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I know that a cross-over experiment (of any number of treatments/levels) necessarily has a repeated-measures design, but that not all repeated-measures designs are cross-overs. However, I wasn't able to find much information on when exactly it is that a repeated-measures design qualifies as cross-over.

Surely the counter-balancing of the order of treatments given to each subject is not the cross-over 'hallmark', as that is also done in a simple repeated-measures experiment.

Does this hallmark perhaps have to do with the longitudinality of the intervention, i.e. each treatment/level has to last for some number of sessions (for some time), as opposed to being just a one-off measurement? Or perhaps with the fact that after each condition, there needs to be a washout period in the cross-over, so as to avoid the effects of condition A spilling over into condition B, whereas for repeated measures there is no such thing?

How does the Venn diagramm look like for these two designs, i.e. when is one also the other and when is it not?

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Let's assume a simple situation, where you have to compare two treatments, A and B.

Repeated measures design

In a repeated measures design, you would randomly assign some of your study subjects to treatment A and some to treatment B. Following assignment, you would repeatedly measure the value of the outcome variable of interest over time until the study concludes. If the study lasts one week, you may measure the outcome variable daily for one week. Each subject in this design takes only one of the two treatments under investigation. Subjects are randomized to treatment.

Cross over design

In a cross over design, you would randomly assign some of your study subjects to the treatment sequence AB and some to the treatment sequence BA. Subjects assigned to the treatment sequence AB would take treatment A for one week, say (Period 1) followed by treatment B for another week (Period 2). Subjects assigned to the treatment sequence BA would take treatment B in Period 1 and treatment A in Period 2. You may decide to have a period of rest between Period 1 and Period 2. On each of those weeks, the outcome of interest may be measured daily for each subject. Each subject in this type of design serves as his/her own control (which means that this type of design generally requires fewer subjects than the repeated measures design) and takes, in sequence, both of the treatments under investigation. Subjects are randomized to treatment sequence.

Repeated measures design versus Cross over design

You can think of the repeated measures design in this simplified example as representing what happens in the first period of the cross over design. However, while the repeated measures design ends after the first week, the cross over design continues for a second week, with subjects switching treatments.

The repeated measures design allows you to investigate changes over time in the outcome value for patients who took treatment A versus those who took treatment B. (The two sets of patients are distinct - patients who took treatment A are distinct from those who took treatment B.)

The cross over design allows you to investigate changes over time in the outcome value for patients who took treatment A and then treatment B, or the other way around. (The same patients took both treatments.)

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  • $\begingroup$ Thanks Isabella for your helpful reply. What I (perhaps incorrectly) meant by repeated measures did not, however, at all assume that "Each subject in this design takes only one of the two treatments under investigation.", but that "repeated" aspect involves them taking treatment B at timepoint #2, after they've taken treatment A at timepoint #1. It is this operationalisation of the repeated measures design that made me wonder what is, in fact, the difference between it and a crossover. But you would argue this is not what repeated measures is, in fact, right? $\endgroup$
    – z8080
    Commented Oct 25, 2018 at 9:22
  • $\begingroup$ ....That is to say, repeated measures doesn't entail different conditions at each repetition, but really the same measure (in the same condition) repeated across time. Did I understand correctly? $\endgroup$
    – z8080
    Commented Oct 25, 2018 at 9:22
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    $\begingroup$ In a general sense, repeated measures designs entail measuring the outcome variable repeatedly over time for the same subject. But within this generic umbrella, you can have numerous variations regarding how subjects are assigned to treatments before their outcome value measurements are recorded. So a cross over design is, in this general sense, a special case of a repeated measures design. The definition of a repeated measures design I used in my simplified example is one that most people will naturally think of when hearing the words "repeated measures". $\endgroup$
    – Isabella Ghement
    Commented Oct 25, 2018 at 15:19
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    $\begingroup$ So "repeated measures" really refers to measuring the same subject repeatedly over time with respect to the outcome variable of interest, but gives no clues about whether or not that subject was assigned to just one treatment/condition throughout the whole study or to a sequence of treatments/conditions, where each treatment covered only a portion of the study and some treatments may have been taken more than once. That is why you need to be specific and name the actual design (e.g., cross over design) to clarify the situation regarding the treatments/conditions taken by each subject. $\endgroup$
    – Isabella Ghement
    Commented Oct 25, 2018 at 15:23

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