pps (probability proportional to size) Systematic Sampling

 Meby Joseph Manoj

Introduction

          Systematic sampling is a type of probability sampling method in which sample members from a larger population are selected according to a random starting point but with a fixed, periodic interval. This interval, called the sampling interval, is calculated by dividing the population size by the desired sample size.

            Probability proportional to size (pps) sampling is a method of sampling from a finite population in which a size measure is available for each population unit before sampling and where the probability of selecting a unit is proportional to its size.

             It is known that sampling clusters with probability proportional to a measure of size often yields considerable reduction in the variance of estimators. William G Madow (1949) suggested that generalization can be made about systematic sampling with equal probabilities. The theory of the systematic selection of several clusters with probability proportionate to a measure of size is used for this.

pps Systematic Sampling

As in the case of cumulative total method, in probability proportional to size systematic sampling, with each unit a number of numbers equal to its size are associated and the units corresponding to a sample of numbers drawn systematically will be selected as sample. That is, in sampling n units with this procedure, the cumulative totals Ti, i = 1.2 .... N, are determined and the units corresponding to the numbers {r + jk}, j = 0, 1, 2, ... (n -1) are selected, where k = T/n = X/n and r is a random number from 1 to k. This procedure is known as pps systematic sampling. The unit U i is included in the sample, if Ti-1 < r + jk ≤ Ti for some value of j = 0.1. 2, ... (n -1). Since the random number, which determines the sample. is selected from 1 to k and since Xi of the numbers are favourable for inclusion of the ith unit in a sample, the probability ℼi of inclusion of the ith population unit is nXi/X provided k < Xi.

Advantages.

  •    Easy to Execute and Understand.
  •  Control and Sense of Process.
  •  Low Risk Factor. 
  • Assumes Size of Population Can Be Determined.


Disadvantages.

  •    Unbiased variance estimator on the basis of a single sample is not possible.
  •   Greater Risk of Data Manipulation.
  • Over- or under-representation of particular patterns

Formulas

            An unbiased estimator of the population total Y, derived by Hartley and Rao is given by


            For fairly large values of N and for values of n relatively small compared to N, the approximate sampling variance of the estimator can be written as

            

           
                 Further, this variance can be estimated by


    Selection of sample using R

    












When to Use.

  • ·         When the size of the variables is inconsistent.
  • ·         When there’s low risk of data manipulation.
  • ·         When there is no pattern in the data










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