Questions: Mixed-Factorial Designs: Between and Within Factors

Question 1

A clinical trial assigns participants to treatment or placebo (between-subjects) and measures them at baseline, 6 weeks, and 12 weeks (within-subjects). The treatment group improves by 15 points from baseline to 12 weeks; the placebo group improves by 14 points. The Group × Time interaction is non-significant. What can you conclude?

• A. The treatment was effective because the treatment group improved by more than the placebo group
• B. The treatment showed no differential efficacy — both groups changed over time by nearly the same amount, with no significant difference in trajectory
• C. The design is flawed because both groups should not improve if the treatment is working
• D. No conclusion is possible without knowing whether the main effect of time was significant

Answer: (B)

The scientifically central question in a treatment × time design is whether the trajectory of change over time differs between groups — this is the Group × Time interaction. A non-significant interaction means both groups changed by approximately the same amount. The 1-point difference between 15 and 14 is not a differential treatment trajectory; it is captured by the group main effect and likely reflects noise. A significant interaction — not a significant group main effect — is the signature of differential treatment efficacy.

Question 2

In a mixed-factorial design, why is the error term for testing the within-subjects factor smaller than for the between-subjects factor?

• A. Because within-subjects effects are measured at more time points, providing more data
• B. Because individual differences (person-level variance) are removed from the within-subjects error term, since each person serves as their own baseline
• C. Because the within-subjects factor always has more levels, spreading variance across more cells
• D. Because researchers choose the within-subjects factor to be the more reliable measurement

Answer: (B)

In within-subjects analysis, each participant contributes measurements at every level of the factor, so person-level variance can be partitioned out of the error term. Two people who both improve 10 points are perfectly consistent even if their baseline scores differ by 30 points — their individual baseline is subtracted out. This partitioning of individual differences is the core statistical efficiency advantage of within-subjects designs. It carries into the within-subjects portion of mixed designs, giving those effects more power than the between-subjects effects.

Question 3

In a mixed-factorial treatment study, a significant main effect of the between-subjects group factor is the primary evidence that the treatment worked.

• T. True
• F. False

Answer: False

A significant between-subjects main effect only tells you that one group scored higher on average across all time points. This could reflect pre-existing group differences rather than treatment-driven change. The Group × Time interaction is the primary evidence of differential treatment efficacy — it tests whether the trajectory of change over time differs between groups. Main effects alone cannot distinguish 'the treatment group was already different' from 'the treatment group changed differently over time.'

Question 4

In a treatment × time mixed design, non-parallel lines when plotting group means across time points indicate a Group × Time interaction.

• T. True
• F. False

Answer: True

Plotting time on the x-axis with separate lines per group is the standard visualization of a mixed design. Parallel lines mean each group changed by the same amount across time — no interaction. Non-parallel lines (different slopes, or crossing lines) indicate the groups differed in their rate or direction of change over time — a Group × Time interaction. This graphical interpretation is one of the most direct ways to communicate whether a differential treatment trajectory exists.

Question 5

What does a Group × Time interaction actually mean in a treatment study, and why is it more informative than the main effects alone?

Model answer: A Group × Time interaction means the effect of time is different depending on which group you are in — equivalently, the effect of group differs depending on when you measure it. In a treatment study, this is the signature that one group's scores changed over time in a pattern the other group did not show. Main effects in isolation cannot capture this: the main effect of group asks whether one group scored higher overall; the main effect of time asks whether scores changed overall. Neither tells you whether the treatment produced a distinctive trajectory of change compared to control.

This is why the interaction is typically reported as the primary finding in treatment research. A treatment that causes both groups to improve equally produces significant main effects of time and possibly group, but a non-significant interaction — meaning no evidence the treatment specifically drove differential change. The interaction directly tests the causal claim.