Blocked and Paired Randomized Experiments
Blocking groups similar units before treatment is assigned. Randomization then occurs within each block, ensuring treatment-control comparisons within important baseline groups.
Stratified or blocked estimator
For stratum $k$, define
\[\hat\tau_k=\bar Y_{1k}-\bar Y_{0k}.\]With $N_k$ units in stratum $k$ and $N=\sum_{k=1}^K N_k$, the sample-average treatment-effect estimator is
\[\boxed{ \hat\tau= \sum_{k=1}^K\frac{N_k}{N}\hat\tau_k }.\]The weights follow the target population. Equal weights would target an average stratum effect and can answer a different question when strata have different sizes.
Paired randomized experiments
A matched-pair design is blocking with two units per block and one treated unit in each pair. For pair $j$, define the treated-minus-control difference
\[D_j=Y_{j,T}-Y_{j,C}.\]For $J$ pairs,
\[\boxed{ \hat\tau=\frac{1}{J}\sum_{j=1}^J D_j }.\]The sample variance of pair differences is
\[s_D^2=\frac{1}{J-1}\sum_{j=1}^J(D_j-\bar D)^2,\]and the standard error is
\[SE(\hat\tau)=\frac{s_D}{\sqrt J}.\]Why blocking can improve precision
When outcomes are similar within blocks, within-block comparisons remove predictable baseline variation. Poorly chosen blocks need not improve precision, but valid within-block randomization still supports unbiased estimation when analyzed according to the design.
Analysis must follow design
- Weight block effects according to the estimand.
- Preserve pair labels in randomization tests and standard errors.
- Do not analyze a paired experiment as if all treated and control observations were independently assigned across the entire sample.
Paired observations versus matched pairs
A paired $t$-test can analyze before-after measurements, naturally paired units, or randomized matched pairs. The algebra is similar, but the causal interpretation depends on how the pairs and treatment assignments were created.