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Probability Proportional to Size (PPS) Sampling

Probability proportional to size (PPS) sampling includes a number of sample selection methods in which the probability of selection for a sampling unit is directly proportional to a size measure, Xit which is known for all sampling units and thought to be approximately proportional to the unknown F;. The Xi must all be strictly greater than 0. In single-stage sampling it can be used to reduce the variance of survey estimates. If the observed values, Yit are exactly proportional to Xi, the variance of an estimated total will be exactly 0. When the F; are approximately proportional to the Xi, the variance can still be greatly reduced relative to equal probability sampling schemes. PPS sampling is also used in the early stages of multi-stage samples to achieve equal probability samples of the final-stage sampling units or EPSEM samples.

In all cases, suppose one is estimating a population total for a variable F; with N units.

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There are a wide variety of sampling schemes that have PPS properties. Only a few of them are discussed here for illustration.

PPS with Replacement Sampling

The simplest PPS sampling method is PPS with replacement sampling. In this case, the single-draw probability of selection, pit on each independent sample draw is proportional to Xi, that is,

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Note that unless N—1, the individual probabilities will be less than 1. The with replacement estimator is

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where yi represent the observed values indexed over the sample of n draws. Note that the same population unit may be selected at more than one draw. The variance of this estimator is

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The replacement variance estimator has a simpler form:

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An unbiased estimate of the variance can be obtained for all samples of size 2 draws or greater.

To select a PPS with replacement sample, the following procedure is followed for each sample draw. Select a uniform (0,1] random number, R. Assume the N population elements are ordered and indexed by I.

Compute the partial sums None Select the unit i if None in simpler terms, the unit selected will be the first one whose partial sum equals or exceeds R.

PPS without Replacement Sampling

In practice, PPS without replacement sampling is more commonly used, particularly in the first stage of multistage samples. Since sampling is without replacement, the sample size n now represents unique population units selected into the sample. Let S be the set of all None samples of size n from the population of size N and is indexed by s and None represents the probability of a particular sample s. The probability of selecting any particular unit i in a sample of size n is formally defined as None Summation is over all samples of size n that contain the population element i. Many creative methods have been developed to select PPS without replacement samples that satisfy the property πi = npi where pi is as defined previously for with replacement sampling. It often happens that npi ≥ 1 for some i. A common fix for this problem is to select unit i with probability 1 and select a PPS without replacement sample of size n − 1 from the remaining elements.

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