This discussion relates to these previous discussions:

Cross-over experiment

  • 3 treatments: rest, relax, activating
  • 2 sequences across 3 periods:
    • rest -> relax -> activating
    • rest -> activating -> relax
  • 20 subjects
  • explanatory varibles:
    • years music education > 5 years (binary)
  • response is impulse reading (continuous)


Consider models that only look at treatments relax and activating

Design-based model (no interaction)

Fixed effects:

  • treatment
  • sequence
  • period

Random effects:

  • subject


Sequence effect is significant.

Design-based model with interaction

Fixed effects:

  • treatment
  • sequence
  • period
  • music education
  • treatment x music education

Random effects:

  • subject


Treatment x sequence interaction is significant.

Modeling with explanatory variable(s)

Fixed effects:

  • treatment
  • sequence
  • period
  • treatment x sequence
  • music education

Random effects:

  • subject




Modeling without explanatory variables

Comparing treatment A to B

Treat the problem has having 5 means:

  • rest in period 0
  • relax in period 1
  • relax in period 2
  • activating in period 1
  • activating in period 2

To assess differences between relax and activating, take contrasts to calculate:

  • the difference between relax and activating in period 1
  • the difference between relax and activating in period 2

Comparing effect of treatment vs control

Due to the effect of sequence, construct contrasts to calculate

  • the difference between relax in period 1 and control
  • the difference between activating in period 1 and control

and the difference between these differences.

Modeling including explanatory variable

Simpler analysis

Ignore period 2 data. For each subject, calculate the difference between the period 1 treatment and control and use this as the response. Build a model for this response including period 1 treatment, music education, and (possibly) their interaction.

More complex analysis

Build a model for all the data with 10 means consisting of every combination of period, treatment, and music education (control treatment in period 0 with the same music education can be combined into a common group). Now construct contrasts to assess the effect of music education.

Modeling including control measurement

Rather than using differences between treatments and control, control measurement could be included as an explanatory variable. This is related to change scores.


These suggestions are mainly focused on the effect of sequence.

Increase wash-out

If there is interest in eliminating the sequence effect, more time should be given between treatments (the wash-out) so that the sequence effect is minimized.

Measure rest between treatments

To assess the effect of period 1 treatment, measure the response between the two treatments.

Study longevity of treatment effect

Measure response a number of times before the treatment. Give the treatment, then measure the response many times after the treatment.

Additional thoughts

Consider a crossover design with two periods and two treatments like we discussed today.
Subjects get one of these two sequences:

  • Sequence 1: Treatment A in Period 1 (A1) and then Treatment B in Period 2 (B2).
  • Sequence 2: Treatment B in Period 1 (B1) and then Treatment A in Period 2 (A2).

I was thinking the Sequence contrast should be

( A1 + B2 ) / 2 - ( B1 + A2 ) / 2,

which is equivalent to

( A1 + B2 ) - ( B1 + A2 ).

This is correct, but this Sequence contrast is equivalent to

( A1 - B1 ) - ( A2 - B2 ),

which is clearly equivalent to the Treatment-by-Period interaction contrast because it is examining whether the treatment difference in Period 1 is the same as the treatment difference in Period 2.