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)

## Model

Consider models that only look at treatments relax and activating

### Design-based model (no interaction)

Fixed effects:

• treatment
• sequence
• period

Random effects:

• subject

#### Issue

Sequence effect is significant.

### Design-based model with interaction

Fixed effects:

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

Random effects:

• subject

#### Issue

Treatment x sequence interaction is significant.

### Modeling with explanatory variable(s)

Fixed effects:

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

Random effects:

• subject

Unknown.

### 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.

### Design

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.

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.