RESEARCH GUIDE
Analysis of Variance Designs Explained
Factorial, Nested, Blocked and Split-Plot Structures
Analysis of variance designs differ in how factors are combined and how units are assigned to them, and that arrangement fixes which error terms the model uses and which effects can be tested. A model that ignores the arrangement can look fine in the output and still answer the wrong question.
Analysis of variance designs specify the factors being compared, how their levels are combined, which units receive which levels and how often each combination is replicated. The design, and not the software menu, decides the model: crossed factors can interact, nested factors occur within only one level of another factor, blocking removes a known source of variation, and a split-plot design gives different factors different error terms because they are applied to units of different size. OnlineSPSS, a specialist statistical consultancy for academic, clinical and professional research, helps researchers write the model from the design, choose the error terms and word the results, while the experiment and its interpretation remain the researcher’s.
DESIGN STRUCTURES
Types of Analysis of Variance Design and Their Error Structure
Factorial and nested ANOVA designs differ in how the factors relate, while blocked and split-plot layouts differ in how units are grouped, and each arrangement carries its own error terms.
Completely Randomized One factor with units allocated to its levels at random, giving the simplest model with a single error term and the fewest assumptions to defend.
Factorial Two or more crossed factors with every combination of levels, which lets interactions be estimated and uses the same units more efficiently than separate experiments.
Randomized Block Units are grouped into blocks of similar units and treatments are randomized within each block, so differences between blocks are removed from the error.
Nested Levels of one factor occur within only one level of another, such as batches within suppliers, and in a nested ANOVA design the nested factor is often treated as random, so the parent factor is tested against the variation among the nested units.
Split-Plot One factor is applied to larger units and another to smaller units within them, so the model uses separate error terms for the whole-plot and subplot factors.
Repeated-Measures The same units are observed under every level of a factor, so observations are correlated and the model must account for the dependence between measurements.
STRUCTURE COMPARISON
Factorial, Nested and Split-Plot Designs Compared
Three structures that are often confused differ in how the factors relate and in which error term tests each effect, and the table sets the differences side by side.
| Feature | Factorial | Nested | Split-plot |
|---|---|---|---|
| Arrangement of factors | Crossed: every level of A appears with every level of B | Hierarchical: each level of B occurs within one level of A | Crossed, with one factor applied to larger units than the other |
| Interaction between the factors | Estimable | Not defined between a nested factor and its parent | Estimable between whole-plot and subplot factors |
| Error term | One residual when units are fully randomized | The parent factor is tested against variation among the nested units | Separate error terms for whole-plot and subplot effects |
| Unit of replication | Each combination of levels | The nested units | The whole plot for the whole-plot factor |
| Typical fault | Main effects read despite an interaction | Nested units treated as independent observations | A single residual used to test every effect |
KEY PLANNING DECISIONS
Design Decisions That Fix the Model in Advance
The model follows from decisions made while the experiment can still change, and answering these in order prevents most later disputes about the analysis.
Identify the Experimental Unit Name the smallest unit that can receive a treatment independently, since replication is counted in these units and not in the measurements taken on them.
Record the Randomization Write down which units were randomized to which factor, since applying one factor to whole batches, plots or sessions and the other within them creates a split-plot structure whether or not it was intended.
Classify Fixed or Random Treat levels as fixed when they are the only ones of interest and as random when they sample a larger set, since the choice changes the error term.
Consider Blocking Block on a known nuisance source of variation, such as batch, site or day, and randomize within blocks to keep the comparison fair.
Plan Replication and Balance Choose the number of units per cell with precision in mind, and aim for equal cell sizes, as unequal sizes complicate the partition of variability.
Prespecify the Contrasts Write down the comparisons of interest before the data arrive, since planned contrasts say more than an overall test followed by every pairwise comparison.
COMMON ISSUES
How Design Structures Get Misread in the Model
Analysis of variance output looks much the same whatever the design, which is why these mismatches between the experiment and the model often pass unnoticed.
Pseudoreplication Repeated measurements on one tank, plot or patient are treated as independent units, which makes the error too small and the effects look more certain.
The Default Error Term Standard menus assume crossed fixed factors with one residual, so nested and split-plot effects need their error terms specified explicitly.
Unbalanced Cells, Unstated Sums of Squares With unequal cell sizes the type of sums of squares can change which hypothesis is tested, so it is chosen deliberately and reported.
Main Effects Beside an Interaction When an interaction is present the effect of one factor depends on the level of another, so main effects are interpreted with care, often through simple effects.
Unequal Spread Across Cells When cells differ in size and the smaller cells are also the more variable, the F test rejects too often, so spread is compared across cells and a weighted or robust approach is considered.
Interaction Depends on the Scale An interaction can appear or vanish when the outcome is transformed, such as to logarithms, so the scale of analysis is chosen for interpretation before the interaction is read.
REPORTING
Describing the Design Precisely in a Methods Section
A reader can trust an analysis of variance only if the design is stated as clearly as the result, and the description is short when it follows a fixed order.
The methods state each factor with its levels, whether factors are crossed or nested, fixed or random, the experimental unit, how units were allocated and how many were used per cell. When cells are unbalanced, the type of sums of squares is named. The results give the F statistic with both degrees of freedom, an effect size with its interval, and the error term used for each effect in nested and split-plot designs. APA 7 style is followed unless the journal names another, and diagnostic checks are described in terms of what was examined and what it showed.
STATISTICAL METHODS
The Analyses Behind Each Design Structure
Once the structure is clear, the nearest named method shows what the analysis involves, and designs with correlated or clustered data lead toward mixed models.
Mixed ANOVA
Between-group and within-unit factors together
Repeated Measures ANOVA
The same units under several conditions
ANCOVA
Group comparison adjusted for a covariate
MANOVA
Several outcomes compared across groups
Linear Mixed Models (Multilevel Modeling)
Nested units and random effects
Comparative & Hypothesis Testing
Choosing among comparison methods
ANALYSIS SERVICES
Support From the Design Stage to the Final Table
The design is settled first, then the model and its error terms, and support follows that order through to the write-up and a final check.
Research Planning & Study Design
Units, factors, blocking, replication and the sample size that gives usable precision, fixed in the analysis plan before the experiment begins.
Quantitative Data Analysis
The model that matches the design, with correct error terms, diagnostics, planned contrasts and effect estimates with confidence intervals.
Results Reporting & Interpretation
Design descriptions, ANOVA tables and follow-up comparisons prepared to APA 7 or your journal's style, with each effect worded for the design.
Independent Statistical Review
An analysis already run is checked for the experimental unit, the error terms used and the handling of interactions and follow-up tests.
FREQUENTLY ASKED QUESTIONS
Questions About Analysis of Variance Designs
Experiments with several factors raise practical design questions, from telling nested factors from crossed ones to Latin squares, blocks too small for every treatment and interactions with blocks.
How can I tell whether two factors are nested or crossed?
Factors are crossed when every level of one appears with every level of the other, and nested when each level of one occurs within only one level of the other. Hospitals and the patients treated in them are nested, since a patient belongs to one hospital, while drug and dose are crossed when each dose is given with each drug. A table of the level combinations present in the data settles the question directly.
When is a Latin square design useful?
A Latin square is useful when two nuisance sources of variation, such as day and operator, must be blocked at once and each has as many levels as there are treatments. Every treatment appears once in each row and each column, which saves units, but the design assumes that treatments do not interact with the blocking factors and leaves few error degrees of freedom when the square is small, so squares are often replicated.
When is an incomplete block design used?
An incomplete block design is used when each block holds fewer units than there are treatments, such as a session that fits three of five conditions. In a balanced incomplete block design every pair of treatments shares a block equally often, so all pairwise comparisons have the same precision, and treatment means are adjusted for the blocks in which each treatment happened to appear.
Does a randomized block design need a block-by-treatment interaction?
With one unit per treatment in each block, the interaction cannot be separated from the error, so the standard model assumes that treatment effects are the same in every block. If blocks may respond differently, units are replicated within blocks so that the interaction can be estimated, and a large interaction is reported because an average treatment effect may then describe individual blocks poorly.
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