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A Computer Implementation of Estimated Variances in Multi-Stage Cluster Sampling Schemes

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A COMPUTER IMPLEMENTATION OF ESTIMATED VARIANCES IN MULTI-STAGE CLUSTER SAMPLING SCHEMES
L. A. Nafiu, L. Idris, A. F. Busari and A. B. Olaniyan
Department of Mathematics and Statistics,
Federal University of Technology, Minna, Niger State
(lanconserv@yahoo.com)

ABSTRACT
The computation of sample variances arising from multi-stage cluster sampling schemes or designs are complex and time-consuming. This paper presents a computer software written with Java programing language for implementing some of the available formulars for estimated variances in multi-stage techniques. The software has the advantages of accessibility, cheapness, and ease of use in computing estimated variances in both one-stage, two-stage and three-stage sampling schemes. A data set for estimating number of diabetic patients in Niger state for 2005 was used for illustration. We recommend that computation involving these estimated variances be done with the aid of this software.
Keywords: Software, Computation, Multi-stage, Estimated Variances, Time, Data and Diabetic Patients.
Introduction
Multistage sampling is where the researcher divides the population into clusters, samples the clusters, and then resample, repeating the process until the ultimate sampling units are selected at the last of the hierarchical levels (Okafor, 2002). For instance, at the top level, states may be sampled (with sampling proportionate to state population size); then cities may be sampled; then schools; then classes; and finally students (Goldstein, 1995). Multistage sampling according to Adams et al (2003) is generally used when it is costly or impossible to form a list of all the units in the target population.
If, after selecting a sample of primary units, a sample of secondary units is selected from each of the selected primary units, the design is referred to as two-stage sampling. If in turn a sample of tertiary units is selected from each selected secondary unit, the design is three-stage sampling. Three-stage sampling involves selecting a sample in three stages (Thompson, 2002). At the first stage, large groups or clusters of population units are selected. These clusters are designed to contain more units than are required for a final sample. At the second stage, units are sampled from the selected clusters to derive the final sample. Importantly, the process of selecting “sub-cluster” within clusters continues until the final sample is achieved (Fink, 2002).
Aims and Objectives
The aim of this paper is to develop a software to solve the manually computed estimation of variances in multi-stage cluster sampling schemes.
The objectives of this automated system are: i. Speed: Instead of having to solve the problem manually, the written program gives instruction to the computer which executes for as many intervals as it is instructed to do so. Thus, it is convenient and saves time. ii. Accuracy: Mistakes can easily be made when manual process is used. With the computer, accurate results are gotten if the right instruction is supplied. iii. Ease of Use: Using the software for estimation of variances in both one-stage, two-stage and three-stage cluster sampling schemes has been made very easy.
Review of the Study
The manual computation of estimeted variances in multi-stage cluster sampling desings is complex, time-consuming and prone to human-error (Cochran, 1977). Horvitz and Thompson (1952) did their computation with advice on calculation by computer. Nafiu (2007) also used computer softaware for the computation of variances when comparing four (4) estimators under sampling without replacement.
Methodology
In this section, we give the procedural steps involved in both one-stage, two-stage and three-stage cluster sampling schemes.
One-Stage Cluster Sampling Scheme
Ai = Sum of Yij
Bi = AiMi
Ci = Bi * ni
D=Add all Ci's
Y1sc = D
Ei = (Ci - Dn)2
F=Add all Ei's
G = F(n-1) n=Add all mi's
N=Add all ni's
F = nN V(Y1sc) = N2 (1-f)Gn

Two- Stage Cluster Sampling Scheme
Aij=Sum of Yijk
Bi= Aimi
Ci= Bi* Mi
Di= Cini
Ei= Di* Ni
Then F=Add all Ei's
Y2cs = F.
Gij = (Yij- Bi)2,
Then Hi=Add all Gij's
Ii= Hi(mi -1)
Then J=Add all Ii's L= Fn
Qi=(Ci's- L)2
Then R = Add all Qi's f2i = miMi f1 = nN
W = Rn-1
Xi = Mi2 (1-f2i)mi
Then X = Add all Xi's
Zi = Ii*Xi
Then Z = Add all Zi's n = Add all ni's N = Add all Ni's
V(Y2sc) = N2(1-f1)Wn + NZn

Three-Stage Cluster Sampling Scheme
Aij=Sum of Yijk
Bij= Aijkij
Cij= Bij* Kij
Then Di=Add all Cij's
Ei= Dimi
Fi= Ei* Mi
Then G=Add all Fi
Y3cs = G.
Hijk=(Yijk- Bij)2
Then Hij=Add all Hijk
Ii= Hij(kij- 1)
Then J=Add all Ii's
P = Jm
Q = P*M
Ri= Di- P2/(mi- 1)
Then R=Add all Ri's
N = 25, n = 5 m=Add all mi's, M=Add all Mi's f1= nN f2i= miMi f3ij= kijKij
Xi= (Fi- G/n)2
Then X=Add all Xi's
W= X(n-1)
Zi= Mi2(1 - f2i)Rimi
Tij = Kij2(1 - f3ij)Iijkij
Ti=Add all Tij's
Ui= Ti* Mimi
Then U=Add all Ui's
And Z=Add all Zi's
V(Y3cs) = N2(1 - f1)Wn + NnZ + NnU

Program Interface and Result
In line with Eckel (1998), we present the interface for the developed software and the results generated from the illustration using data from both Niger State (2005) and National Bureau of Statistics (2007):

Figure 1: Showing the Format of Dataset on the Program Interface for one-stage.

Table 1: Showing Parameters and Estimates for one-stage.

Figure 2: Showing the Graph of Various Estimates for one-stage.

Figure 3: Showing the Format of Dataset on the Program Interface for two-stage.

Table 2: Showing the Parameters and the Estimates for two-stage.

Figure 4: Showing the Graph of Various Estimates for two-stage.

Figure 5: Showing the Format of Dataset on the Program Interface for three-stage.

Table 3: Showing the Parameters and the Estimate for three-stage.

Figure 6: Showing the Graph of Various Estimates for three-stage.
Discussion of Results
Computing variances in one-stage, two-stage and three-stage manually with the formulars specified for each of the stages in the methodology is slow, tedious, time-consuming and prone to human-error but the application of software developed in this paper for computing these variances in each of the stages has alleviated the problems. Rigorous testing done on the software has helped to eliminate most known bug in the software which has in turn helped in getting better and more accurate result with the software as shown in the comparison of the result set of the estimated variances in tables 1, 2 and 3. It is obvious that the two-stage cluster sampling of the dataset yields a better result compared to that of one-stage while the result of the three-stage cluster sampling proves to be the best. The process of computing these result set has been made simple, less-prone to error and easy with the aid of the software developed.

Conclusion and Recommendations
Considering the fact that the objective of any automated system is speed, accuracy and ease of use (Deitel and Deitel, 2004). The software developed in this paper meets these objectives with respect to computing variances in multi-stage cluster sampling schemes. Hence, we recommend that computation involving the estimation of variances in one-stage, two-stage and three-stage cluster sampling schemes be done with the aid of this software.

References
Adams, J. L., Wickstrom, S. L., Burgess, M. J., Lee, P. P. and Escarce, J. J. (2003). “Sampling Patients within Physician Practices and Health Plans: Multistage Cluster Samples in Health Services Research”. The Global Journal for Improving Health Care Delivery and Policy. 38: 1625-1640.
Cochran, W.G. (1977). Sampling Techniques. Third Edition. New York: John Wiley and Sons.
Fink, A. (2002). How To Sample In Surveys. Thousand Oaks, C.A.: Sage Publications.
Goldstein, H. (1995). Multilevel Statistical Models. New York: Halstead Press.
Horvitz, D. G. and Thompson, D. J. (1952). “A Generalization of Sampling without Replacement from a Finite Universe”. Journal of American Statistical Association. 47: 663-685.
Deitel, H. M. and Deitel, P. J. (2004). Java How To Program. Sixth Edition. New Jersey: Prentice Hall PTR.
Eckel, B. (1998). Thinking in Java. Fifth Edition. New Jersey: Prentice-Hall Inc.
Nafiu, L. A. (2007). “Comparison of Four Estimators under Sampling without Replacement”. Unpublished M. Sc. Dissertation, University of Ilorin, Ilorin, Nigeria. National Bureau of Statistics (2007). Directory of Health Establishments in Nigeria. Nigeria: Abuja Printing Press. Niger State (2005). Niger State Statistical Year Book. Nigeria: Niger Press Printing & Publishing. Okafor, F. (2002). Sample Survey Theory with Applications. Nigeria: Afro-Orbis Publications. Thompson, S. K. (2002). Sampling. New York: John Wiley and Sons.

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