Two-stage sampling
Survey samplingSampling techniques
𝑁 number of primary sampling units
𝑛 number of selected primary sampling units
M number of subunits (final units) per primary sampling unit
𝑦𝑖𝑗 value of the jth subunit in the ith primary unit
ത𝑦𝑖 = σ𝑗=1𝑚 𝑦𝑖𝑗
𝑚sample mean per subunit in ith primary unit
ധ𝑦 = σ𝑖=1𝑛 ത𝑦𝑖
𝑛over-all sample mean per subunit
𝑣 ധ𝑦 =1−𝑓1𝑛
𝑠12 +
𝑓1 1−𝑓2𝑚𝑛
𝑠22
𝑠12 =
σ𝑖=1𝑛 ത𝑦𝑖−ധ𝑦 2
𝑛−1𝑠22 =
σ𝑖=1𝑛 σ𝑗=1
𝑚 𝑦𝑖𝑗−ത𝑦𝑖2
𝑛 𝑚−1
Two-stage clusters sampling: clusters of equal size
Optimisation of two stage area sample designs
• First phase: compute the optimum SSU size for a single stage sampling
• Second phase: determine the optimum set of size of PSUs and number of PSUs and SSU to be selected
• This procedure assumes that PSUs are much larger than SSUs thus clustering SSUs within PSUs does not affect the optimum size of SSUs
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Two phase and simultaneous optimisation• First phase: compute the optimum SSU size for a single
stage sampling
• Second phase: determine the optimum set of size of PSUs and number of PSUs and SSU to be selected
• This procedure assumes that PSUs are much larger than SSUs thus clustering SSUs within PSUs does not affect the optimum size of SSUs.
• Although this assumption is reasonable, we explore the possibility of simultaneous optimization for the set of variables involved in two stage area sample designs
• “Simultaneous Optimization for Two Stage Area Sampling” (with D. Giuliani and A. Carfagna), Atti della XLIV Riunione Scientifica della Società Italiana di Statistica, Università della Calabria, Arcavacata, 25-27 June 2008, pp. 1-2 http://old.sis-statistica.org/files/pdf/atti/rs08_spontanee_a_4_4.pdf
Simultaneous optimisation of two stage area sample designs
Cost function:
Where:
a1 fixed cost per PSU, a2 variable cost per PSU a3 fixed cost per SSU,a4 variable cost per SSU
m number of PSUs in the sample
n number of SSUs sampled from each selected PSU
size of PSUs
size of SSUs
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Two phase versus simultaneous optimisation
• Two phase optimization procedure assumes that PSUs are much larger than SSUs
• simultaneous optimization requires an iterative procedure
• simultaneous optimization where within and between variances are variables
• very complicated and assumes independency
• Both types of simultaneous optimization procedure do not guaranty that all its solutions are acceptable
• When some solutions are not acceptable, the two phase optimization procedure gives better results than forcing unacceptable solutions of the simultaneous optimization