Stratified sampling

Can improve on SRS given more information.

Basic idea is to split population into strata:

• divide population into non-overlapping groups (strata)
• SRS each strata
• produce estimate for each strata
• overall estimates weighted average of strata estimates

Advantages:

• protects against getting a really bad sample
• improves precision of estimates
• can use difference sampling methods in each stratum
• separate estimates in each stratum
• some protection against non-response

General theory

Population mean = weighted average of stratum means.

\[ \bar{y} = \sum_l W_l \bar{y}_l \]

Population SE(standard error) = weighted average of statrum SE’s

\[ Var[\bar{y}_st] = \sum_l W_l^2 Var[\bar{y}_l] \]

Stratified sampling does well when $\sigma^2_l$ are small (values in each stratum are similar).

Allocation

\[n_l \propto \frac{W_n \sigma_n}{\sqrt{cost_n}} \]

Post-stratification

• eg. can’t identify stratum unit is from before sampling
• cannot benefit from sample design
• can use correct stratum weight $W_l$
• if sample size large, post-stratification will be close to proportional allocation
• biggest benefit is overcoming some effects of non-response

How many strata?

Goal of stratification is to reduce within stratum variation, so we want more strata, but this comes with increased complexity and cost. Additionally var $\propto \sigma^2_l / n_l$ so decreasing strata sizes will increase variance.

If number of strata, $L$ , chosen, then choose stratum boundaries to minimise $\sum W_l \sigma_l$ (need preliminary smaple of distribution of $y$ ). Approximate rule is to choose $y_h$ so that $W_h(y_h - y_{h-1})$ is constant. With optimal choice of boundaries var $\propto 1/L^2$ . Increasing L too far – within stratum variance dominates.

Conclusion: $L$ often small, < 6.