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Systematic Sampling (SYS)

Systematic Sampling (SYS). Up to now, we have only considered one design: SRS of size n from a population of size N New design: SYS

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Systematic Sampling (SYS)

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  1. Systematic Sampling (SYS) • Up to now, we have only considered one design: • SRS of size n from a population of size N • New design: SYS • DEFN: A 1-in-ksystematic sample is a sample obtained by randomly selecting one sampling unit from the first k sampling units in the sampling frame, and every k-th sampling unit thereafter

  2. Sampling procedure for SYS • Have a frame, or list of N SUs • Assume SU = OU for now • Determine sampling interval, k • k is the next integer after N/n • Select first SU in the list • Choose a random number, R , between 1 & k • R-th SU is the first SU to be included in the sample • Select every k-th SU after the R-th SU • Sample includes unit R, unit R + k, unit R + 2k, …, unit R + (n-1)k

  3. Example • Telephone survey of members in an organization abut organization’s website use • N = 500 members • Have resources to do n = 75 calls • N / n = 500/75 = 6.67 • k = 7 • Random number table entry: 52994 • Rule: if pick 1, 2, …, 7, assign as R; otherwise discard # • Select R = 5 • Take SU 5, then SU 5+7 =12, then SU 12+7 =19, 26, 33, 40, 47, …

  4. Example – 2 • Arrange population in rows of length k = 7

  5. Properties of systematic sampling – 1 • Number of possible SYS samples of size n is k • Only 1 random act - selecting R • After select 1st SU, all other SUs to be included in the sample are predetermined • A SYS is a cluster with sample size 1 • Cluster = set of SUs separated by k units • Unlike SRS, some sample sets of size n have no chance of being selected given a frame • A SU belongs to 1 and only 1 sample

  6. Properties of systematic sampling – 2 • Number of possible SYS samples of size n is k • Are these samples equally likely to be selected? • Probability of selecting a sample • P{S } = 1/k • Inclusion probability for a SU • P{ SU i S } = 1/k

  7. Properties of systematic sampling – 3 • Plan for sample size of n , but actual sample size may vary • If N / k is an integer, then n = N / k • If N / k is NOT an integer, then n is either the integer part of (N / k ) or the integer part of (N / k ) + 1

  8. Properties of systematic sampling – 4 • Because only the starting SU of a SYS sample is randomized, a direct estimate of the variance of the sampling distribution can not be estimated • Under SRS, variance of the sampling distribution was a function of the population variance, S2 • Have no such relationship for SYS

  9. Estimation for SYS • Use SRS formulas to estimate population parameters and variance of estimator

  10. Properties of systematic sampling – 5 • Properties of SRS estimators depends on frame ordering • SRS estimators for population parameters usually have little or no bias under SYS • Precision of SRS estimators under SYS depends on ordering of sample frame

  11. Order of sampling frame • Random order • SYS acts very much like SRS • SRS variance formula is good approximation • Ordered in relation to y • Improves representativeness of sample • SRS formula overestimates sampling variance (estimate is more precise than indicated by SE) • Periodicity in y = sampling interval k • Poor quality estimates • SRS formula underestimates sampling variance (overstate precision of estimate)

  12. Example – 3 • Suppose X [age of member] is correlated with Y [use of org website] • Sort list by X before selecting sample

  13. Practicalities • Another building block (like SRS) used in combination with other designs • SYS is more likely to be used than SRS if there is no stratification or clustering • Useful when a full frame cannot be enumerated at beginning of study • Exit polls for elections • Entrance polls for parks

  14. Practicalities – 2 • Best if you can sort the sampling frame by an auxiliary variable X that is related to Y • Improve representativeness of sample (relative to SRS) • Improve precision of estimates • Essentially offers implicit form of stratification

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