Two-stage Cluster Sampling

Outline

Readings

• Handout on Carmen with mathematical details on variance estimation (OPTIONAL)

Assignments

• Quiz 3 due Thursday 10/9/2025 11:59pm via Carmen
• Group Progress Report due Thursday 10/9/2025 11:59pm via Carmen (only one group member needs to upload this)

Two-Stage Cluster Sampling

• In one-stage cluster sampling, we sample all SSUs within the sampled PSUs

• In two-stage cluster sampling, we only sample some SSUs within the sampled PSUs

• Why might we do this?

Stages of Two-Stage Cluster Sampling

1. Select an SRS of \(n\) PSUs from population of \(N\) PSUs

2. Select an SRS of \(m_i\) SSUs from the \(M_i\) SSUs in PSU \(i\)

• The extra stage of sampling complicates the notation and estimators

  • Two samples:
    • \(\mathcal{S}\) = the sample of PSUs
    • \(\mathcal{S}_i=\) the sample of SSUs from PSU \(i\)

• Note that quite often in large-scale surveys something other than SRS is used at the first stage – often it is probability proportional to size (PPS) sampling

• We will return to this concept next lecture (but it is exactly what it sounds like!)

Example (con’t)

• Result is a total of 65 SSUs (schools)
  • Some sampled PSUs (districts) had only 1 SSU (school)
• API \((y)\) by PSU (sampled SSUs only):

• Number of SSUs per PSU is either 1 or 2 (by design)

• We want to use this 2-stage cluster sample to estimate the overall mean API

Activity 13.1 (Part 1)

Two-Stage Cluster Sampling (Part 1)

Step 1: Cluster-Specific Estimates

Start with cluster-specific estimates:

• These look messy – but are just the SRS formulae applied within each PSU!

• Note that the total, \(\hat{t}_i\), has a “hat” on it (its an estimate) – different from 1-stage sampling

• If we have taken an SRS within PSU, then by properties of SRS, estimates are unbiased:

  • \(E[\bar{y}_i] = \bar{y}_{Ui}\)

  • \(E[\hat{t}_i] = t_i\)

  • \(E[s_i^2] = S_i^2\)

Step 2: Overall Estimates

Then combine cluster-specific estimates to get overall estimates:

• Note that these steps look just like the steps for one-stage cluster samples – but with (additional) “hats” on the pieces involving \(t\) (e.g., \(\hat{t}_i, \bar{\hat{t}}\) )

Steps, Combined

• So to estimate the mean of \(y\) we…

  • calculate the mean of \(y\) for each sampled PSU \((\bar{y}_i)\),

  • multiply this average by the number of SSUs per PSU (\(M_i\)) to get the estimated totals in each PSU \((\hat{t}_i)\)

  • calculate the average of the estimated cluster totals \(\hat{t}_i\) across the sampled PSUs \((\bar{\hat{t}})\),

  • multiply this average by the number of PSUs (\(N\)) to get a grand total \((\hat{t}_{clus})\)

  • and divide by the total number of SSUs \((M_0)\) to get the unbiased estimate \((\hat{\bar{y}}_{clus})\)

  • OR, divide by the estimated total number of SSUs \((\widehat{M}_0)\) to get the ratio estimate \((\hat{\bar{y}}_r)\)

Expectation of the Estimators (con’t)

Example (con’t)

Step 1: Cluster-Specific Estimates: For each of the \(n=\) 38 sampled PSUs (districts):

Step 2: Combine to get Overall Estimates:

• Mean PSU Total = \(\bar{\hat{t}} = \frac{7800 + 7525 + 10222 + \dots}{38} = 4252.013\)

• Overall Estimated Total = \(\hat{t}_{clus}= N\bar{\hat{t}} = 757 \times 4252.013 = {3218774}\)

• Overall Mean, Unbiased Estimator = \(\frac{\hat{t}_{clus}}{M_0} = \frac{{3218773.8}}{6194} = {519.7}\)

• For Ratio Estimator: Estimated population size = \(\widehat{M}_0 = \sum_{i \in \mathcal{S}} \sum_{j \in \mathcal{S}_i} w_{ij}\) = sum of weights (sampled SSUs)

Sampling Weights for Two-Stage Cluster Samples

• For two-stage cluster sampling with SRS at first and second stage:

Sample weight for sampled SSU \(j\) in PSU \(i\):   \(\displaystyle w_{ij} = \frac{1}{\pi_{ij}} = \frac{N}{n} \frac{M_i}{m_i}\)

• Intuition:

  • Each sampled PSU “represents” \(\frac{N}{n}\) PSUs

  • And within a PSU each sampled SSU “represents” \(\frac{M_i}{m_i}\) SSUs

  • So in total a sampled SSU “represents” \(\frac{N}{n} \times \frac{M_i}{m_i}\) SSUs

• Weight is the inverse of the probability of selection:

  \[\begin{align} \pi_{ij} &= P(\text{SSU $j$ of PSU $i$ in sample}) = P(\text{SSU $j$ selected}|\text{PSU $i$ selected}) \times P(\text{PSU $i$ selected})\\ &= \frac{m_i}{M_i} \times \frac{n}{N} = \frac{n}{N}\frac{m_i}{M_i} \end{align}\]

Example (con’t)

• To get the ratio estimator, we need the estimated population size:

  • \(\widehat{M}_0 = \sum_{i \in \mathcal{S}} \sum_{j \in \mathcal{S}_i} w_{ij}\) = sum of weights, sampled SSUs = 4900.579 (from software)¹

• Overall Mean, Ratio Estimator = \(\displaystyle \frac{\hat{t}_{clus}}{\widehat{M}_0} = \frac{3218773.8}{4900.6} = 656.8\)

• We can also write estimates in terms of weights (that’s how software is calculating these!)

  • \(\displaystyle \hat{t}_{clus}= \sum_{i \in \mathcal{S}} \sum_{j \in \mathcal{S}_i} w_{ij} y_{ij}\) = weight \(\times\) y, added up for all sampled SSUs

  • \(\displaystyle \hat{\bar{y}}_r= \frac{\sum_{i \in \mathcal{S}} \sum_{j \in \mathcal{S}_i} w_{ij} y_{ij}}{\sum_{i \in \mathcal{S}} \sum_{j \in \mathcal{S}_i} w_{ij}} = \frac{\text{ weight $\times$ y, added up for all sampled SSUs}}{\text{sum of weights for all sampled SSUs}}\)

Activity 13.1 (Part 2)

Two-Stage Cluster Sampling (Part 2)

Why We Focus on Estimating the Total

• We will spend a bit of time calculating the variance of the estimated total (and how to estimate it)

• But – our main interest is often not in the totals!

  • Possible exception: estimating the number of people with a condition

• So why do we focus on the total?

Variance Estimation for the Estimated Total

• As it turns out, \(V(\hat{t}_{clus})\) under two-stage cluster sampling is equal to:

  • the variance under one-stage cluster sampling, plus

  • an extra term to account for the fact that we are estimating the \(\hat{t}_i\)
    (as opposed to directly observing the true \(t_i\) in one-stage cluster sampling)

• True (theoretical) variance of the estimated total¹: \[V(\hat{t}_{clus}) = \underbrace{N^2 \left(1-\frac{n}{N}\right)\frac{S^2_t}{n}}_{V_{stage1}} + \underbrace{\frac{N}{n} \sum_{i=1}^N M_i^2 \left(1-\frac{m_i}{M_i}\right) \frac{S_i^2}{m_i}}_{V_{stage2}}\]

  \(S^2_t\) = true (population) variance of the PSU totals, \(t_i\)
  \(S_i^2\) = true variance of the \(y_{ij}\) within PSU \(i\)

• Note that both these pieces look like SRS formulae – we take an SRS at each stage.

Variance Estimation for the Estimated Total (con’t)

Example (con’t)

Estimator of the Total

• Analysis including information about both stages of sampling

des <- svydesign(id = ~dnum + snum, weight = ~wt, fpc = ~N + Mi, data=samp)
svytotal(~api00, design = des)

        total     SE
api00 3218774 612674

• Analysis ignoring second stage of sampling (“ultimate PSU analysis”)

des_ign <- svydesign(id = ~dnum, weight = ~wt, fpc = ~N, data=samp)
svytotal(~api00, design = des_ign)

        total     SE
api00 3218774 612593

• Point estimates identical

• Standard error estimates very similar

  • Most variability is from sampling PSUs (not sampling SSUs within PSUs)

Variance Estimation for the Ratio Estimator

• For the unbiased estimator, since \(M_0\) is a constant, we have \(V(\hat{\bar{y}}_{clus}) = \frac{1}{M_0^2} V(\hat{t}_{clus})\)

• But we know from 1-stage cluster sampling that we can do better (smaller variance)

• We usually use the ratio estimator of the mean for 2-stage cluster sampling: \(\displaystyle \hat{\bar{y}}_r= \frac{\hat{t}_{clus}}{\widehat{M}_0}\)

  • Same as one-stage cluster sampling with unequal cluster sizes

  • Linearization is again used to calculate the theoretical variance, \(V(\hat{\bar{y}}_r)\)

    • We can use the same general result from last time (1-stage cluster sampling): \[V(\hat{\bar{y}}_r) \approx \frac{1}{M_0^2}\left[V(\hat{t}_{clus}) + \frac{t_U^2}{M_0^2} V(\widehat{M}_0) - 2\frac{t_U}{M_0} \text{Cov}(\hat{t}_{clus},\widehat{M}_0) \right]\]

    • We just now have a different expression for \(V(\hat{t}_{clus})\) because of the 2-stage sampling

  • As we just discussed, often \(V(\hat{t}_{clus})\) is approximated by only the stage 1 variance

Variance Estimation for the Ratio Estimator (con’t)

• Variance Estimator: \[\widehat{V}(\hat{\bar{y}}_r) = \underbrace{\frac{1}{n \hat{\bar{M}}^2} \left(1-\frac{n}{N}\right)\frac{\sum_{i \in \mathcal{S}} (\hat{t}_i - M_i\hat{\bar{y}}_r)^2}{n-1}}_{\widehat{V}_{stage1}} + \underbrace{\frac{1}{nN\hat{\bar{M}}^2} \sum_{i \in \mathcal{S}}M_i^2 \left(1-\frac{m_i}{M_i}\right) \frac{s_i^2}{m_i} }_{\widehat{V}_{stage2}}\]

• The first piece, \(\widehat{V}_{stage1}\), is almost the same as 1-stage cluster sampling

  • Only difference is the estimated PSU total \(\hat{t}_i\) in place of true PSU total \(t_i\)

  • Uses the variability of the estimated PSU totals \((\hat{t}_i)\) around a PSU-specific estimated total \((M_i\hat{\bar{y}}_r)\), instead of the (estimated) overall average \((\bar{\hat{t}})\) – thus has lower variance than unbiased estimator

• As with the variance of \(\hat{t}_{clus}\), the second term \(\widehat{V}_{stage2}\) is usually very small compared to the first term

  • Thus, we often only calculate the first term, \(\widehat{V}_{stage1}\) (“ultimate PSU analysis”)

Example (con’t)

Ratio Estimator of the Mean

• Analysis including information about both stages of sampling

des <- svydesign(id = ~dnum + snum, weight = ~wt, fpc = ~N + Mi, data=samp)
svymean(~api00, design = des)

        mean     SE
api00 656.82 32.651

• Analysis ignoring second stage of sampling (“ultimate PSU analysis”)

des_ign <- svydesign(id = ~dnum, weight = ~wt, fpc = ~N, data=samp)
svymean(~api00, design = des_ign)

        mean     SE
api00 656.82 32.587

• Point estimates identical

• Standard error estimates very similar

  • Most variability is from sampling PSUs (not sampling SSUs within PSUs) (Look back at plot of API)

Example (con’t)

Unbiased Estimator of the Mean

• With Second Stage Variance:

  • \(\widehat{V}(\hat{\bar{y}}_{clus}) = \frac{1}{M_0} \widehat{V}(\hat{t}_{clus}) = \frac{1}{6194} (612674^2) = 60{,}602{,}104\)

  • \(\widehat{SE}(\hat{\bar{y}}_{clus}) = \sqrt{\widehat{V}(\hat{\bar{y}}_{clus})} = \sqrt{60{,}602{,}104} = {7{,}785}\)

  • Compare to \(\widehat{SE}(\hat{\bar{y}}_r) = {32.65}\)

• Without Second Stage Variance:

  • \(\widehat{V}(\hat{\bar{y}}_{clus}) = \frac{1}{M_0} \widehat{V}(\hat{t}_{clus}) = \frac{1}{6194} (612593^2) = 60{,}586{,}081\)

  • \(\widehat{SE}(\hat{\bar{y}}_{clus}) = \sqrt{\widehat{V}(\hat{\bar{y}}_{clus})} = \sqrt{60{,}586{,}081} = {7{,}784}\)

  • Compare to \(\widehat{SE}(\hat{\bar{y}}_r) = {32.59}\)

• Ratio estimator has much smaller SEs!

• SE is over 200 times larger for unbiased estimator!!

Activity 13.1 (Part 3)

Two-Stage Cluster Sampling (Part 3)

A Note on Notation for Variance of Ratio Estimator

• We have the variance estimator for the ratio estimator for 2-stage cluster sampling: \[\widehat{V}(\hat{\bar{y}}_r) = \frac{1}{n \hat{\bar{M}}^2} \left(1-\frac{n}{N}\right)\frac{\sum_{i \in \mathcal{S}} (\hat{t}_i - M_i\hat{\bar{y}}_r)^2}{n-1} + \frac{1}{nN\hat{\bar{M}}^2} \sum_{i \in \mathcal{S}}M_i^2 \left(1-\frac{m_i}{M_i}\right) \frac{s_i^2}{m_i}\]

• We can also write that first stage piece as:

\[\widehat{V}_{stage1}(\hat{\bar{y}}_r) = \frac{1}{\hat{\bar{M}}^2} \left(1-\frac{n}{N}\right)\frac{{\sum_{i \in \mathcal{S}} (\hat{t}_i - M_i\hat{\bar{y}}_r)^2}}{n{(n-1)}} = \frac{1}{\hat{\bar{M}}^2} \left(1-\frac{n}{N}\right)\frac{{s_r^2}}{n}\]

• \({s_r^2} = \frac{1}{n-1} \sum_{i \in \mathcal{S}} (\hat{t}_i - M_i\hat{\bar{y}}_r)^2\) = estimated variance of the residuals

  • Dfference between estimated PSU totals, \(\hat{t}_i\), and predicted total based on the ratio estimator, \(M_i\hat{\bar{y}}_r\)

  • No “model” involved, but a residual nonetheless