On The Efficiency of Some Techniques For Optimum Allocation In Multivariate Stratified Survey.

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1 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) O The Efficiecy of Some Techiques For Optimum Allocatio I Multivariate Stratified Survey. AMAHIA, G.N Departmet of Statistics, Uiversity of Ibada, Nigeria, FAWEYA, O. Departmet Of Mathematical Scieces, Ekiti State Uiversity, Ado-Ekiti, Nigeria. Abstract AYOOLA, F.J Departmet of Statistics, Uiversity of Ibada, Nigeria I multivariate stratified samplig, the major cocer is o the problem of estimatio of more tha oe populatio characteristics which ofte make coflictig demads o samplig techique. I this type of survey, a allocatio which is optimum for oe characteristic may ot be optimum for other characteristics. I such situatios a compromise criterio is eeded to work out a usable allocatio which is optimum for all characteristics i some sese. This study is focuses o the efficiecy of some techiques for optimum sample allocatio which are Yates/Chatterjee, Booth ad Sedrask ad Vector maximum criterio (VMC) o the set of real life data stratified ito six strata ad two variates with desired variaces usig: (i) method of miimum variace with fixed sample size ad (ii) a arbitrary fixig of variaces. The stratum sample sizes h amog the classes were obtaied to examie the criterio that will produce the smallest. I this paper, it was discovered that VMC ad Booth ad Sedrask are superior to Yates/Chatterjee. Eve though, o uiversal coclusio ca be draw, the work clearly brigs out the fact that the best allocatio is ot always obvious ad that sufficiet care is ecessary i the choice of allocatio of the sample sizes to differet strata with several items. Keywords:- Stratified Survey; Optimum Allocatio; Vector Maximum Criterio (VMC); Yates/Chatterjee;Booth ad Sedrask Itroductio I multivariate samplig, more tha oe populatio characteristics are estimated. These characteristics may be of coflictig ature (Sukhatme, 970). Whe stratified samplig is used, a procedure that is likely to decrease the variace of the estimate of oe characteristic may very well icrease the estimate of aother. The problem of optimum allocatio of sample sizes i a sample survey whe a sigle characteristics is beig studied uder a give samplig procedure is well defied; It is that which miimize the cost of the survey for a desired precisio or the variace of the sample estimate for a give budget of the survey. Meawhile, typical uivariate optimum sample allocatio strategy failed whe a umber of characteristics are simultaeously uder study as i the most survey situatio where the possibility of irrecocilable idividual allocatio betwee characters (variables) become real. 54

2 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) I light of this, several optimality criteria have bee developed over the years by differet authors i a survey where may variables are uder study, these icludes: Neyma (934), Daleius (957) Yates (960); Koka ad Kha (967), Chatterjee (968), Adelaku (00) etc. However, this study is achored o the efficiecy of some techiques of optimum sample allocatio whe desired variaces are set to see which method is superior i producig the best (optimal) allocatio for a give desired variace. I stratified samplig, the values of the sample size h i the respective strata are chose by the sampler. They may be selected to miimize variace of (V(ȳ st ))for a specified cost of takig the sample or to miimize the cost for a specified value of V(ȳ st ). Thus, it is geerally cosidered i these two forms:- (i) (ii) Fixed precisio-least cost formulatio:- Give the cost fuctio C = C o + C h h ad some prescribed value (V) for the variace of the stratified sample meas, we determie the stratum sample sizes such that the cost fuctio C = C o + C h h is miimized. Fixed cost-best precisio formulatio:- Give the cost fuctio C = C o + C h h ad a fixed budget c, we determie the stratum sample size such that the variace of the stratified sample mea is at the miimum. Several optimality criteria are foud i the literature which were cotributed by various authors to allocatio problems through compromise solutio, loss fuctio ad iterative solutio. The use of liear programmig i sample survey to determie allocatios whe several characters are uder study was first suggested by Daleius (953), ad Nordbotte (956) illustrated this approach by a umerical example. No liear programmig techiques ca also be used to solve allocatio problems i sample surveys, ad this possibility was briefly metioed by Daleius (957) who suggested miimizig a weighted average of precisio. Koka (963) defie a optimum allocatio for a multivariate survey as oe that miimizes the cost of obtaiig estimates with error smaller tha previously specified umbers at a previously specified cofidece level. He the showed how iformatio o the various stratum variaces could be used to obtai ear optimum allocatio. The multivariate samplig problem was proposed as a o-liear multi-objective programmig problem by Koka ad Kha (967). A compromise allocatio was suggested by Cochra (977) for various characters, whereas Omule (985) used dyamic programmig to obtai a compromise allocatio. Kha et al; (997) used iteger programmig to obtai a compromise solutio i multivariate stratified samplig. Daiz etal (006) proposed stochastic programmig approach to the allocatio problem. Yates (960) i his approach suggested that the sampler specifies the variaces that he wats for the estimates of each variate while Chatterjee (968) followig Yates (960), et al illustrates a method of allocatio i multivariate surveys that miimize the cost of obtaiig estimates with variaces ot bigger tha previously specified umbers. The various stratum variaces are assumed kow. I a related problem, Booth ad Sedrask (969) poited out that i default of a computer program a good approximatio to the solutio of Yates ca ofte be obtaied. Adelaku (00) proposed a alterative solutio see as a compromise solutio desiged as vector maximum criterio (VMC) which is a modificatio of the criterio advaced by Yates 55

3 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) (96, 967) ad Chatterjee (97, 987). I the VMC, ormalized vectors are to be used as weight rather tha arbitrary umbers used by Yates ad assumed by Chatterjee VMC differs from Yates criterio because it provides a base for decidig the choice of allocatio whe the sample size is specified i advace. It also provides a basis for judgig the reasoableess of specified precisio..0 THE DATA AND METHODS USED FOR THE EMPIRICAL STUDY (MATERIAL AND METHODS). Data Source I this empirical study, five sets of real life data were used. Each set of data was divided ito two variates ad was stratified ito six strata. The data were draw from a survey o educatio ad icidece. The first set of data show i table deals with performace of studet i Mathematics by sex i Abuja Secodary School JSCE for the year 00/0. The percetage of male that passed (X ) was take as the first variate while that of female was take as the secod variate (X ) ad the data were stratified ito six strata as show below:- Table : Studet performace i Mathematics by sex i Abuja secodary school JSCE for year 00/00 Stratum No N h W h S h S h Abaji () Muicipal () Gwagwalada (3) Kuje (4) Kwali (5) Bwari (6) The secod set of data show i Table deals with percetage passed i Eglish (X ) ad Mathematics (X ) mock result i Abuja Secodary Schools for years 00 ad 004. The proportio passed i Eglish (X ) was take as the first variate while that of Mathematics (X ) was take as secod variate ad the data were stratified ito six strata as show below:- Table : Percetage passed i Eglish ad Mathematics Mock Result i Abuja secodary school for year 00 ad 004. Stratum No N h W h S h S h Abaji () Muicipal () Gwagwalada (3) Kuje (4) Kwali (5) Bwari (6) The third set of data show i Table 3 deals with poverty icidece by state for the year 996 (X ) ad 004 (X ). The data o year 996 (X ) was take as the first variate ad that of the year 004 (X ) was take as the secod variate. The data were stratified ito six strata as show below:- 56

4 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) Table 3: Poverty Icidece by state for the year 996 ad 004. Stratum No N h W h S h S h NW () NE () NC (3) SW (4) SE (5) SS (6) The fourth set of data show i Table 4 deals with primary erolmet ratio by State ad Sex. The male ratio (X ) was take as the first variate ad female ratio (X ) as the secod variate while the data were stratified ito six strata as follows:- Table 4: Primary Erolmet ratio by state ad sex. Stratum No N h W h S h S h NW () NE () NC (3) SW (4) SE (5) SS (6) The last set of data show i Table 5 deals with sets of scores of 80 studets i the promotio examiatio which were radomly selected from six schools i Muicipal Area Coucil of Abuja. The scores i Eglish (X ) was take as the first variate while that of Physics (X ) was used as secod variate ad the data were stratified ito six strata as show below:- Table 5: Scores of 80 studets i the promotio Examiatio from six schools i Muicipal Area Coucil, Abuja. Stratum No N h W h S h S h Statistical Aalysis For each of the five sets of data used, desired variaces were set usig:- (i) Method of miimum variace calculatio with a give sample of size i.e. V mi (ȳ st ) = ( W hs ih ) ( W hs ih) (i =,) N Where W h is the stratum weight 57

5 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) S h is the true variace. N is the total umber of uits. (ii) Arbitrary variaces fixed. The three techiques amely:- (a) (b) (c) Yates/Chatterjee techiques Booth ad Sedrask techiques Vector maximum criterio (VMC) were applied to each set of the data with their set desired variaces... Yates/Chatterjee procedure:-. Set the desired variaces to be used. Obtai the resultig variaces for a sample of size i.e. 3. Obtai h λv(ȳ st ) = W hs ih h = W hs h W h S ih ad h = W hs ih h () = W hs h W h S ih (3) 4. Obtai the values of λ 5. Obtai h = λ( h) +( λ)( h) (4) λ( h) +( λ)( h).. Booth ad Sedrask Procedure:-. Set the desired variaces to be used.. Obtai a = V V + V ad a = V V + V.where V ad V are the variaces for variate ad respectively. 3. Obtai V = V V V + V..(5) 4. Equate L = a V(ȳ st ) + a V(ȳ st ) = V..(6) Where L is the quadratic loss fuctio. 5. Obtai ( W h A h ) where A h = i= a i S ih (7) 6. Obtai = ( W h A h ) L (8) 7. Hece, obtai h = ( W ha h W h A h ) (9)..3 Vector Maximum Criterio (VMC) procedures.. Obtai the value of the efficiet feasible poit for a total sample size of. i.e. 58

6 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) h = ( i= α is ih) N h ( i= α i S ih) = ( α is ih) W h L h= W h ( α i S p i= ih) Where α i is the weight for the variate i such that α i = V(ȳ st ) = W hs ih h.(0) = ( W hs h) W h ( α i S ih) () W h ( α i S ih ). For several values of α i obtai correspodig values of V(ȳ st ), V(ȳ st ) ad V(ȳ st ) V(ȳ st ). 3. Preset the values i a table called efficiet poit tables. 4. Set the desired variaces. 5. Obtai the actual values of V V. 6. Obtai the value of α i correspodig to the values of V V. 7. Draw the graphs of V, V ad V V agaist α i o the same axis. 8. O the graphs, trace the values of the relative variaces set to V V. 9. Obtai the value of α i ad trace it to the other two curves of V ad V. 0. Through the value of α i obtai the correspodig values of V ad V respectively. The substitute ito 3.0 DATA ANALYSIS 3. Aalysis of data set by fixig :- h = ( α is ih) N h () N h ( α i S ih ) 3.. Usig Yates/Chatterjee procedures based o settig relative variaces with =7 Table 4.8: Calculatio of W h S h ad W h S h Stratu m N h W h S h S h W h S h W h S h (W h S h ) W h S h (W h S h ) W h S h W h S h W h S h TOTA L

7 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) V V V (y st ) = (4.0874) V (y st ) = (4.35) 60 = = If allocatio is used with =9.4747, the variace obtaied for y is V (y st ) = (4.35)(35.54) =9.339 This value is larger tha specified for V, hece we seek a compromise allocatio that satisfies both toleraces exactly. Usig lagrage multipliers λ ad λ, we fid the values of h. For ay value of λ, we have V(y st ) = φ (λ), V(y st) = φ (λ) Hece, we shall fid λ ad such that φ (λ) φ (λ) = V = = V = For φ (λ), whe λ =, φ (λ) = , whe λ = 0, φ (λ) = This gives a parabolic approximatio as: φ (λ) = ( λ) = () = λ λ = () For φ (λ) whe λ=, φ (λ)=59.90, whe λ = 0, φ (λ)= Also this gives a parabolic approximatio as φ (λ) = λ λ = = (3) λ = (4) Substitutig from (4) i () ad solvig for λ ad we obtai

8 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) λ= , λ=0.48 Let r = λ ( h ) + ( λ) ( h ) Where h = W hs h ad h = W hs h W h S h W h S h We have table 4.9 below :- Table 4.9: Calculatio of h Stratum ( h ) ( h ) r r h h = h W hs h W hs h r h h TOTAL To obtai the resultig variace from the sample, we use Thus λv(y st ) = =.63. λv(y ist ) = W hs ih h = = λv(y st ) = = W hs ih h = The two values of are so close that we accept this allocatio ad take = Usig Booth ad Sedrask procedure based o settig relative variace with =7 V ad V a = V V + V = =

9 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) a = V V + V = = V = V V V + V = L = a V(y st ) + a V(y st ) = V = V(y st ) V(y st ) = Usig A h = a S h + a S h ad h = (W ha h ) W h A h Where = ( W ha h ) We have the table 4.0:- L Table 4.0: Calculatio of h Stratum W h S h S h A h W h A h h TOTAL = (5.39) Usig Vector Maximum Criterio Procedure (V.C.M) h = ( α is ih) N h N h( αi S ih ) = ( α is ih) W h W h( αi S ih ) Let A h = ( α i S ih) W h A h h = A h 6

10 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) The we have table 4.:- Table 4.: Calculatio of A h values for various α i α = 0 α = 0. α = 0. α = 0.3 Stratum W h S h S h W h S h α = α = 0.9 α = 0.8 α = TOTAL α = 0.4 α = 0.5 α = 0.6 α = 0.7 α = 0.8 α = 0.9 α = α = 0.5 α = 0.78 Stratum α = 0.6 α = 0.5 α = 0.4 α = 0.3 α = 0. α = 0. α = 0 α = 0.48 α = TOTA L Usig V(y st ) = We obtaied the table 4.3:- L h= (W hs ih) = (W hs ih)( A h ) h A h V(y st ) = (W hs ih)( A h ) A h 63

11 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) Table 4.3 VMC TABLE OF EFFICIENT POINTS α i V(y st ) V(y st ) V(y st ) Vy st Usig VMC based o settig relative variaces with =7 Thus, V V = V(y st ) V(y st )

12 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) Graphs of V,V ad V V agaist efficiet poit (α i ) v,v v /v efficiet poits (αi) 0 v v v /v From the graph, we obtai the values of α,v, V correspodig with V V = as 0.5, ad respectively. For V, For V, For α = 0.5, α = 0.48 = 67.8 = = 646 = = = = = = = = = = = = =

13 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) With the use of Yates/Chatterjee, Booth ad Sedrask, ad Vector Maximum Criterio (VMC) procedures for optimum allocatio problems i multivariate survey o five sets of umerical real life data, the summary of the tabulated results are show below TABULATED RESULTS OF THE DATA ANALYSIS Table shows the results obtaied o the distributio of the sample sizes with relative variaces based o give from the five data sets. Data set Techiques Strata Total Yates/Chatterjee 4 Booth & Sedrask 4 0 VMC 4 0 Yates/Chatterjee 3 0 Booth & Sedrask 3 0 VMC 3 0 Yates/Chatterjee Booth & Sedrask VMC Yates/Chatterjee Booth & Sedrask VMC Yates/Chatterjee Booth & Sedrask VMC TABLE shows the results obtaied o the distributio of the sample size o settig arbitrary variaces. Data set Techiques Strata Total Yates/Chatterjee Booth & Sedrask VMC Yates/Chatterjee Booth & Sedrask VMC Yates/Chatterjee Booth & Sedrask VMC Yates/Chatterjee Booth & Sedrask VMC Yates/Chatterjee Booth & Sedrask VMC

14 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) TABLE 3:- Sample sizes geerated for differet data classified by techiques ad types of relative variace. Based o give Based o arbitrary variace Techiques D D D 3 D 4 D 5 D D D 3 D 4 D 5 Yates/Chatterjee Booth ad Sedrask VMC SUMMARY AND CONCLUSION Summary, the results i the tables show that VMC ad Booth ad Sedrask procedures are superior to Yates/Chatterjee i the sese that the procedures domiate Yates/Chatterjee i majority of the results i terms of sample sizes with relative variaces based o give ad o settig arbitrary variaces. CONCLUSION I this research work, we discovered based o the set of data collected ad used for the empirical studies, that VMC ad Booth ad Sedrask are superior to Yates/Chatterjee. It was also discovered that some strata has oe observatio i some tables, hece there will be o eed to estimate. The, we collapse the affected strata to form a stratum. Eve though, o geeral coclusio ca be draw, the study clearly brigs out the fact that the best allocatio is ot always obvious ad that sufficiet care is ecessary i the choice of allocatio of the sample sizes to differet strata with several items. REFERENCES Adelaku, A.A. (00) Optimum Allocatio i Multi-item sample Desig: Some Cotributios. Upublished Ph.D. Thesis, Uiversity of Ibada, Ibada. Attila, C. (997) Optimum Allocatio i Stratified Radom Samplig Via Holder s iequality. The Statisticia (997) Vol. 46. No 3 pp Bee, F.S ad David O.Y. (97) Allocatio i Stratified Samplig as a Game.Joural of the America Statistical Associatio September (97), Vol. 67. No 339 pp Booth, G. ad Sedrask, J.,(967) Desigig several factor aalytical surveys, Proceedig of the Social Statistical Associatio, Chatterjee,.S. (968) Multivariate Stratified Surveys. Joural of the America statistical Associatio Vol 63, No.3, PP Cochra, W.G. (963) Samplig Techiques Joh Wiley ad Sos New York. Secod Editio. Daleius, T. (950). The problem of optimum stratificatio. Skadiavisk Aktuarietidskrift 33, Daleius, T. (953). The multivariate samplig problem. Skadiavisk Aktuarietidskrift 36,

15 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-05 (Olie) Kha, M.F. et al (0). Allocatio i Multivariate Stratified Surveys with No-liear Radom Cost Fuctio. America Joural of Operatios Research, Vol. pp Koka, A.R. ad Kha, S. (967). Optimum Allocatio i Multivariate surveys: A aalytical solutio. Joural of the Royal Statistical Society, Series B, 9, 5-5. Kozak, M. ad Wag H. (00). O stochastic optimizatio i sample allocatio amog strata. METRON-Iteratioal Joural of Statistics, Vol.68 No pp Omule,.S. A. Y. (985). Optimum desig i Multivariate samplig. Biometrical Joural 7, Sukhatme, P. V., Sukhatme, B. V., Sukhatme, S., ad Asok, C. (984). Samplig Theory of Surveys with Applicatios. Third editio. Ames, Iowa: Iowa State Uiversity Press. Wikler, W. E. (009). Sample Allocatio ad Stratificatio. U.S. Bureau of Cesus, Statistical Research Divisio Report RRS Yates, F. (966). Samplig Methods for Cesuses ad Surveys. Third Editio. Lodo Griffi. 68

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