COMPARATIVE ANALYSIS OF VARIANCE UNDER SAMPLES ALLOCATION FOR DEVELOPMENTAL PROGRAM EVALUATION

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COMPARATIVE ANALYSIS OF VARIANCE UNDER SAMPLES ALLOCATION FOR DEVELOPMENTAL PROGRAM EVALUATION
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     Rev. Bras. Biom ., São Paulo, v.29, n.2, p.198-203, 2011   198 COMPARATIVE ANALYSIS OF VARIANCE UNDER SAMPLESALLOCATION FOR DEVELOPMENTAL PROGRAM EVALUATION Rajiv PANDEY 1       ABSTRACT: Evaluation of program, implemented at large scale in different phases can beascertained by proper allocation of sampling units from each phase of program for further analysis. Method has been proposed for sample allocation in different strata (phase) withconsideration that impacts in successive phases (implemented at different time) follow arithmetic progression (Pandey and Verma, 2008). Present paper proposes variance for the design under the allocation method. Comparison has also been made and discussed with the variance of  proportional and optimum allocation methods.    KEYWORDS :  Impact evaluation; sample allocation; arithmetic progression; weight; efficiency.   1   Introduction The estimation of any parameter of interest for a large population depends primarilyon sampling theory, which deals with the properties of the estimates from a statisticalsample (Cochran, 1977). However, for the heterogeneous population the allocation of sample size in different homogeneous strata is crucial for estimation of parameters(Aoyama, 1954).For a heterogeneous population, the samples are allocated into various stratadepending on the nature of the population. Ideally, the sample allocation should beoptimized so that the precision is maximized within the cost constraint. The simplest formof optimal allocation is to make the sampling fraction in stratum proportional to thestandard deviation in the stratum, and inversely proportional to the square root of the costof including a unit from the stratum in the sample. That is, more heterogeneous andcheaper strata are sampled at higher rates (Dalenius and Hodges, 1959; Cochran, 1977;Sukhatme  et al. ,   1984). In stratified sampling, for a given sample size, approximate minimization of variation depends on number of strata, sample allocation within strata, populationvariance within strata, population size within strata, and strata boundary break points(Cochran, 1977). However, Pandey and Verma (2008) and Pandey (2010) have discussedthe case of sample allocation by assigning weights for those designs, where allinformation, except sample allocation within strata is known apriori. These design dealtwith the case of implementation of developmental program in phased manner, under theassumptions that the units of different strata received different impacts and follows some 1 Indian Council of Forestry Research & Education, Dehradun, Utterakhand, 248006, India. E-mail: rajivfri@yahoo.com     Rev. Bras. Biom ., São Paulo, v.29, n.2, p.198-203, 2011   199 trend such as arithmetic and geometric. Such program includes various governments andnon-governments sponsored programs for large region and implemented at time intervalssuch as educational mission, health (child care, maternal care, AIDS eradication), povertyeradication. The impact of these programs differs in successive phases and may followknown and unknown trend such as arithmetic, geometric and others.This paper addresses the properties of the estimator and compare with theconventional proportional and optimum allocation method for sample allocation withinstrata through integration of weight of response under different phases with assumptionthat the response in successive phases has additive effect. The efficiency of estimator forphased evaluation under combined arithmetic and geometric impacts has been dealtelsewhere (Pandey, 2011). Actual practical examples are not being readily discussed,however, chances of availability of such situation is very high. 2   Sample allocation and variance as per Pandey and Verma (2008) Pandey and Verma (2008) consider the case of sample allocation for population of size N containing the population units h  N  N  N  ...,,, 21 of a development programmeimplemented into “h” different phases under the consideration that the impact of theprogramme is uniformly distributed within each phases and response follows arithmetictrend with respect to different phases. Therefore, the sample weight is as follows:(1) ii Hy hiN  N  ϖ   − += ,(1)whereh - number of development programme phasei - stratum number, i = 1, 2, 3, …, h i  N  - Actual numbers of units (beneficiaries) in the i th stratum  Hy  N  - Total numbers of units in the population adjusted by the impact or phase factorwith product of actual population size in each stratum i.e. ]}1{[ i  N ih +−  .Under this case, the unbiased estimator of population mean is as follows 1 histmii  yy ω  = =  ,(2)where stm  y is the mean of the character and i ω  is weight.And, the variance of  stm  y is as 22211 (). hhiiiistmiiii SSVynN  ω ω  = = = −     (3)     Rev. Bras. Biom ., São Paulo, v.29, n.2, p.198-203, 2011   200 3   Properties of estimator The estimated variance of  stm  y may be rewritten as follows after the substitution of value of  i ω  as per Pandey and Verma (2008). ii nn ω  =   With substitution of equation (1), the expression may be written as,  [ ][ ] 1 1.1 iiihii nhiN nnhiN  ω  = − += =− +    (4)With substitution of equation (4) in equation (3), the estimated variance may bewritten as iihiiihiistm  N SnnSnn yV  21212 .)(  == −=   = { }( ) { } ( ) { } ( ) { } 221111 11...[1][1] hhiiihhiiiiii hihiN SSnhiNhiN  = == = − +− +−− + − +      (5)To be more simple, substitute ]}1{[ i  N ih +−  =  Hy  N    (Say)   .Therefore, the above equation of estimated variance may be written as follows = )( stm  yV  { }( ) { } 2211 11... hhiiiii HyHy hihiN SS NnN  = = − +− +−     (6)On the other hand, the variance for simple random sampling (SRS) will be asfollows: 2 ()(1). sr  nSVy Nn = −   (7)Where, 2 S is sample variance under SRS.And variance under conventional proportional allocation method 22Pr11 111()()(1). hhiiiiopii nVySNSnNnNN  ω  = = = − = −     (8)   Rev. Bras. Biom ., São Paulo, v.29, n.2, p.198-203, 2011   201 And variance under Optimum allocation method 2211 11()(). hhiiii Neyii VySSnN  ω ω  = = = −     (9)Then this method is efficient to SRS and stratified sampling with conventionalproportional and Neyman allocation, if and only if, the )()( stmsr   yV  yV  − ; )()( Pr stmop  yV  yV  − ; and )()( stm Ney  yV  yV  − is positive.Under comparison, the difference of variance between conventional proportional andproposed proportional allocation method may be as follows )()( Pr stmop  yV  yV  − −−=  = hiii S N  N nnN  12 )1(1   { }( ) { } 2211 11... hhiiiii HyHy hihiN SS NnN  = = − +− ++     (10)And, if finite population correction (fpc) terms ignored then, the equation (10) maybe written as follows: )()( Pr stmop  yV  yV  − −=  = hiii S N nN  12 1 ( ) { } 21 11. hiii Hy hi NSnN  = − +    (11)Equation 11 may be written as follows )()( Pr stmop  yV  yV  −−=  = hii Snh 12 1 21 1(1). hiii Hy hiNSnN  = − +    (12)This will be a positive quantity, if, the first term is more than second term for the lefthand side (LHS). This may be also true, if the individual coefficient of  2 i S for first term of LHS is greater than the corresponding coefficient of  2 i S for second term of LHS for eachcoefficient.  Hyi  N  N ihh )1(1 +−>      ii  N ihh N ih )}1{(])1([ +−>+−     iiii ihN iN hN  N h N  Nh −>−−+  2    iii ihN iN hhN h N  −>+−+  )1()1(    )())(1( iii hN  N iiN iN hN  N h −+−>−+     (1)(). ii hiNhNiNiN  − + − > −    (13)     Rev. Bras. Biom ., São Paulo, v.29, n.2, p.198-203, 2011   202 Thus, the proposed estimator will be efficient than the conventional proportionalmethod )()( Pr stmop  yV  yV  ≥ , if and only if, equation 13 holds. The all phase (strata)will have equal number of units.Based on the similar analogy, for )()( stm Ney  yV  yV  − should also be a positivequantity and may be written as follows, without inclusion of the finite populationcorrection (fpc) term )()( stm Ney  yV  yV  −−=  = hiii S N nN  122 )(1 21 1(1). hiii Hy hiNSnN  = − +    (14)That is, Eq. (14) can be written as follows, )()( stm Ney  yV  yV  −−=  = hii Snh 122 )(1 21 1(1). hiii Hy hiNSnN  = − +    (15)This equation is sensitive in respect to total number of strata h, and the differencewill be positive, if and only if, for individual i , following will be true.  Hyi  N  N ihh )1(1 2 +−>    2 [(1)]{(1)}. ii hiNhhiN  − + > − +    (16)This condition is essentially required for the proposed estimator to be more efficientthan the Optimum allocation. Conclusion It can be concluded that the proposed estimator is case sensitive as far as efficiencyis concerned with respect to conventional proportional and Optimum allocation methodand logically, with the simple random sample too. However, theoretically sounds wellthan these all. The proposed estimator will be equal to the simple random sample, if eachstratum contains same number of units, however, less efficient than proportional andoptimum allocation method.PANDEY, R. Análise de variância comparativa sobre alocação de amostras na avaliaçãode programas de desenvolvimento.  Rev. Bras. Biom ., São Paulo, v.29, n.2, p.198-203,2011.    RESUMO: Avaliação de programa implementado em larga escala em diferentes fases pode serdeterminada por uma distribuição adequada de unidades de amostragem de cada fase doprograma para análise posterior. Métodos tem sido propostos para a alocação da amostra emdiferentes estratos (fases), considerando que os impactos nas diversas fases (implementado emhorários diferentes) crescem em progressão aritmética (Pandey e Verma, 2008). O presentetrabalho propõe para a verificação do projeto o método de repartição. Comparação também foifeita e discutida com a variação dos métodos de alocação proporcional e ótima.    PALAVRAS-CHAVE: Avaliação de impacto; alocação de amostra; progressão aritmética; peso;eficiência.
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