Multistage sampling

As with cluster sampling, we select c of 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:

Disadvantages:

Basic results

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

Let fi=cC and f2i=miMi , 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 μ^R is approximately normally distributed.

Note: If f1 is very small, then V^(μ^R)~s12/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 f1 is small can pretend we are sampling with replacement and treat clusters like individuals.

Performance is similar to SRS subsampling (but need to know Mi 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 Mi=M for all clusters then both estimators reduce to mean of cluster means.

If mi=m as well then variance reduces to:

V^(μ^R)=(1f1)s12c+f1(1f2)s22cm , where s12=c(y¯iy¯)2c1 , and s22=c(yijy¯i.)2m1 .

Can obtain these results from a standard one-way ANOVA where s12=sB2/m and s22=sw2

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 y¯ . Suppose cost = k1c×k2cm . Variance of y¯=(1f1)σ2_1c+(1f2)σ2_2cm . Minimised when m=k1k2ft(σ2σuright) .

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.