The Central Idea
Random assignment prevents systematic treatment selection, but chance can still place more high-risk, high-performing, older, larger, or otherwise different units in one treatment group. If a pre-treatment characteristic is strongly related to the outcome, researchers can account for it in the design.
In blocking, experimental units are grouped by a relevant pre-treatment characteristic, and treatments are randomly assigned within each group. In stratification, a population or sample is divided into meaningful subgroups, and sampling or analysis proceeds within those subgroups.
\[ \boxed{ \text{group similar units} \;\longrightarrow\; \text{compare or sample within groups} \;\longrightarrow\; \text{combine the evidence} } \]Both methods separate variation that is already understood from the variation used to evaluate the main question.
Blocks in Experiments
A block is a group of experimental units expected to respond similarly because they share an important pre-treatment characteristic. Units should be relatively similar within a block and meaningfully different across blocks.
Examples include grouping:
- students by prior achievement level;
- patients by disease severity;
- agricultural plots by soil type or field location;
- machines by model or production line;
- stores by region or sales volume; and
- laboratory runs by day or batch.
Treatment assignment then occurs separately within each block. The randomization is preserved, but it is constrained to maintain useful balance.
The Blocking Procedure
A randomized block design follows a clear sequence:
- Identify a pre-treatment characteristic strongly related to the outcome.
- Group experimental units into blocks using that characteristic.
- Randomly assign treatments within every block.
- Measure outcomes using the same procedures across treatments.
- Compare treatments within blocks.
- Combine the within-block comparisons appropriately.
The defining feature is not merely placing units into categories. The design must use those categories in the treatment assignment and analysis.
Example: Blocking by Prior Achievement
A school compares two algebra review programs. Students are classified into three blocks using a pretest:
- developing preparation;
- established preparation; and
- advanced preparation.
Within each block, half of the students are randomly assigned to program (A) and half to program (B).
\[ \begin{array}{c|cc} \text{block} & \text{program }A & \text{program }B\\ \hline \text{developing} & \text{random half} & \text{random half}\\ \text{established} & \text{random half} & \text{random half}\\ \text{advanced} & \text{random half} & \text{random half} \end{array} \]Each treatment receives students from every preparation level. The treatment comparison is made among students with similar starting points rather than allowing pretest differences to dominate the outcome variation.
Why Blocking Can Improve Precision
Suppose prior preparation strongly predicts the final score. In a completely randomized design, the treatment-group variation contains both treatment differences and large preparation differences.
Blocking isolates preparation-level variation. The treatment comparison is then based on differences within preparation blocks:
\[ d_b = \overline{y}_{A,b} - \overline{y}_{B,b}, \]where (d_b) is the treatment difference in block (b). The overall effect can be estimated by combining the block differences using appropriate weights:
\[ \widehat{\tau} = \sum_{b=1}^{B} w_b d_b, \qquad \sum_{b=1}^{B}w_b=1. \]When blocks are internally similar and the blocking variable predicts the outcome, the within-block comparisons are less noisy than an unstructured overall comparison.
Separating Treatment and Block Effects
A simple additive model for outcome (Y_{ij}), measured under treatment (i) in block (j), is
\[ Y_{ij} = \mu+\tau_i+\beta_j+\varepsilon_{ij}, \]where
- (\mu) is an overall baseline;
- (\tau_i) is the treatment contribution;
- (\beta_j) is the block contribution; and
- (\varepsilon_{ij}) is remaining unexplained variation.
This model expresses the design's central purpose: explain predictable block differences separately so they do not obscure the treatment comparison. More advanced models can also allow treatment effects to vary across blocks.
Complete and Incomplete Block Designs
A complete block design places every treatment within every block. If an experiment has four treatments, each block contains units assigned to all four.
An incomplete block design places only some treatments within each block. Incomplete blocks may be necessary when a block cannot accommodate every treatment, such as when one participant can compare only a few products or one laboratory batch has limited capacity.
Incomplete designs require careful planning so treatment comparisons remain connected across blocks. They are not created by casually omitting inconvenient treatment combinations.
Matched Pairs as Blocks of Two
A matched-pairs design forms pairs of similar units and randomly assigns the two treatments within each pair. Each pair is a block containing two units.
If (Y_{A,j}) and (Y_{B,j}) are outcomes for the two units in pair (j), the pair difference is
\[ d_j=Y_{A,j}-Y_{B,j}. \]The treatment comparison is based on the collection of pair differences. Matching is effective when the paired units are similar on characteristics that strongly predict the outcome.
A participant can also serve as a block when receiving both treatments in a randomized order, provided carryover and time effects are addressed. That repeated-treatment structure is different from merely taking repeated measurements under one treatment.
Blocks Are Not Treatments
A treatment is a condition deliberately imposed to answer the main causal question. A block is a preexisting or design-based grouping used to improve comparison.
In the student example, review program is the treatment and preparation level is the blocking variable. Researchers do not interpret preparation level as though it were randomly assigned.
Blocks can be scientifically interesting, but their primary role in a blocked design is to organize variation and treatment assignment.
Blocks and Clusters Are Different
| Feature | Block | Cluster |
|---|---|---|
| Purpose | Group similar units for comparison | Represent an intact group that is sampled or assigned together |
| Treatment pattern | Usually contains multiple treatments | Often receives one treatment as a whole |
| Example | A soil zone containing plots from both fertilizer groups | A classroom assigned entirely to one curriculum |
Blocking creates within-group treatment comparisons. Cluster assignment creates treatment groups from intact units and must be analyzed at the cluster-aware level.
Strata in Sampling
A stratum is a subgroup of a population whose members share a characteristic relevant to sampling or analysis. In stratified random sampling, the population is divided into nonoverlapping strata, and a random sample is drawn separately from each.
If a university wants representation from every academic level, it might form four strata:
\[ \text{first-year},\quad \text{sophomore},\quad \text{junior},\quad \text{senior}. \]Sampling within each stratum guarantees that all four levels enter the sample. A simple random sample of the whole student body might include too few students from a smaller level merely by chance.
The Stratified Sampling Procedure
- Define the target population.
- Choose a characteristic known for every population member.
- Divide the population into nonoverlapping strata covering the population.
- Select a probability sample independently within each stratum.
- Calculate stratum-level estimates.
- Combine them using weights that represent the target population.
If stratum (h) contains (N_h) members and the whole population contains (N), the population weight is
\[ W_h=\frac{N_h}{N}. \]An overall mean can then be estimated by
\[ \overline{y}_{\mathrm{st}} = \sum_{h=1}^{H}W_h\overline{y}_h. \]Proportional and Disproportionate Allocation
In proportional allocation, each stratum receives a sample size proportional to its population size:
\[ n_h = n\frac{N_h}{N}. \]If a population is (60\%) urban and (40\%) rural, a proportional sample of (500) contains approximately
\[ 0.60(500)=300\quad\text{urban members} \]and
\[ 0.40(500)=200\quad\text{rural members}. \]In disproportionate allocation, smaller or especially important strata may be sampled at higher rates. This can improve subgroup estimates, but population-level results must use appropriate weights.
Stratification in Analysis
Researchers may also stratify data during analysis. For example, an observational study comparing an exposure and an outcome might estimate the association separately within age groups.
Within-stratum comparisons can reduce confounding when units in the same stratum are more comparable. However, analysis-stage stratification is not equivalent to random assignment. It can address only the variables measured and represented adequately in the strata.
Very coarse strata may leave important differences within groups. Very narrow strata may contain too few observations for stable comparisons.
Blocking and Stratification: Related but Context-Specific
| Context | Usual Term | Action Within Groups | Primary Goal |
|---|---|---|---|
| Experiment | Blocking | Randomly assign treatments | Improve balance and treatment precision |
| Probability sample | Stratification | Randomly sample population members | Ensure representation and improve estimation |
| Observational analysis | Stratification | Compare exposure groups | Control measured differences or describe heterogeneity |
Some fields use the words more flexibly. The operational question is more important than the label: were treatments assigned, population members sampled, or existing observations compared within the groups?
Choosing a Blocking or Stratification Variable
A useful grouping variable should be:
- measured before treatment or exposure;
- strongly related to the outcome or sampling objective;
- defined consistently for all relevant units;
- available before randomization or sampling;
- divisible into groups large enough for the intended comparisons; and
- substantively meaningful rather than chosen only after seeing results.
Blocking on a variable unrelated to the outcome may provide little benefit. Creating many unnecessary blocks can complicate assignment and leave too few units in each treatment-by-block combination.
Use Pre-Treatment Information
A blocking variable must be known before treatment assignment and must not be caused by the treatment. Grouping units by a post-treatment measurement can distort the comparison because treatment may influence which group a unit enters.
For example, a study of a training program may block on a baseline skill test. It should not define blocks using motivation measured after the program if the program itself may change motivation.
The safe ordering is
\[ \text{measure baseline} \;\longrightarrow\; \text{form blocks} \;\longrightarrow\; \text{randomize treatments} \;\longrightarrow\; \text{measure outcomes}. \]When Treatment Effects Differ Across Blocks
Blocking may reveal that treatment effects are heterogeneous. Suppose a new tutoring method improves outcomes in the developing-preparation block but has little effect in the advanced-preparation block.
That pattern is a treatment-by-block interaction: the treatment difference changes across blocks. An overall average remains useful, but it can hide meaningful differences in response.
Subgroup comparisons require enough replication within each block. A design created only to control variation may not have enough units to estimate many block-specific effects precisely.
The Analysis Must Respect the Design
Blocking changes how randomization occurs, so the analysis should preserve the within-block comparisons. Ignoring blocks can discard the precision gained by the design and may use an inappropriate reference for chance variation.
Likewise, disproportionate stratified samples require weights when estimating population totals, proportions, or means. Treating an oversampled small stratum as though it represented the same population share as a large stratum can bias the combined estimate.
Design and analysis form one system. A carefully structured design needs a correspondingly structured analysis.
Example: Blocking Across a Field
A field has a moisture gradient from north to south. A researcher wants to compare three seed varieties using 24 plots.
Form the Blocks
Divide the field into eight narrow east-west blocks so plots within each block have similar moisture conditions. Each block contains three plots.
Randomize Within Blocks
Randomly assign the three varieties to the three plots in every block. Each variety appears once per block.
Compare Within Blocks
Differences among varieties are evaluated within the same moisture region. The north-south gradient is represented by the blocks rather than being mixed into the treatment comparison.
Why Not Assign by Region?
Giving variety (A) only to northern plots and variety (B) only to southern plots would confound variety with moisture. Blocking places every variety throughout the field.
Example: A Disproportionate Stratified Sample
A workforce contains 900 day-shift employees and 100 night-shift employees. A researcher samples 100 from each shift to compare workplace experiences.
The sample is intentionally disproportionate. Night-shift employees are (10\%) of the population but (50\%) of the sample. The equal stratum sample sizes provide a stronger night-versus-day comparison.
For an overall workforce estimate, the stratum weights should reflect the population shares:
\[ W_{\text{day}}=\frac{900}{1000}=0.9, \qquad W_{\text{night}}=\frac{100}{1000}=0.1. \]If the sample means are (72) for day-shift employees and (60) for night-shift employees, the weighted workforce estimate is
\[ 0.9(72)+0.1(60)=70.8. \]The unweighted mean of (66) would incorrectly treat the two shifts as equal shares of the workforce.
When Blocking or Stratification Helps Less
Grouping is not automatically beneficial. It may add little or create problems when:
- the grouping variable is weakly related to the outcome;
- measurements used to form groups are unreliable;
- blocks or strata are defined after outcomes are examined;
- too many groups leave sparse treatment cells;
- important units cannot be classified consistently;
- the analysis ignores the design or uses incorrect weights; or
- the chosen categories conceal important variation within groups.
The objective is not to create the maximum number of categories. It is to use a small number of scientifically relevant distinctions that materially improve the design.
A Blocking and Stratification Checklist
- State the purpose. Experimental balance, sampling representation, or observational adjustment?
- Choose a baseline variable. Is it known before treatment and related to the outcome?
- Define the groups. Are they nonoverlapping, meaningful, and sufficiently large?
- Preserve chance. Will treatments or sample members be randomized within groups?
- Ensure coverage. Does every treatment or relevant population segment appear as intended?
- Plan replication. Are there enough units for within-group comparison?
- Choose weights. How will group estimates be combined?
- Match the analysis. Will the model or estimator reflect the grouping design?
- Check interpretation. Are group-specific and overall conclusions distinguished?
Common Errors
Blocking After Treatment
Blocks should be formed from pre-treatment information. A variable influenced by treatment cannot safely define the original randomization structure.
Putting One Treatment in Each Block
Blocks usually contain multiple treatments so comparisons occur within blocks. Assigning one treatment exclusively to each location confounds treatment with location.
Confusing Blocks with Clusters
Blocks contain units from several treatment groups. Clusters are intact groups often assigned or sampled as wholes.
Ignoring Weights After Oversampling
Disproportionate stratified sampling can improve subgroup information, but unweighted population estimates may overrepresent the oversampled strata.
Creating Too Many Groups
Extremely narrow groups may contain too few units for valid randomization or stable estimation. Similarity must be balanced against usable group size.
Guided Practice
Problem 1
A clinical experiment groups participants by baseline disease severity and randomly assigns two treatments within each severity group. Is severity a treatment, block, or outcome?
Show the solution
Baseline severity is a blocking variable. It is measured before treatment and used to organize the random assignment. The two assigned conditions are the treatments, and the later health measurement is the outcome.
Problem 2
A study assigns curriculum (A) to one school district and curriculum (B) to another, then calls the districts blocks. Why is this not an effective blocked comparison?
Show the solution
Each district contains only one treatment, so curriculum is confounded with district. A blocked design would include both curricula within each district, with schools or classrooms randomly assigned inside the districts when feasible.
Problem 3
A population contains 2,000 members in stratum (1) and 3,000 in stratum (2). What are the population weights?
Show the solution
The total is (5{,}000). Therefore, (W_1=2{,}000/5{,}000=0.4) and (W_2=3{,}000/5{,}000=0.6).
Problem 4
A researcher measures stress after an intervention and then forms “low-stress” and “high-stress” blocks for the treatment analysis. What is the design problem?
Show the solution
Post-treatment stress may be affected by the intervention. Grouping on it can distort the treatment comparison. Blocks should be defined from baseline information before random assignment.
Problem 5
Within two blocks, the estimated treatment differences are (d_1=4) and (d_2=10). If the appropriate weights are (w_1=0.75) and (w_2=0.25), compute the combined estimate.
Show the solution
The combined estimate is (0.75(4)+0.25(10)=3+2.5=5.5).
A Final Grouping Check
Before using a group in the design, ask:
\[ \boxed{ \text{Does this preexisting difference predict the outcome strongly enough to organize the comparison?} } \]If yes, group first and preserve chance within the groups. If no, the added complexity may not improve the study.
Summary
- Blocking groups similar experimental units and randomizes treatments within each group.
- Stratification divides a population or dataset into meaningful subgroups for sampling or analysis.
- Effective grouping variables are pre-treatment, reliably measured, and strongly related to the outcome or sampling objective.
- Treatment effects are compared within blocks and then combined using appropriate weights.
- Matched pairs are blocks of two; blocks are distinct from clusters.
- Disproportionate stratified samples require population weights for overall estimation.
- The analysis must respect the grouping, assignment, and sampling structure established by the design.
Next Step
Blocking improves the assignment structure, but expectations and knowledge of treatment can still influence behavior or measurement. The next lesson develops blinding—the separation of treatment identity from participants, providers, and outcome assessors when the research setting allows it.