This question asks about the relationship between within-cluster variance (\(S_w^2\)) and between-cluster variance (\(S_b^2\)) in cluster sampling, specifically when the units within the same cluster are highly correlated.

Understanding Cluster Sampling and Variance

Cluster sampling is a sampling technique where the population is divided into groups called clusters. Instead of sampling individuals directly, a random sample of these clusters is selected, and then all units within the selected clusters are typically included in the sample.

In the context of cluster sampling, we often consider two main components of variance:

• Within-cluster variance (\(S_w^2\)): This measures how much the individual units vary *within* each specific cluster. If all units inside a cluster are very similar, the within-cluster variance will be low.
• Between-cluster variance (\(S_b^2\)): This measures how much the averages (or totals) of different clusters vary *from each other*. If the clusters are very different from one another in terms of their average values, the between-cluster variance will be high.

Analyzing the Effect of High Within-Cluster Correlation

The question specifies a scenario where the units within the same cluster are highly correlated. This means that if one unit in a cluster has a certain characteristic or value, other units in the same cluster are very likely to have a similar characteristic or value. For example, if sampling households in neighborhoods (clusters), and income is highly correlated within neighborhoods, then households in the same neighborhood tend to have similar incomes.

Let's think about what high within-cluster correlation implies for the variances:

• Because units within a cluster are very similar (highly correlated), there is little variation among them. This directly leads to a low within-cluster variance (\(S_w^2\)).
• If units within a cluster are similar, the main differences in the population values will likely exist *between* the clusters. For example, one neighborhood (cluster) might have generally high incomes (all similar within that cluster), while another neighborhood might have generally low incomes (all similar within that cluster). The difference in average income between these two clusters would be large. This scenario results in a high between-cluster variance (\(S_b^2\)).

Relationship Between \(S_w^2\) and \(S_b^2\) with High Within-Cluster Correlation

When units within clusters are highly correlated, the variability *within* clusters is suppressed, while the variability *between* clusters becomes more pronounced. Therefore, in this scenario, the within-cluster variance (\(S_w^2\)) tends to be smaller than the between-cluster variance (\(S_b^2\)).

Mathematically, high within-cluster correlation is often associated with a large Intraclass Correlation Coefficient (ICC), denoted by \(\rho\). A positive ICC indicates similarity within clusters. For simple random sampling within clusters, the expected within-cluster variance is related to the total variance and ICC. When \(\rho\) is large (high positive correlation), \(S_w^2\) is reduced relative to \(S_b^2\). Specifically, the total variance can be decomposed into components related to \(S_w^2\) and \(S_b^2\), and high \(\rho\) makes the between-cluster component dominant.

Thus, in the case of high within-cluster correlation:

\(S_w^2 \le S_b^2\)

This means the within-cluster variance is lesser than or equal to the between-cluster variance.

Evaluating the Options

Let's look at the given options based on our analysis:

1. \(S_w^2\) is greater than and equal to \(S_b^2\): This contradicts our finding that high correlation within clusters reduces \(S_w^2\).
2. \(S_w^2 \) is equal to \(S_b^2\): This is a specific case and not generally true when correlation is high; \(S_w^2\) is typically *less* than \(S_b^2\).
3. \(S_w^2\) is lesser than and equal to \(S_b^2\): This aligns with our conclusion that high within-cluster correlation leads to a smaller within-cluster variance compared to the between-cluster variance.
4. \(S_w^2 \) and \(S_b^2\) are not comparable: These are both measures of variance and are directly comparable; their relationship depends on the population structure and sampling method.

Therefore, the option that correctly describes the relationship when units within the same cluster are highly correlated is that \(S_w^2\) is lesser than and equal to \(S_b^2\).

Revision Table: Cluster Sampling Concepts

Additional Information: Variance Components and Sampling Efficiency

Understanding the components of variance, \(S_w^2\) and \(S_b^2\), is crucial for evaluating the efficiency of cluster sampling compared to other methods like simple random sampling (SRS).

• In SRS, the variance of an estimate depends directly on the total population variance.
• In cluster sampling, the variance of an estimate is influenced by both \(S_w^2\) and \(S_b^2\), and importantly, the relationship between them, which is captured by the intraclass correlation coefficient (\(\rho\)).
• When units within clusters are highly correlated (\(\rho\) is large and positive), \(S_w^2\) is low, and \(S_b^2\) is high. In this scenario, the clusters are internally homogeneous but externally heterogeneous. Sampling only a few clusters means you might miss the variability that exists *between* clusters, leading to a higher sampling variance compared to SRS for the same number of elements sampled. Cluster sampling is typically less efficient than SRS when within-cluster correlation is high and positive.
• Conversely, if units within clusters were uncorrelated or even negatively correlated (very rare in practice), \(S_w^2\) would be high and \(S_b^2\) low. Clusters would be internally heterogeneous but externally homogeneous. Sampling a few clusters would capture a lot of the population variability, making cluster sampling potentially more efficient than SRS.
• Therefore, high positive within-cluster correlation is generally undesirable for precision in cluster sampling estimates, as it concentrates the variability between clusters rather than within them.