# How to analyse this semi-cross over study design?

I have this semi-cross over study design. It can be depicted as two different groups - control and treatment. Patients are initially randomized into either control or treatment group. The control group received placebo in the first 12 months, then received treatment in the next 12 months, and then received the same treatment in another 12 months. The treatment group received treatment for the first 12 months, and then same treatment for the next 12 months then stops.

Control group: [Period 1: Placebo][Period 2: Tx]*[Period 3: Tx]*
Treatment group: [Period 1: Tx][Period 2: Tx]*


*No blinding of assigned treatment/control during this period.

What would be the most appropriate statistical testing for this study design, assuming the outcome is average biochemical level (in numeric value).

Along the lines of @Bernhard's answer: you needn't use a different model than you normally would in most other settings.

To be a pedant, I might describe this design as "stepped wedge" because no patient reverted from active treatment to control. A proper cross-over design has the advantage of measuring a possible washout effect which you cannot do here. I think open label phases are questionable. I think the placebo effects can be residual. I would not say you can account for that because there are no control patients crossing over into an open-label phase. I think for an efficacy analysis, I would only use the data from the blinded phase, and unfortunately that's just a parallel design.

For parallel designs, you can use log-rank tests for sample proportions for binary outcomes after a certain survival time. For recurring events, consider a mixed logistic model. For continuous outcomes, a mixed linear model.

I'm surprised by two open label phases for control patients where they're assigned to open treatment. Crossover isn't even an aspect of the design as I reckon.

• Can you clarify "there are no control patients crossing over into an open-label phase"? I thought we have control patients crossing into open-label treatment, but no Tx patient crossing into control. No? – KubiK888 Mar 21 '18 at 21:24
• @KubiK888 they are blinded to control, the open label phase is just them taking the active treatment. I mean to say there is no instance of control crossing over into control with an open label. – AdamO Mar 21 '18 at 21:48
• I agree the most meaningful comparison is the parallel design. But say, we try to model all periods into one modeling, can we include a Tx and Time variable to capture the outcome difference (numeric), and include periods 1 - 3 in a linear mixed effect model? – KubiK888 Mar 21 '18 at 21:54
• @KubiK888 You can do that too, but my point is just to be aware of the impact on interpretation. I would compare that to the blinded, parallel design (not formally) to be sure to describe any differences. – AdamO Mar 22 '18 at 14:05

It depends on the exact question, but probably a linear mixed effects model with the proband id as random effect and a dummy variable that is $0$ if the proband did receive nothing (pretest values) or placebo in the last period and $1$ if the proband received true therapy in the last period, and otherwise $0$, as fixed effect. The biochemical level being the dependent variable.

Depending on the values and your expectiations on the durability of the drug effect you might consider another dummy variable on whether there was drug influence in the last two periods or you might consider the log of the measurement instead of the measurement itself or if you are faced with floor or ceiling effects you might consider the same as logistic regression. It really depends on your expectations on the drug. However, the above given model will be a good start to improve upon.

• Would linear mixed effects model be better than repeated ANOVA? – KubiK888 Mar 21 '18 at 21:41