# 15.8 - Analysis - Continuous Outcome

The statistical analysis of normally-distributed data from a 2 × 2 crossover trial, under the assumption that the carryover effects are equal ( λ_{A} = λ_{A} = λ ), is relatively straightforward.

Remember the statistical model we assumed for continuous data from the 2 × 2 crossover trial:

[Design 11] | Period 1 |
Period 2 |

Sequence AB |
μ_{A} + ν + ρ |
μ_{B} + ν - ρ + λ_{A} |

Sequence BA |
μ_{B} - ν + ρ |
μ_{A} - ν - ρ + λ_{B} |

For a patient in the AB sequence, the Period 1 vs. Period 2 difference has expectation μ_{AB} = μ_{A} - μ_{B} + 2ρ - λ .

For a patient in the BA sequence, the Period 1 vs. Period 2 difference has expectation μ_{BA} = μ_{B} - μ_{A} + 2ρ - λ .

Therefore, we construct these differences for every patient and compare the two sequences with respect to these differences using a two-sample t test or a Wilcoxon rank sumtest. Thus, we are testing:

H_{0} : μ_{AB} - μ_{BA} = 0

The expression:

μ_{AB} - μ_{BA} = 2( μ_{A} - μ_{B} )

so testing H_{0} : μ_{AB} - μ_{BA} = 0, is equivalent to testing:

H_{0} : μ_{A} - μ_{B} = 0

To get a confidence interval for μ_{A} - μ_{B} , simply multiply each difference by ½ prior to constructing the confidence interval for the difference in population means for two independent samples.

**SAS Example **( 16.1_-_2x2_crossover__contin.sas )

This is an example of an analysis of the data from a 2 × 2 crossover trial. The example is taken from Example 3.1 from Senn's book (Senn S. *Cross-over Trials in Clinical Research *, Chichester, England: John Wiley & Sons, 1993). The data set consists of 13 children enrolled in a trial to investigate the effects of two bronchodilators, formoterol and salbutamol, in the treatment of asthma. The outcome variable is peak expiratory flow rate (liters per minute) and was measured eight hours after treatment. There was a one-day washout period between treatment periods.

The estimated treatment mean difference was 46.6 L/min in favor of formoterol (*p *= 0.0012) and the 95% confidence interval for the treatment mean difference is (22.9, 70.3). The Wilcoxon rank sumtest also indicated statistical significance between the treatment groups (*p* = 0.0276).