extra Pealt Term Var st Var prop st + [ / ] σ

55 Stratificatio Brief Summar Slides

56 Stratified Radom Samplig-W Stratif? Te stratum parameters are of iterest i teir ow rigt Icrease efficiec of estimators of overall populatio parameters b coice of strata tat are omogeeous over te samplig uits witi eac. To make te surve easier to admiister operatioall.

57 Stratified Radom Samplig-Advatages Advatages ad Disadvatages -Icreased efficiec -Coveiece -Focus o Subpopulatios of special iterest Over sample tem Disadvatages - Ma ot alwas ave te iformatio ou eed o our frame so ou kow stratum sizes. - Ma ot kow stratum st deviatios or costs well to use optimal allocatio

58 Stratified Radom Samplig- We is it Better? Te Stratum meas differ widel from eac oter. Te stratum stadard deviatios are small. Tat is te variatio witi eac of te stratum are small. Recall motivatig example

59 Compariso of Sample Allocatio Rules c c / / eed Pilot Ifo Costs, Allocatio Uequal Optimal Geeral eed Pilot Ifo Costs, ema Allocatio Equal AllocatioMost Commo Proportioal AllocatioRarel Used Equal σ σ σ σ