9.2 - Two Stages with Primary Units Selected by Probability Proportional to Size and Secondary Units Selected with S.R.S.

Multi-stage design with primary units selected with p.p.s. and secondary units selected with simple random sampling.

Using the Hansen-Hurwitz estimator, we get the following:

To estimate the population total:

\(\hat{\tau}_p=\dfrac{M}{n}\sum\limits_{i=1}^n \dfrac{\hat{y}_i}{M_i}=M \dfrac{\sum \bar{y}_i}{n}\), where \(\bar{y}_i=\dfrac{\hat{y}_i}{M_i}\)

\(\hat{V}ar(\hat{\tau}_p)=\dfrac{M^2}{n(n-1)} \sum (\bar{y}_i-\hat{\mu}_p)^2\)

To estimate the population mean:

\(\hat{\mu}_p=\dfrac{\sum \bar{y}_i}{n}\)

[since \(\hat{\mu}_p=\left(\dfrac{\hat{\tau}_p}{M}\right)\) and thus it becomes \(\dfrac{\sum \bar{y}_i}{n}\)]

\(\hat{V}ar(\hat{\mu}_p)=\dfrac{1}{n(n-1)}\sum (\bar{y}_i-\hat{\mu}_p)^2\)

Example

There are 36 departments in a small liberal arts college. One wants to estimate the average amount of money the students spent on textbooks last semester. Since the size of each department varies very much, a two-stage cluster sampling using probability proportional to size for the primary unit is carried out. The results are listed in the table below.

Department \(\mathbf{M_i}\) \(\mathbf{m_i}\) Textbook expenses in $ for last semester
1 10 4 326, 400, 423, 443
2 20 8 278, 312, 450, 350, 227, 438, 512, 403
3 30 12 512, 256, 332, 402, 512, 309, 411, 610, 422, 630, 550, 470
4 15 6 426, 312, 512, 440, 342, 533

 Minitab output:

Decriptive Statistics: dept1, dept2, dept3, dept4
Variable Mean SE Mean StDev Variance
dept1 398.0 25.6 51.1 2612.7
dept2 371.3 34.1 96.3 9277.4
dept3 451.3 33.9 117.6 13828.8
dept4 427.5 36.1 88.4 7815.9

 

Try it!

Estimate the population mean using probability proportional to size estimator (Hansen-Hurwitz) and estimate the variance of that estimator.

\(\hat{\mu}_p=\dfrac{\sum \bar{y}_i}{n}=\dfrac{398+371.3+451.3+427.5}{4}=412.025\)

\begin{align}
\hat{V}ar(\hat{\mu}_p) &= \dfrac{1}{n(n-1)}\sum (\bar{y}_i-\hat{\mu}_p)^2\\
&= \dfrac{1}{4\times 3}\left[(398-412.025)^2+(371.3-412.025)^2+(451.3-412.025)^2+(427.5-412.025)^2\right]\\
&= 303.12\\
\end{align}