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A cluster sampling design is one where some sampling units include more than one element.
Steps of a cluster sampling design:
Note: This is what is called one-stage cluster sampling. Later we will discuss two-stage and multi-stage cluster sampling. But until we discuss these designs, it will be implied that we are referring to a one-stage design.
Example: Consider the following sampling design. The population, sampling units, and their respective sizes are as follows. \[ \mathcal{P} = \{ \underbrace{\mathcal{E}_1, \mathcal{E}_2, \mathcal{E}_3}_{\mathcal{U}_1}, \underbrace{\mathcal{E}_4, \mathcal{E}_5, \mathcal{E}_6, \mathcal{E}_7}_{\mathcal{U}_2}, \underbrace{\mathcal{E}_8,\mathcal{E}_9,\mathcal{E}_{10},\mathcal{E}_{11},\mathcal{E}_{12}}_{\mathcal{U}_3}\}, \ N=3, \ M=12 \] \[\begin{align*} \mathcal{U}_1 & = \{\mathcal{E}_1,\mathcal{E}_2,\mathcal{E}_3\}, \ m_1 = 3 \\ \mathcal{U}_2 & = \{\mathcal{E}_4,\mathcal{E}_5,\mathcal{E}_6,\mathcal{E}_7\}, \ m_2 = 4 \\ \mathcal{U}_3 & = \{\mathcal{E}_8,\mathcal{E}_9,\mathcal{E}_{10},\mathcal{E}_{11},\mathcal{E}_{12}\}, \ m_3 = 5 \end{align*}\] Note that \(m_i\) denotes the number of elements in the \(i\)-th sampling unit or cluster.
If we were to apply simple random sampling to these sampling units to select \(n\) = 2 clusters, the possible samples and their probabilities are as follows. \[\begin{align*} \mathcal{S}_1 & = \{\mathcal{U}_1,\mathcal{U}_2\} = \{\underbrace{\mathcal{E}_1,\mathcal{E}_2,\mathcal{E}_3}_{\mathcal{U}_1},\underbrace{\mathcal{E}_4,\mathcal{E}_5,\mathcal{E}_6,\mathcal{E}_7}_{\mathcal{U}_2}\}, P(\mathcal{S}_1) = 1/3 \\ \mathcal{S}_2 & = \{\mathcal{U}_1,\mathcal{U}_3\} = \{\underbrace{\mathcal{E}_1,\mathcal{E}_2,\mathcal{E}_3}_{\mathcal{U}_1},\underbrace{\mathcal{E}_8,\mathcal{E}_9,\mathcal{E}_{10},\mathcal{E}_{11},\mathcal{E}_{12}}_{\mathcal{U}_3}\}, P(\mathcal{S}_2) = 1/3 \\ \mathcal{S}_3 & = \{\mathcal{U}_2,\mathcal{U}_3\} = \{\underbrace{\mathcal{E}_4,\mathcal{E}_5,\mathcal{E}_6,\mathcal{E}_7}_{\mathcal{U}_2},\underbrace{\mathcal{E}_8,\mathcal{E}_9,\mathcal{E}_{10},\mathcal{E}_{11},\mathcal{E}_{12}}_{\mathcal{U}_3}\}, P(\mathcal{S}_3) = 1/3. \end{align*}\] This would be one possible cluster sampling design.
Examples of sampling units, elements, and target variables where cluster sampling might be used.| Sampling Unit | Element | Target Variable |
|---|---|---|
| box | widget | weight |
| block | household | income |
| county | farm | acres of wheat |
| classroom | student | test score |
| hour | minute | number of fish |
| plot | tree | volume |
What are the potential advantages of cluster sampling (relative to SRS)?
What are the potential disadvantages of cluster sampling (relative to SRS)?
How is cluster sampling different from stratified random sampling?1
How could we view simple random sampling as a special case of cluster sampling?
| \(i\) | \(m_i\) | \(y_{ij}\) | \(y_i\) |
|---|---|---|---|
| 1 | 3 | \(y_{11},y_{12},y_{13}\) | \(y_1 = y_{11} + y_{12} + y_{13}\) |
| 2 | 4 | \(y_{21},y_{22},y_{23},y_{24}\) | \(y_2 = y_{21} + y_{22} + y_{23} + y_{24}\) |
| 3 | 5 | \(y_{31},y_{32},y_{33},y_{34},y_{35}\) | \(y_3 = y_{31} + y_{32} + y_{33} + y_{34} + y_{35}\) |
Note that the three clusters have sizes of \(m_1\) = 3, \(m_2\) = 4, and \(m_3\) = 5.
The mean and total of a target variable for all elements in the population can be computed as \[ \mu = \frac{1}{M}\sum_{i=1}^N y_i \ \ \ \text{and} \ \ \ \tau = \sum_{i=1}^N y_i, \] respectively, noting that \(M = \sum_{i=1}^N m_i\) is the number of elements in the population.
Note: Much of the estimation theory of cluster sampling (assuming simple random sampling of clusters) is essentially treating the clusters as elements where the cluster total \(y_i\) is the target variable. But one key difference is that \(\mu = \tau/M\) and not \(\mu = \tau/N\).
Note that we can write \(\mu\) as \[ \mu = \frac{1}{M}\sum_{i=1}^N y_i = \frac{\sum_{i=1}^N y_i}{\sum_{i=1}^N m_i}. \] We can also write this as \[ \mu = \frac{\frac{1}{N}\sum_{i=1}^N y_i}{\frac{1}{N}\sum_{i=1}^N m_i} = \frac{\mu_y}{\mu_m}, \] where \(\mu_y\) is the mean cluster total across all clusters (which is not necessarily the same as \(\mu\)) and \(\mu_m\) is the mean cluster size across all clusters (which is also called \(\bar{M}\) and equals \(M/N\)). Assuming simple random sampling of clusters, this suggests that we might estimate \(\mu\) with \[ \hat\mu = \frac{\bar{y}}{\bar{m}} = \frac{\frac{1}{n}\sum_{i \in \mathcal{S}} y_i}{\frac{1}{n}\sum_{i \in \mathcal{S}} m_i} = \frac{\sum_{i \in \mathcal{S}} y_i}{\sum_{i \in \mathcal{S}} m_i}, \] (i.e., the ratio of the totals of the clusters totals and the cluster sizes for the sampled clusters). Where have we seen this kind of estimator before?
Example: A cluster sampling design selects \(n\) = 3 boxes using simple random sampling of the boxes. The number of widgets in these boxes are \(m_1\) = 3, \(m_2\) = 4, and \(m_3\) = 5. The total weight of the widgets in these boxes are \(y_1\) = 6.2, \(y_2\) = 7.5, and \(y_3\) = 10.3. What is the estimate of \(\mu\)?
The estimated variance of the estimator \[ \hat\mu = \frac{\sum_{i \in \mathcal{S}} y_i}{\sum_{i \in \mathcal{S}} m_i}, \] assuming simple random sampling of clusters, is \[ \hat{V}(\hat\mu) = \frac{1}{\bar{M}^2}\left(1 - \frac{n}{N}\right) \frac{s_r^2}{n} \ \ \ \text{where} \ \ \ s_r^2 = \frac{\sum_{i \in \mathcal{S}}(y_i-\hat\mu m_i)^2}{n-1}, \] where \(\bar{m}\) can be used in place of \(\bar{M}\) if it is unknown.
Example: Assume that in the previous example that there are a total of 100 boxes, and that the total number of widgets in all those boxes is 425. What is the variance and bound on the error of estimation for \(\hat\mu\)?
Recall that \[ \mu = \frac{\mu_y}{\mu_m}, \] where \(\mu_y\) is the mean cluster total across all clusters and \(\mu_m\) is the mean cluster size across all clusters. If we know \(\mu_m\) we might use it instead of \(\bar{m}\) and therefore use the estimator \[ \hat\mu = \frac{\bar{y}}{\mu_m} \] instead of the estimator introduced earlier which can be written as \(\hat\mu = \bar{y}/\bar{m}\). These estimators are not equivalent unless all clusters are of the same size (in which case \(\mu_m = \bar{m}\)). Should we use this alternative estimator? Probably not. Why? Consider our discussion of two estimators of a ratio of totals.
Note: The symbol \(\bar{y}_U\) in that diagram is the population mean (\(\mu\)).↩︎
