We have learned about cluster sampling where one selects the primary units and then all of the cases from the secondary units. With multi-stage sampling, we will only select some of the units from the secondary stages.

For example, in two-stage sampling:

• 1st stage samples n primary units
• 2nd stage, for the ith primary unit, selects m_i (not all) secondary units

Multistage designs are used in many practical cases. These are just a few:

1. Large surveys involving the sampling of housing units - The U.S. Census Bureau selects geographical areas within each state and then selects housing units within each selected geographical area.
2. Practical quality control problems often involve two (or more) stages of sampling. For example, Ford wants to inspect the quality of a supplier of air filters. They first sample some cartons and then inspect some air filters inside these selected cartons.
3. Gallop poll samples approximately 300 election districts. In the second stage, they select 5 households per district.

Notation:

• N: number of primary units in the population
• \(\mathbf{M_i}\): number of secondary units in the ith primary unit
• \(y_i=\sum\limits_{j=1}^{M_i}y_{ij}\)
• population total : \(\tau=\sum\limits_{i=1}^N \sum\limits_{j=1}^{M_i}y_{ij}\)
• \(\mu=\dfrac{\tau}{M}\) where \(M=\sum\limits_{i=1}^N M_i\)
• n: number of primary units selected in the first stage
• \(\mathbf{m_i}\): number of secondary units selected in the second stage

Multistage Design

This is something that arises in practice quite often. As a result, we need to be able to figure out how this type of sampling design is implemented. Most of the time this deals with two stages of sampling with simple random sampling at each stage.

Let's take a look at this graph as a means of understanding how this type of sampling design plays out.

N = 50 for both graphs

Here is another graph for another example of a two-stage sample

Two-stage cluster sampling with simple random sampling at each stage

We will discuss two possible estimators for this sampling design: unbiased estimator and ratio estimator.

Since simple random sampling is used in the second stage, an unbiased estimator of the total y-value for the ith primary unit is:

\(\hat{y}_i=M_i \dfrac{\sum\limits_{j=1}^{m_i}y_{ij}}{m_i}=M_i \bar{y}_i\) where \(\bar{y}_i=\dfrac{\sum\limits_{j=1}^{m_i}y_{ij}}{m_i}\)

The first part of this formula is also known as the expansion estimator.

Also, since simple random sampling is used in the first stage, an unbiased estimator for the population total is:

\(\hat{\tau}=N\cdot \dfrac{\sum\limits_{i=1}^n\hat{y}_i}{n}=N \cdot \dfrac{\sum\limits_{i=1}^n M_i \bar{y}_i}{n}\)

Now we have the expansion estimators from each stage. The next thing we need is the variance.

The estimated variance of \(\hat{\tau}\) is:

\(\hat{V}ar(\hat{\tau})=N(N-n)\dfrac{s^2_u}{n}+\dfrac{N}{n}\sum\limits_{i=1}^n M_i (M_i-m_i) \dfrac{s^2_i}{m_i}\)

\(s_u^2\) is the sample variance among the primary unit totals,
\(s_i^2\) is the sample variance within the ith primary unit, here

\(s^2_u=\dfrac{1}{n-1}\sum\limits_{i=1}^n \left(\hat{y}_i-\dfrac{\sum\limits_{i=1}^n \hat{y}_i}{n}\right)^2\), and \(s^2_i=\dfrac{1}{m_i-1}\sum\limits_{j=1}^{m_i}(y_{ij}-\bar{y}_i)^2\)

To estimate the population mean μ =\(\tau\)/ M, the estimators, and the estimated variance are:

\(\hat{\mu}=\dfrac{N}{M}\cdot \dfrac{\sum\limits_{i=1}^n \hat{y}_i}{n}\), and \(\hat{V}ar(\hat{\mu})=\dfrac{1}{M^2}\hat{V}ar(\hat{\tau})\)

Let's take a look at an example where we can compute both the estimates and their variances.

For the population total, the ratio estimator and its estimated variance are:

\(\hat{\tau}_r=\dfrac{\sum\limits_{i=1}^n \hat{y}_i}{\sum\limits_{i=1}^n M_i}\cdot M=\hat{r}M\)

\(\hat{V}ar(\hat{\tau}_r)=\dfrac{N(N-n)}{n}\cdot \dfrac{1}{n-1}\sum\limits_{i=1}^n(\hat{y}_i-M_i\hat{r})^2+\dfrac{N}{n}\sum\limits_{i=1}^n M_i(M_i-m_i)\dfrac{s^2_i}{m_i}\)

A similar question can be asked of the population mean. Therefore, for the population mean, the ratio estimator and its estimated variance are:

\(\hat{\mu}_r=\hat{r}\)

\(\hat{V}ar(\hat{\mu}_r)=\dfrac{1}{M^2}\hat{V}ar(\hat{\tau}_r)\)

Recall: \(M=\sum\limits_{i=1}^N M_i\)

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 \(\dfrac{\sum \bar{y}_i}{n}\)]

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