# Modification To Selection Procedures In Probability Proportional To Size Sampling Without Replacement

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Probability proportional to size sampling (or unequal probability sampling) is a probability sampling in which every unit in the population has different probabilities of being selected in the sample. It can be with or without replacement. Most sample selection procedures in probability proportional to size sampling without replacement (PPSSWOR) focus on selection of sample size two which is the challenge of this study. A procedure involves selection of the first unit with probability proportional to without replacement, then the second unit is selected with probability , where and are the non â€“ zero known probability values assigned to each unit of the population at any specific draw for , and N is the total number of units. Another procedure involves selection of the first unit with probability proportional to without replacement, and then a random sample of size n-1 from the remaining N-1 units is selected , where Î± and Î² are integer constants. This study aimed at modifying the sample selection procedures in PPSSWOR described above to produce more efficient ones that can be used for any sample size. The specific objectives of this study were to: (i) obtain two modified sample selection procedures (I and II) in PPSSWOR for any sample size; (ii) derive the probability of inclusion of single unit Ï€_i and the joint probability of inclusion of units Ï€_ij in the sample for procedures I and II; and (iii) determine the most efficient sample selection procedure with minimum variance using both real life and simulated data.rnModified Procedure I involves selection of the first unit with probability proportional to P_i/( 2-P_i )without replacement, and then, a random sample of size n-1 from the remaining N-1 units will be selected while the modified procedure II involves selection of the first unit with probability proportional to (P_i^Î±)/( 2-P_i^Î² ) without replacement, and then a random sample of size n-1 from the remaining N-1 units is selected. The optimum values of Î± and Î² would be determined through empirical studies.rnThe findings were: rn(i). two modified sample selection procedures (I and II) in PPSSWOR were obtained. Both procedures can be used to select sample of any size for specified values of two constants ; rn(ii). the probabilities of inclusion of single unit, Ï€_i, and the joint probability of inclusion of units Ï€_ij in the sample in PPSSWOR for these modified procedures were derived; andrn(iii). using both real life and simulated data, modified procedure II was found to have minimum variance, hence, most efficient sample selection procedure for any sample size.rnThis study concluded that when and , modified procedure II is better than the two existing procedures. It is therefore recommended for use in probability proportional to size sampling without replacement.