Crossover Experimental Design

Imagine designing an experiment to compare the effects of two different treatments. These two treatments could be, for example, two newly synthesized drugs, a placebo and an experimental medication, or simply two separate tasks that you’d like for the subjects of the experiment to complete. For this test, you’ve recruited a total of 6 individuals, and you’ve decided to test both treatments with each subject. In other words, you’re going to perform some sort of repeated measures design experiment. But there are a number of ways you could proceed. You could, in theory, provide Treatment A to all six subjects, record the outcome for each, wait an appropriate amount of time for any effects of Treatment A to dissipate, and then provide Treatment B to all six subjects and record the outcomes.

This seems like a fairly straightforward experimental design, and you would likely analyze this data using a paired t test with the data structure shown below:

However, this approach presents some shortcomings. Namely, in the description of this experimental design, it was stated that all six subjects would receive Treatment A first, and then receive Treatment B. However, it’s possible that Treatment A had some effect on the outcomes recorded after Treatment B. In the design, it was noted that there would be a waiting time between Treatment A and Treatment B to allow the effects of A to dissipate. However, in some cases, this may not be possible. Going back to our experimental drug example, this waiting period may be to allow for the drug to be cleared by the body. However, it’s possible that this drug induced permanent changes (an immune response, for example) that would affect the outcome of Treatment B compared to the situation in which a subject had never been given Treatment A. Even something as simple as the subject being more comfortable, less anxious, or more familiar with the procedure during the second treatment could potentially affect the outcomes. This is known as a carryover effect, and is important to address in experimental designs such as this one.

So how can we deal with the problem of the carryover effect? One way would be to recruit 12 subjects instead of 6, and randomly assign six subjects to each treatment. However, by doing this, you lose statistical power due to the fact that this is no longer a repeated measures design. Instead, you could use a different method known as a Crossover experimental design. But first, some terminology:

• Treatment: we’ve already discussed this one, and it is generally the effect of the treatment that is your primary interest in performing the experiment. Typically, for Crossover designs, treatments are represented by capital letters (A, B, etc.)

• Sequence: the order that the treatments will be administered by individuals within a group. An example could be “AB” meaning Treatment A will be administered first, then Treatment B

• Period: this represents the time that the treatments are administered. For example, in the “AB” sequence, Treatment A would be administered during Period 1, while Treatment B would be administered during Period 2.

With that out of the way, we can discuss the most popular crossover experimental design: the 2x2 Crossover. This is a 2-treatment, 2-sequence, 2-period design in which each patient is assigned to either sequence AB or BA. The reason this is called a Crossover experimental design is because each subject receives one treatment before "crossing-over" to receive the second treatment. As mentioned previously, the primary drawback to this experimental design is a carryover effect (the first treatment given to a subject may affect the result of the second treatment given to the same subject), but the 2x2 Crossover design has a number of ways of mitigating this problem. 

First, how subjects are assigned to sequences can help to eliminate carryover effects. Assigning half the subjects to sequence AB (A then B) and the other half to sequence BA (B then A) can reduce bias due to a carryover effect. However, the precision of the analysis will still be affected unless the carryover effect is explicitly accounted for in the analysis.

In Prism, analysis of Crossover designs aren’t (currently) included as a default analysis. However, the two-way repeated measures ANOVA tool (from the Grouped data table format) can be used to test and account for carryover effects in our proposed 2x2 Crossover experiment. The data can be entered into Prism in a grouped table as follows:

In this table, you can see that Treatment has been assigned to the columns of the data table (Treatment A in the first and Treatment B in the second), and Sequence has been assigned to the rows (Sequence AB in the first, and Sequence BA in the second).

The data shown here represent the same six subjects with the same measurements for each subject as were given in the column table earlier on this page. Using this new format, you can see that Subject 1 was assigned to Sequence AB (they were given Treatment A in Period 1, then Treatment B during Period 2). The values for Subject 1 are 10 (row 1 A:Y1) and 25 (row 1 B:Y1). In comparison, Subject 4 was assigned to Sequence BA (they were given Treatment B during Period 1, then Treatment A during Period 2). The values for Subject 4 are 15 (row 2 A:Y1) and 20 (row 2 B:Y1). Mapping out the rest of the data, it can be seen that Subjects 1-3 were assigned to Sequence AB and Subjects 4-6 assigned to Sequence BA. To analyze this data properly in Prism, first click the Analyze button in the toolbar and select “Two-way ANOVA (or mixed mode)” and click “OK”. On the first tab (the RM Design tab), select the first matching option, “Each column represents a different time point, so matched values are spread across a row.” as shown below.

Next, in order to make the results of the analysis a bit easier to interpret, you’ll want to switch to the “Factor Names” tab of the dialog. On this dialog, you’ll want to change the name of the factor that defines column to “Treatment” and the name of the factor that defines rows to “Sequence” as shown:

That’s it! If you entered your data in the correct format, you can click “OK” and Prism will perform the analysis, generating a table of results that contain the values you’ll need to interpret. Specifically, this analysis will generate four P values. They correspond to the following tests:

1. Treatment: this compares the overall effect of Treatment A vs. Treatment B. You can think of this as being equivalent to a carryover-adjusted result from the paired t-test that we considered at the beginning of this page.

2. Sequence: this compares the data collected in Sequence AB to the data collected in Sequence BA. A small P value indicates that "A following B" is different than "B following A". In our example, the average of all the values in the AB sequence is 18 ([10+10+11+25+26+26]/6 = 18) and the average of the BA sequence is also 18 ([15+16+16+20+20+21]/6 = 18). Thus, "A following B" has the same effect as "B following A", and this is reflected by the calculated P value (see below).

3. Sequence x Treatment interaction: this is testing for an order effect. For example, in our data table, look at the response values in the Treatment A column. When Treatment A is given first (sequence AB), the response values are consistently lower than when Treatment A is given second (sequence BA). Similarly, looking at the Treatment B column, we see that the responses are consistently lower when Treatment B is given first (sequence BA) compared to when Treatment B is given second (sequence AB). This is what is meant by an order effect. A small P value for this interaction term indicates an order effect is present and the effectiveness of the second treatment given to a subject has been altered due to some carryover effect from the first treatment given to that same subject (as shown with the data in this example).

4. Subject: this test examines the variation between subjects.