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

mncncC 21

1).(

2

1

c

cmopt

).().(

21 moptcc

Cnopt

111

)(V̂ ˆ2

mSnm

fy

M

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

mnAmAC sp )()( 4321 aaaa

pA

sA

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