Multistage sampling

As with cluster sampling, we select ${\displaystyle c}$ of ${\displaystyle C}$ clusters, but now instead of sampling all units in each cluster, we take a random sample. Most large surveys carried out this way.

Advantages:

• cost and speed
• convenience (only need list of clusters and individuals in selected clusters)
• usually more accurately than cluster for same total size

Disadvantages:

• less accurate than SRS of same size (but more accurate for same cost)
• further analysis is difficult

Basic results

\[ \hat{\mu}_R = \frac{\sum_C M_i \bar{y}_i}{\sum_C M_i} \]

Let ${\displaystyle f_i=\frac{c}{C}}$ and ${\displaystyle f_{2i}=\frac{m_i}{M_i}}$ , then

\[ \hat{V}(\hat{\mu}_R) = \frac{1-f_1}{c} \sum_sample \frac{(M_i / \bar{M})^2 (\bar{y}_i - \hat{\mu}_R)^2}{c-1} + \frac{f_1}{c} \sum_sample \frac{(M_i / \bar{M})^2 (1-f_{2i})s^2_{2i}}{c m_i} \]

If number of sampled clusters is reasonably large, then ${\displaystyle {\hat{\mu }}_R}$ is approximately normally distributed.

Note: If ${\displaystyle f_1}$ is very small, then ${\displaystyle \hat{V}\left({\hat{\mu }}_R\right)~s_1^2/c}$ . This result holds for more general subsampling schemes than SRS; only need a scheme with unbiased sample mean. Using systematic sampling is common.

Sampling with probability proportional to size

Similar to PPS for cluster sampling, and if ${\displaystyle f_1}$ is small can pretend we are sampling with replacement and treat clusters like individuals.

Performance is similar to SRS subsampling (but need to know ${\displaystyle M_i}$ for every cluster). Is intuitively appealing because if we take same-sized sample from every cluster then every unit has same chance of being selected.

Equal cluster sizes

If ${\displaystyle M_i=M}$ for all clusters then both estimators reduce to mean of cluster means.

If ${\displaystyle m_i=m}$ as well then variance reduces to:

${\displaystyle \hat{V}\left({\hat{\mu }}_R\right)=(1‐f_1)\frac{s_1^2}{c}+f_1(1‐f_2)\frac{s_2^2}{cm}}$ , where ${\displaystyle s_1^2=\sum _c\frac{{({\overline{y}}_i‐\overline{y})}^2}{c‐1}}$ , and ${\displaystyle s_2^2=\sum _c\frac{{(y_{i}j‐{\overline{y}}_{i.})}^2}{m‐1}}$ .

Can obtain these results from a standard one-way ANOVA where ${\displaystyle s_1^2=s_B^2/m}$ and ${\displaystyle s_2^2=s_w^2}$

Estimating proportions

As with cluster sampling, formulae don’t simplify much. See formula sheet for details.

Optimal sub-sample sizes

For simplicity, we’ll only deal with equal cluster and sample sizes, when all estimators reduce to ${\displaystyle \overline{y}}$ . Suppose cost = ${\displaystyle k_{1}c\times k_{2}cm}$ . Variance of ${\displaystyle \overline{y}=(1‐f_1)\frac{{\sigma }^2\_1}{c}+(1‐f_2)\frac{{\sigma }^2\_2}{cm}}$ . Minimised when ${\displaystyle m=\sqrt{\frac{k_1}{k_2}}\le ft\left(\frac{{\sigma }_2}{{\sigma }_u}right\right)}$ .

Stratified multistage sampling

In most large surveys first-stage sample will be stratified. Introduces no new problems, use results results above to estimate mean and se for each clutser, then weighted average to get overall results.