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Sequential Sampling

For survey sampling applications, the term sequential sampling describes any method of sampling that reads an ordered frame of N sampling units and selects the sample with specified probabilities or specified expectations. Sequential sampling methods are particularly well suited when applied with computers. They can also be applied for selecting samples of a population resulting from some other process: for example, cars coming off an assembly line, patients arriving at a clinic, or voters exiting the polls. Examples of sequential sampling schemes discussed in this entry include simple random sampling, systematic sampling, and probability proportional to size (PPS) sequential sampling.

Simple Random Sampling (Without Replacement)

Simple random sampling without replacement is defined as selecting one of the possible distinct samples of size n from a population of size N. There are

such possible samples, and each has an equal probability of being selected. Other methods generally involve selecting random numbers between 1 and N, discarding any repeats, and retaining the first n distinct units selected. Random ordering before selecting a pre-specified chunk is often used in computerized selection of simple random samples. The sequential procedure requires selecting a random number, Ri, for each population element and comparing it to a conditional probability based on what has occurred up to this point. Select Unit 1 if R1 < n/N. If Unit 1 is selected, select Unit 2 if R2 < (n - 1)/{N - 1); if Unit 1 is not selected, select Unit 2 if R2 < n/(N - 1). Proceed through the list decreasing the denominator for each new unit but decreasing the numerator only when a selection occurs.

Systematic Sampling

For the simplest case, where sampling is with equal probabilities and k = N/n is an integer, a random integer, I, between 1 and lc is drawn and when the 7th element is encountered it is included in the sample. I is then incremented by k, and the (I + k)th element is included in the sample when encountered. The process continues until n units have been designated for the sample.

A more general form of systematic sampling can be applied where sampling is with unequal probabilities and/or k ≠ N/n. Define the desired probabilities of selection for each unit as πi for i − 1,2,…,N. For an equal probability design, πi = n/N. For unequal probability designs, it is only necessary that 0 < πi≤ 1 and

To select the sample sequentially, it is necessary to draw a uniform (0,1) random number, R. For Unit 1, define S = π1. If R < S, then select Unit 1 and increment R by 1. For subsequent Unit i increase S by π1∗ and if R < S, then select Unit i and increment R by 1.

PPS Sequential

PPS sequential sampling is defined for probability minimum replacement (PMR) sampling. Sampling without replacement is then shown as a special case. Rather than working with the probability of selection, PMR selection schemes work with the expected number of hits or selections for each unit designated by E(ni), where ni is the number of times unit i is selected for a particular sample. When a size measure Xi is used, the expected number of selections per unit i is set as

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