Sample Size Requirements For Stratified Random Sampling Of Pond Water With Cost Considerations Using A Bayesian Methodology

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1 Sample Size Requiremets For Stratified Radom Samplig Of Pod Water Wit Cost Cosideratios Usig A Bayesia Metodology A.A. Bartolui a, A.D. Bartolui b, ad K.P. Sig a Departmet of Biostatistis, Uiversity of Alabama at Birmigam, Birmigam, Alabama USA, Albartol@uab.edu, Tel: , FAX: b Departmet of Psyology, Uiversity of Georgia, Ates, Georgia 3060 USA Departmet of Biostatistis, Sool of Publi Healt, Uiversity of Texas Healt Siee Ceter at Fort Wort, Fort Wort, Texas USA Abstrat: Estimatig average evirometal pollutio oetratios ad it=s variae is a fairly straigt forward task i stratified radom samplig. A more allegig oept is te itrodutio of te ost fator ito tis evirometal model. Traditioal statistial teiques ave iorporated osts from samplig witi a stratum as well as stratum weigts to determie te stratum size ad overall required sample size. Iformatio i te form of iformative prior distributios to determie a more oeret variae i te system yield a more preise Bayesia approa to te sample size ad ost alulatios. Tis approa results i a more effiiet samplig strategy i terms of ost we osiderig a pre speified margi of error for te samplig mea as well as te more ompliated situatio of orrelatio amog te strata samples. Keywords: Stratified; radom samplig; ost; Bayesia. INTRODUCTION Te Traditioal statistial approaes to alulatig overall ad stratum sample sizes i a stratified radom sample are fairly straigt forward. Te proedure is somewat ompliated wit te iorporatio of ost as well as te possibility of orrelatio amog te stratum samples. Appliatios of su approaes employig several moitorig strategies are well kow (Torto et. al, 98, Nelso ad Ward, 98, Rekow ad Capra, 983, ad Gilbert, 987). Our fous ere is to osider a pod water eviromet i wi te strata are basially dept levels. Weigtig of te strata as well as te overall variae of te sample mea are te mai ompoets i our derived statistis to determie sample size witi te stratum. Te tree situatios osidered are tat of pre speified margi of error, pre speified fixed ost ad orrelatio amog te strata samples. Cost effiiey is see for most situatios wit te itrodutio of Bayesia metodology (Dayal ad Dikey, 976, Bartolui ad Dikey, 977, Bir ad Bartolui, 983 ad Bartolui et. al., 998). Te trust of te Bayesia approa is troug te derivatio of te posterior estimate of te variae derived from oeret iferee o a ormal variae i te Bere=s Fiser otext (Dayal ad Dikey, 976, Bartolui et. al., 998). Comparisos of te traditioal or lassial ad Bayesia metodologies are preseted usig summary data from determiig te posporous oetratio i a pod water samplig eviromet.. TRADITIONA SETUP et N=total umber of populatio uits i te target populatio. N is te umber of populatio uits witi ea of te stratum, =,... Clearly N= N. Wit referee to te sample, =total umber of samplig uits i te target sample. ikewise as above, = =. We defie te weigt of te stratum,, 874

2 as W =N N. Te mea, µ, of te populatio of N uits is: ( N) N W () were µ is te mea of te stratum ad is estimated by m ( ) xi i () were x i = it observatio i stratum,. A ubiased estimate of u is m st et N N = i all strata, te m Wm. (3) m st ( ) i x i. (4) We defie Var(m )= were is te ( ) W variae of te stratum. We estimate te stratum variae by s. It a be ( ) ( i x ) m i sow tat for large N, sm ( ) st Ws. (5) It will be importat to ote te robustess of tis sample mea variae i te Bayesia otext. were for N, z W s d (8) ad z - is te usual00(-) ritial value of te stadard ormal distributio. Tus te optimum for te t stratum is. Ws Ws ii) Pre Speified Fixed Cost We defie te overall ost of te samplig as Cos t C Co C (9) were is te ost per populatio uit i te t stratum ad o is te fixed overead ost. Tis is a stadard ost represetatio. Tus te optimum a be derived as (Azel,999), ( C Co) Ws Ws As above te optimum per stratum is W s W s (0) () Oe a examie equatio (0) i terms of its sesitivity to ages i te PMOE. et W =. 3. COMPUTING THE OPTIMUM We give a brief overview of tree metods to ompute te optimum. i) Pre Speified Margi of Error (PMOE) ettig d= m st -µ, we deote d as te pre speified margi of error, (Gilbert, 987). Te value d is su tat Te (0) a be rewritte as ( C C o ) s s () If we assume uequal PMOE, d, for samplig witi stratum te we a write =(Z - s d ). (See Cora, 977, Azel, 999). Tus equatio () a ow be examied wit respet to sesitivity to ages i d. iii) Correlatio amog Dept Stratum for small. tus P( m st -µ d)= (6) et = average orrelatio amog all possible lags i te dept samplig eviromet. For Te optimum (Cora, 977) is example if is te umber of strata or depts ad l =te orrelatio of te lt lag, te (7) z Ws d, ( ) z W s d N l. (3) l 875

3 If is te umber to be sampled i ea of te strata or = stratum size, te [ z W d ][ ( )]. s 4. BAYESIAN CONSIDERATIONS (4) Examiig equatios (7), (0), () ad (4) we see tat tey all ivolve te expressio for te stratum variae, s. We reevaluated tese expressios addig a prior struture to te variae (Dayal ad Dikey, 977, Bartolui et. al. 998) ad te estimatig te posterior expressio for te variae, ormal. We assumed a uderlyig ormal distributio wit bot mea, µ, ad variae, ukow. I tis otext we defie te likeliood futio for observatios: l(, ) ( ) exp[ ( ( m) s ] (5) for v=-, m=x +x +...+x, vs =(x -m) + (x -m) +...(x -m) ad deotes a proportioal relatiosip. Cosider te t-desity, s s ( ) ( x; ) [ Beta(, )] ( ( x s) ) z a b were Beta( a, b) ( z) dz ad, s>0. Te prior for µ is for o. Te prior for is p( ) o p ( ) (, ) 0 ms o o (6) (7) g, >0, g>0 (8) Were is i square o degrees of freedom. Tus osiderig expressios (5), (7), ad (8) te posterior variae is =(s +g )B (9) were B=+. Tus substitutig s for i (7), (0), (), ad (4) yields te Bayesia estimates of ad. Tus i te followig setio we apply te Bayesia aalysis to tese expressios to demostrate ad determie te effiiey of tese expressios i terms of te sample size requiremets ad ost of samplig. 5. EXAMPE We wis to estimate te average posporous oetratio (µg00 ml) i pod water. Te oetratio of 00 ml aliquot from ea liter sample will be measured. Te statistis for a lassial represetatio of te data usig te pre speified margi of error (PMOE) are give i Table. Tere are 5 dept strata to te pod i wi N=total umber of 00 ml water samples i te pod. N is te umber of aliqots i stratum. Te weigts, umber samples from ea strata, mea ad variae of ea strata are give as well all derived from our previous formulatios above. We ave assiged osts to ea strata. For te sake of simpliity ad witout loss of geerality we ave redued te osts to iteger uits. Te ost for samplig stratum ad are ea. Te osts assiged to strata 3, 4, ad 5 are,, ad 3 respetively - te assumptio beig tat osts irease as te dept ireases. Tus te overall ost of samplig is 74 uits. Usig te PMOE approa i Table demostrates te Bayesia results usig empirial prior samplig iformatio ad iorporatig tat ito te variae alulatio overall. Oe sees tat for realisti prior assigmets of, ad g i (3) tat oe realizes a redutio i assiged umber per strata overall as well as a ost redutio. I Table 3 usig pre speified overall ost did ot yield ay savigs usig te lassial (top row) vs. te Bayesia approa (bottom row) tis makes sese somewat i tat te ost is already fixed. However, we did examie tese results usig () i wi we varied te PMOE, d, to determie te effet o ost usig sesitivity ages ad te lassial ad Bayesia results remaied fairly equal. Table 4 summarizes te data results itroduig orrelatio amog te strata as per (4). Te average orrelatio is i te first olum. Oe a see tat as you irease te average orrelatio, (3), te te required umber sampled witi ea strata will irease, but at a slower rate i te Bayesia otext. 876

4 Overall it appears tat: Compared to te lassial samplig aalysis for te pre speified margi of error approa as well as te orrelatioal approa, te Bayesia aalysis resulted i a redutio i required samples tus lowerig te ost, espeially we realisti (empirial) prior yperparameters are utilized. Also tere was o serious impat o te posterior stadard error of te estimates of te mea oetratio. However, tere were o real differees betwee te lassial ad Bayesia approaes i te pre speified fixed ost aalysis. Give te urret omputatioal tools te Bayesia alulatios proved to be fairly straigt forward. Also give te urret availability of databases, future Bayesia approaes to evirometal samplig sould be give serious osideratio espeially were osts are oered. Evirometal Modelig ad Software, 3, 5-9, (998). Bir, R. ad Bartolui, A.A., Determiatio of te yperparameters of a prior probability model i survival aalysis, Computer Programs i Biomediie, 7, 89-84, (983). Cora, W.G., Samplig Teiques,. Wiley Pub., 3 rd editio, New York, (977). Gilbert, R.O., Statistial Metods For Evirometal Pollutio Moitorig, Va Nostrad Pub., New York, (987). REFERENCES Azel, A.D.,. Complete Busiess Statistis, MGraw Hill, Bosto, (999). Bartolui, A.A. ad Dikey, J.M, Comparative Bayesia ad traditioal iferees for Gamma modeled survival data. Biometris, 3(), , (977). Bartolui, A.A., Blaard, P.D., Howell, W.M., ad Sig, K.P., A Bayesia Bere=s Fiser solutio to a problem i taxoomy, Nelso, J.D. ad Ward, R.C., Statistial osideratios ad samplig teiques for groud-water quality moitorig, Groud Water, 9, 67-65, (98). Rekow, K.H. ad Capra, S.C., Egieerig Approaes for ake Maagemet, Volume, Data Aalysis ad Empirial Modelig, Butterwort, Bosto, (983). Torto, K.W., Keedy, R.H., Magou, A.D. ad Saul, G.E., Reservoir water quality samplig desig. Water Resoures Bulleti, 8, , (98). 877

5 Table. Data for stratified radom samplig to estimate samples per strata (PMOE) Classial Approa ( =, =0, g =) s (m st ) = 0.040, Cost=74 Strata N W m s 4.5M M M M M Total 5.99M Table. Bayesia Results (PMOE) (,,g) Total s (m st ) Cost 35,, ,, ,, ,35, ,35, Table 3. Pre speified fixed ost C- 0 g

6 Table 4. Example Usig te Correlatio Struture,. prior (,,g) Classial (35,,0.5) (0,,0.) (40,35,.) Cost Cost Cost Cost