Sample Survey Design

Transcription

1 Sample Survey Desg

2 A Hypotetcal Exposure Scearo () Assume we kow te parameters of a worker s exposure dstrbuto of 8-our TWAs to a cemcal. As t appes, te worker as four dfferet types of days wt regard to te ature of te job tasks e performed. For eac type of workday, tere s a uque dstrbuto of 8-our TWA exposure values. Deote a 8-our TWA by te symbol C, ad deote a C value assocated wt eac of te four dfferet types of days by C, C, C ad C, respectvely.

3 A Hypotetcal Exposure Scearo () Assume Day type # comprses 0% of all workdays, wle #, # ad # comprse, respectvely, 0%, 0% ad 0% of all workdays. Tese percetages are expressed by decmal fracto tme wegts, deoted by W, W, W ad W. Te artmetc mea ad varace for te 8-our TWA dstrbuto assocated wt eac type of day are sow te followg table. Day Type Mea Varace VarC Tme Wegt W 5 ppm 5 ppm 0. 5 ppm 00 ppm ppm 600 ppm 0. 5 ppm 9950 ppm 0.

4 A Hypotetcal Exposure Scearo () Over a workg lfetme, te worker s total dstrbuto of 8-our TWAs s a mxture of tese four day-specfc dstrbutos. Tat s, f te worker works 0,000 days a lfetme, we ca expect 000 wll come from te dstrbuto assocated wt day type #, 000 wll come from te dstrbuto assocated wt day type #, ad so fort.

5 Artmetc Mea of te Total Dstrbuto To compute te artmetc mea of te total dstrbuto, deoted by EC, we eed to take te expectato by codtog: Equato : E X E X Y y p( Y y) all y E C w ppm 5

6 Varace of te Total Dstrbuto To calculate te varace of te total dstrbuto, deoted by VarC, we eed to use te equato for te varace by codtog: Equato : Var X Var X all y Y y p( Y y) all y ( E X Y y E X ) p( Y y) Var C Var C W ( E C) W Var C W ppm ( E C) W (5 ) (50 ) 6ppm 0. (5 ) 0. (5 )

7 Result Te worker s total dstrbuto of 8-our TWAs for te cemcal as: EC = ppm VarC = 68 ppm I effect, te expected value EC ca be regarded as te worker s log-term artmetc mea exposure level to te cemcal. Te mea exposure ca be also deoted by. Te varace VarC s te worker s exposure varablty over a lfetme wc ca be expressed as. 7

8 Smple Radom Samplg () Now, a perso wo as o dea about ts worker s exposure profle wats to estmate te worker s log-term artmetc mea exposure level,. Oe approac e ca use s to select a smple radom sample of workdays, measure te 8-our TWA values o tose days, ad compute te sample mea. Because te terest s estmatg te mea of 8-our TWA values, tus te ssue of ow to deal wt a sample mea dstrbuto s ecoutered. 8

9 Smple Radom Samplg () Uder smple radom samplg, te mea ca be estmated by te followg equato: E ad. Te varace assocated wt te estmator ca be calculated by te followg equato: I ts case, VarC = VarX. Var X Var X 9

10 Smple Radom Samplg () To detfy te estmate of as beg derved from smple radom samplg, let te estmator be deoted by. srs Terefore, Var srs Var C If = 0 workdays are cose ad 0 8-our TWAs are measured, te 68 Var srs ppm 0

11 Stratfed Radom Samplg () Aoter way wc ca be used to radomly select a sample of workdays s stratfed radom samplg. Recall tat te worker s total dstrbuto of 8-our TWAs s mxture of four separate dstrbutos. Tese four dstrbutos are cosdered to be four dfferet strata of 8-our TWA values. Alteratvely, we ca say tat tere are four dfferet strata of workdays. Hece, to estmate te worker s log-term artmetc mea exposure level, oe mgt take a smple radom sample of workdays from eac strata of workdays ad compute a tme wegted average of te sample meas of tese four strata were te tme wegts are te W values.

12 Stratfed Radom Samplg () Te sample meas for te four strata are computed by: Te terms,, ad are te umber of workdays selected or te umber of 8-our TWAs measured for eac stratum., X # : Stratum, X : # Stratum, X : # Stratum, X : # Stratum

13 Stratfed Radom Samplg () Te estmate of uder stratfed radom samplg s represeted by ad s computed by te followg equato: Te expectato of : Terefore, s also a ubased estmator of te worker s logterm artmetc mea exposure level. str str W str,, str str W E E ad E E E E W E W W E E str

14 Stratfed Radom Samplg () str Te varace of te deoted by Var str s computed by te followg equato: Var str X VarW VarW W Var W ( X ( X,, X X,,... X... X W ) W ) Because te samples are radomly selected from eac stratum, t s ot ureasoable to cosder all X values parwse depedet, all X values parwse depedet, ad so fort.,,, ( X ( X,, X X,,... X... X,, ) )

15 5 Stratfed Radom Samplg (5) Furtermore, we ca also cosder tat te values take from eac stratum are depedet of tose from oter strata. Terefore, Te above equato ca be rewrtte as: str C Var W C Var W C Var W C Var W Var str C Var W Var

16 Stratfed Radom Samplg wt Proportoal Allocato It s mpossble to cotrol te values of W ad VarC. But, we ca cotrol te values of, wc are te umber of workdays selected ad te umber of 8-our TWAs measured wt eac stratum. Oe way to select te s proporto to te overall tme wegts W. Terefore, f te total sample sze s ; Te, te umber of samples eac stratum ca be determed by: W Ts sample survey desg s termed stratfed radom samplg wt proportoal allocato. For clarty, deote te estmate of ad te varace of te estmate obtaed by ts sample survey desg as str P ad Var, respectvely. str P 6

17 Stratfed Radom Samplg wt Optmum Allocato () I te stratfed radom samplg proportoal allocato desg, te strata wt ger wt-stratum varaces ted to cotrbute more amouts to te varace of te estmate of,.e., Var. Oe way to reduce te varace of te estmate of s to crease te values for tese g-varace strata. Te equato for optmally allocatg te sample elemets betwee te strata s: were W SD C W SD C SD C Var C str P 7

18 Stratfed Radom Samplg wt Optmum Allocato () Ts sample survey desg s termed stratfed radom samplg wt optmum allocato. For clarty, deote te estmate of ad te varace of te estmate obtaed by ts sample survey desg as ad Var, respectvely. stro stro For te four strata of 8-our TWA dstrbutos descrbed at te begg of ts secto, ad for a total sample sze of = 0, te allocato of te sample betwee strata s: Day Type Number of Samples ( ) Proportoal Optmum 8

19 Varace of Stratfed Radom Samplg Approaces () For bot stratfed desgs, te varace assocated wt te estmate of s: Var str P Var C W ( ) 0. ( ) ppm 600 ( ) ( ) Var stro Var C W ( ) 0. ( ) ppm 600 ( ) ( ) 9

20 Varace of Stratfed Radom Samplg Approaces () From te result, we ca see te varace assocated wt te estmate of s less ta Var srs 68ppm. Sce all tree sample survey desgs provde a ubased estmate of, te best desg wt regard to statstcal precso ad accuracy s stratfed radom samplg wt optmum allocato. Now, f oe were actually performg a sample survey, t would be ecessary to ave a mmum of elemets for eac stratum, tat s,. Te reaso s tat oe would eed to estmate te varace for eac stratum. I ts case, f = 0, te wt proportoal allocato, we mgt pck: =, =, =, =; wereas wt optmum allocato we mgt pck: =, =, =, =. 0

21 Varace of Stratfed Radom Samplg Approaces () Tese ew umbers would cage te varace calculated above, but te relatve order would stll be: Var Var Var stro str P srs Te reaso for ts order ca be uderstad as follows. Te varace of te sample mea dstrbuto reflects te dfferece betwee te values of all possble sample meas.

22 Importat Messages () If eac sample s costructed suc tat a wde rage of elemetal values s draw to te sample, tere wll be less of a dfferece betwee sample mea values. Tat s, by creasg varablty wt te samples, we decrease varablty betwee samples wt regard to te value of te sample mea. Uder smple radom samplg wt = 0, t s possble to ave a sample of 0 8-our TWAs assocated wt oe type of day. Clearly, te sample mea based o 0 8-our TWAs draw from te day type # dstrbuto would be very dfferet from te sample mea based o 0 8-our TWAs draw from te day type # dstrbuto. Hece, smple radom samplg permts large dffereces betwee values of te sample mea.

23 Importat Messages () I cotrast, uder te stratfed survey desgs, t s assured tat a wde rage of elemetal values s bult to te composte sample. It s stll possble to obta a sample wt 0 g values or 0 low values, but suc samples arse muc less frequetly compared to smple radom samplg. Furter, te optmum allocato desg assures more varablty wt te composte sample ta te proportoal allocato desg, because te optmum desg samples more eavly from tose strata wc ave greater wt-stratum varablty.