## Completely randomized vs cross-over design

Compare power for cross-over versus completely randomized design.

### Completely randomized design

There are two treatments and individuals will be randomly assigned to those two treatments.

### Cross-over design

Each individual will be put on both treatments with the order of treatments being randomly assigned and wash-out period between the two treatments.

We need to know

• type I error ($\alpha=0.05$)
• power ($1-\beta = 0.80$)
• effect size ($\Delta$)
• subject-to-subject variability
• within subject variability, i.e. measurement error
• carry-over effect (assume this to be zero?)

The key with a cross-over design is that it eliminates the subject-to-subject variability.

There are some practicalities that likely override statistical considerations, e.g.

• cost to enroll an individual
• drop-out rate for both designs
• drop-out timing for cross-over design

If sample size is expected to be low, then cross-over design is (likely) preferred because it will (likely) have greater power for the same number of individuals. It is only likely because some degrees of freedom will be lost due to the more sophisticated model that is required.