BB2025 Chap.6 Factorial, Repeated and Multivariate Models
Factorial, Repeated and Multivariate Models
Read interactions before averages
A factorial model estimates main effects for each factor and an interaction asking whether one effect changes across levels of another. When the interaction is meaningful, an averaged main effect can describe no actual group well.
Translate the interaction into simple contrasts and report their estimates and intervals rather than narrating a significance table.
Preserve repeated observations
Measurements from one participant are correlated. Repeated-measures ANOVA separates within-person change from between-person variability and introduces assumptions about covariance, including sphericity in traditional forms.
Corrected degrees of freedom can adjust inference when that condition fails. Mixed models can represent unequal spacing, individual trajectories and incomplete visits, but still require an appropriate time function and missing-data assumption.
Count outcomes before predictors
ANCOVA models one outcome with covariates.
MANOVA analyses several outcomes jointly, and MANCOVA adds covariates to that multivariate comparison. A significant multivariate result says outcome vectors differ and must be followed by planned analyses that show which outcomes and contrasts carry the pattern. Adjustment should follow the conceptual model because controlling a mediator or collider can distort interpretation.
What this chapter covers
- 01
Main effects and interaction terms
- 02
Simple effects and planned contrasts
- 03
Difference-in-change interpretation
- 04
Within-person covariance
- 05
Sphericity and corrected inference
- 06
ANCOVA baseline adjustment
- 07
MANOVA outcome vectors
- 08
MANCOVA and covariate choice
Worked example · free
Recover an interaction from four means
- 3Calculate change within each group.
- 2Subtract the two changes.
- 3Interpret the interaction and design condition.
Key terms
- Main Effect
- An average factor contrast across levels of other factors included in a model.
- Interaction Effect
- A contrast showing that one factor's association changes across levels of another factor.
- Simple Effect
- The effect of one factor evaluated at a particular level of another factor.
- Repeated Measures
- Multiple linked observations collected from the same experimental or observational unit.
- ANCOVA
- A model comparing groups on one outcome while accounting for selected covariates.
- MANOVA
- A multivariate model testing group differences across a joint vector of continuous outcomes.
Factorial, Repeated and Multivariate Models FAQ
What does a significant interaction mean?
It means the effect or association of one factor varies across levels of another under the fitted model. It does not prove that every simple contrast is non-zero. Show the cell means or profile plot, estimate the planned simple effects with intervals and avoid presenting an overall main effect as the whole conclusion.
Why are repeated observations not independent?
Measurements from one participant share stable characteristics and a common history, so their errors tend to be related. Treating visits as unrelated exaggerates information and discards within-person structure. The analysis should represent that covariance through a repeated-measures or mixed approach chosen for the design and observation schedule.
How do MANOVA and MANCOVA differ?
Both analyse several continuous outcomes jointly. MANCOVA additionally adjusts the outcome-vector comparison for selected covariates. ANCOVA has one outcome, which is the fastest distinction. After a significant multivariate result, examine pre-specified outcome contrasts and uncertainty because the global test does not identify which variables drive the difference.
Exam move
Draw a small table of cell means for every factorial scenario and compute the key comparison of comparisons before reading output. For repeated measures, write who supplies each visit and which time contrast represents the clinical question. Build a separate card for ANCOVA, repeated ANOVA, MANOVA and MANCOVA using number of outcomes, dependence and covariates.
Always state a planned contrast in words so a significant omnibus term cannot replace the actual research question. Sketch parallel and non-parallel profile plots from invented means, then describe the interaction without using a p value. For covariate problems, return to the conceptual model and label each candidate as pre-exposure confounder, precision variable, mediator or collider.
That classification should determine adjustment before software output is visible. Change one cell mean at a time and predict which main effect or interaction moves. This isolates the meaning of each model term more effectively than memorising an output table. Practise reading each output twice: first at the population-averaged level and then within the relevant subgroup or visit.
If an interaction is present, explicitly limit the main-effect summary. The qualification is part of the answer, not an optional caveat.
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