Derivation of DEFF for One-Stage Cluster Sampling

PUBHBIO 7225

Assume:

• Sample \(n\) of \(N\) PSUs

• Equal size clusters, i.e., # SSUs per cluster = \(M_i = M\)

• One-stage cluster sampling, so if PSU selected, all \(M\) SSUs sampled

• Total # of SSUs in population = \(M_0=NM\)

• \(S_t^2\) = variability of PSU totals \(t_i\) about the mean PSU total \((\bar{t}_U)\)

• \(S^2\) = variability of SSUs \(y_{ij}\) about the overall mean \(\bar{y}_U\)

Goal is to estimate:

\(\displaystyle \text{\textbf{Design Effect}} = \frac{\text{variance of statistic under 1-stage cluster sampling}}{\text{variance of statistic under SRS of same \# SSUs}}\)

Variance of the estimated mean from a cluster sample of \(n\) PSUs: \[\begin{aligned} V_{clus}(\hat{\bar{y}}_{clus}) &{}= \frac{N^2}{M_0^2} \left(1-\frac{n}{N}\right)\frac{S_t^2}{n} &{} \text{(Lecture 11 slide 12)} \\ &{}=\frac{N^2}{(NM)^2} \left(1-\frac{n}{N}\right)\frac{S_t^2}{n} &{} \text{(Substitute $M_0=NM$)}\\ &{}= \frac{1}{M^2} \left(1-\frac{n}{N}\right)\frac{S_t^2}{n} &{} \text{(Cancelling terms)} \end{aligned}\] We can write \(S_t\) in terms of sums of squares as: \[\begin{aligned} S_t^2 &{}= \frac{1}{N-1} \sum_{i=1}^N (t_i - \bar{t}_U)^2 = \frac{1}{N-1} \sum_{i=1}^N (M \bar{y}_{iU} - M \bar{y}_U)^2 = \frac{1}{N-1} M^2 \sum_{i=1}^N (\bar{y}_{iU} - \bar{y}_U)^2 = M \frac{SSB}{N-1} %= M \times MSB \end{aligned}\] Plus this in to the variance of the estimated mean: \[\begin{aligned} V_{clus}(\hat{\bar{y}}_{clus}) &{}= \frac{1}{M^2} \left(1-\frac{n}{N}\right)\frac{S_t^2}{n} \\ &{}=\frac{1}{nM^2} \left(1-\frac{n}{N}\right)S_t^2 &{} \text{(Rearrange terms)}\\ &{}=\frac{1}{nM^2} \left(1-\frac{n}{N}\right)M \frac{SSB}{N-1} &{} \text{(Substitute in $S_t^2 = M \frac{SSB}{N-1}$)}\\ &{}=\frac{1}{nM} \left(1-\frac{n}{N}\right)\frac{SSB}{N-1} &{} \text{(Cancelling terms)} \end{aligned}\]

The cluster sample has \(n\) PSUs, each with \(M\) SSUs \(\rightarrow\) nM total SSUs

Variance of the estimated mean from an SRS of \(nM\) SSUs (out of population of \(NM\) SSUs): \[\begin{aligned} V_{srs}(\bar{y}) &{}= \left(1 - \frac{nM}{NM}\right) \frac{S^2}{nM} &{} \text{(standard SRS formula sampling $nM$ from $NM$)}\\ &{}= \frac{1}{nM} \left(1-\frac{n}{N}\right)S^2 &{} \text{(Cancelling terms/rearranging)}\\ &{}= \frac{1}{nM} \left(1-\frac{n}{N}\right)\frac{SST}{NM-1} &{} \text{(Substitute in $S^2 = \frac{SST}{NM-1}$)}\\ \end{aligned}\]

And finally, we are going to need the following rearranging of \(ICC\) in terms of \(SSB\) and \(SST\): \[\begin{aligned} ICC &{}= 1 - \frac{M}{M-1} \frac{SSW}{SST}\\ (M-1) ICC &{}= (M-1) - M \frac{SSW}{SST}\\ (M-1) ICC &{}= M-1 - M \left(1-\frac{SSB}{SST}\right)\\ (M-1) ICC &{}= M-1 - M +M \frac{SSB}{SST}\\ (M-1) ICC &{}= M \frac{SSB}{SST}-1\\ 1+(M-1) ICC &{}= M \frac{SSB}{SST}\\ \frac{1}{M} [1+(M-1) ICC] &{}= \frac{SSB}{SST} \end{aligned}\]

Thus, the design effect (DEFF) for the estimated overall mean from a one-stage cluster sample is: \[\begin{aligned} \textcolor{red}{\text{deff}(\hat{\bar{y}}_{clus})} &{}= \frac{V_{clus}(\hat{\bar{y}}_{clus})}{V_{srs}(\bar{y})} \\ &{}= \frac{\frac{1}{nM} \left(1-\frac{n}{N}\right)\frac{1}{N-1} \times SSB}{\frac{1}{nM} \left(1-\frac{n}{N}\right)\frac{SST}{NM-1} } &{} \text{(Plug in expressions above)}\\ &{}= \frac{ \frac{1}{N-1} \times SSB}{ \frac{SST}{NM-1} } &{} \text{(Cancelling terms)}\\ &{}= \frac{NM-1}{N-1} \frac{SSB}{SST} &{} \text{(Rearranging)}\\ &{}=\frac{NM-1}{N-1} \frac{1}{M} \left[1+(M-1) ICC \right] &{} \text{(Plugging in for $\frac{SSB}{SST}$)}\\ &{}=\frac{NM-1}{NM-M}[\textcolor{red}{1+(M-1)ICC}] &{} \text{(Rearranging)}\\ &{}\approx \textcolor{red}{1+(M-1)ICC} \quad \text{[if $N$ is large so }NM-1 \approx NM-M] \end{aligned}\]