Crossover study is an experimental designed where each subject received all the treatments, so that the different outcome caused by different treatments within the same subject can be observed

This research model are increasingly popular in clinical research. The two major advantages are firstly it exposes every subject to all of the alternative treatments, so reducing the overall sample size, and secondly by comparing the effects of the interventions against the very much smaller within subject variations, the model is much more powerful.

Crossover studies however are prone to two major areas of bias. These are period and carry over effects

• The treatments are given at different times (periods), and the effect of time may contaminate the results. For example, different treatment for asthma may coincide with seasonal variations in pollen count and thus the frequency and severity of attacks. It is also possible that the progression of illness, the gaining of experience, and other time related effect on the individual subjects, may interfere with the effects of interventions.
• It is also possible that carry over (residual) effects of the earlier treatment may affect observations related to current treatment, so careful design to include a "wash out" period is important. However this is often difficult to achieve, either because of time constraints, or in some studies (e.g. the effect of educational or social intervention) may have a more permanent effect on the subject, and carry over effects may persist.

There are therefore three basic models for the analysis of the crossover experimental design, each attempts to deal with the risks of bias and confounding in a different way.

• Model 1 follow the arguments provided by Senn in his classic text on crossover trials (see references). He suggested that a complex mathematical approach is both unnecessary and does not necessarily identify carry over effects. From the statistical point of view, carry over effects are difficult to separate from the sequence (group) effect and the interaction between period and treatment. Senn's book suggested that researchers should design their studies to avoid carry over effects as much as possible, then use the period and interaction terms to check that these are not significant, thus ensure that no significant carry over effect exists. The approach is therefore to use the standard multiway Analysis of Variance, like that used in the Latin Square Model, with an initial check on the interaction between periods and treatment. If, as expected, there is no significant interaction, then the variance from interaction can be incorporated with that of error, and a final Analysis of Variance presented.
• Model 2 is that presented in many statistical text book as the standard Multiway Analysis of Variance, suitable for the Latin Square model, where the variance is partitioned into rows (subjects), columns (periods), and treatment. In the standard Latin Square model, the rows and columns are partitioned to improve the efficiency of the model, and statistical significances are usually not tested. However when used in a crossover study, statistical significance are tested as a way of detecting bias.
• Model 3, as developed by Williams and presented by Cochran and Cox (see references), argues that the carry over effects are sometimes unavoidable, because of time constraints, or the nature of intervention (such as education), and in some experimental situations, carry over effects may be relevant and need to be identified. The carry over effects should therefore be accepted as part of the information in the research model, identified, estimated, and used to adjust the conclusions of the analysis.

Grp = group Sub = subject Prd = period Tmt = treatment Res = result or measurement

Grp Sub Prd Tmt Res 1 1 1 1 5.6 1 1 2 2 16.7 1 1 3 3 5.4 1 2 1 1 3.6 1 2 2 2 8.4 1 2 3 3 6.6 2 3 1 1 9.7 2 3 2 3 17.1 2 3 3 2 5.6 2 4 1 1 1 2 4 2 3 0.1 2 4 3 2 14.9 3 5 1 2 4 3 5 2 3 29.7 3 5 3 1 4.8 3 6 1 2 6.2 3 6 2 3 28.3 3 6 3 1 3.2 4 7 1 2 13.2 4 7 2 1 0.1 4 7 3 3 28.4 4 8 1 2 9.4 4 8 2 1 2.8 4 8 3 3 12.1 5 9 1 3 12.7 5 9 2 1 0.4 5 9 3 2 7.7 5 10 1 3 24.4 5 10 2 1 6.8 5 10 3 2 2.3 6 11 1 3 21.4 6 11 2 2 13.6 6 11 3 1 0.7 6 12 1 3 12.9 6 12 2 2 5.1 6 12 3 1 9.8

We carried out a crossover study, using three treatments, and crossover in each subject over three periods of time (6 groups or sequences). Each group or sequence is replicated twice, so there are 12 subjects.

The dataset may look something like to the left. Please note that the data is arranged by order of subjects (col 2), There are 3 rows for each subject, each row is a period with a treatment for that subject.

In the template data, the sequence or groups (col 1) are also arranged in order to demonstrate the structure of the data. In real situations, the groups would be randomised.

	n	Mean	SD
Total	36	9.9	8.3
Groups or Sequence
1	6	7.7	4.7
2	6	8.1	7.1
3	6	12.7	12.7
4	6	11.0	10.0
5	6	9.1	8.7
6	6	10.6	7.2
Subjects
1	3	9.2	6.5
2	3	6.2	2.4
3	3	10.8	5.8
4	3	5.3	8.3
5	3	12.8	14.6
6	3	12.6	13.7
7	3	13.9	14.2
8	3	8.1	4.8
9	3	6.9	6.2
10	3	11.2	11.7
11	3	11.9	10.5
12	3	9.3	3.9
Period
1	12	10.3	7.1
2	12	10.8	10.4
3	12	8.5	7.5
Treatment
1	12	4.0	3.4
2	12	8.9	4.7
3	12	16.6	9.9
Interaction Period:Treatment
1:1	4	5.0	3.7
1:2	4	8.2	4.0
1:3	4	17.9	6.0
2:1	4	2.5	3.1
2:2	4	11.0	5.2
2:3	4	18.8	13.7
3:1	4	4.6	3.8
3:2	4	7.6	5.3
3:3	4	13.1	10.6

Please note that 3 treatments means 6 groups, each group has 2 cases, each case has 3 periods (1,2,3) and each period has a different treatment. Group 1 has treatment order of 1/2/3, and group 2 is 1/3/2, and so on.

The table of data and table of means and standard deviations are presented as to the right.

The analysis of variance checks to see if there is a significant difference between the groups or sequences, and if there is a significant interaction between period and treatment. This is shown as below.

Model 1 : Analysis of Variance with Interaction
df = degress of freedom
SSq=sums of square
MSq = mean sums of squares or variance
Groups = groups or patterns of allocation
Subjects = Research subjects, patients
Period = period or order of treatment
Treatment = treatments (the effect of interest)
Interaction = interaction between periods and treatments
Residual = residual or within subject

Source	df	SSq	MSq	F	p
Groups	5	110.1	22.0	0.90	0.54
Subjects	6	146.5	24.4
Period	2	36.0	18.0	0.27	0.77
Treatment	2	960.5	480.3	7.1	0.01
Interaction	4	77.2	19.3	0.29	0.88
Residual	16	1075.9	67.2
Total	35	2406.2

Please note that, in the table of analysis of variance, the group or sequence mean sum squares is compared with that of between subjects, while that of period and treatment are compared with the within subject residuals (error).

Once analysis of variance has been carried out, the residual variance can be used to calculate the least significant difference or the confidence intervals between differences of the treatment groups using the post hoc analysis program in the Unpaired Difference Programs Page .

The results showed there is no significant difference between groups or sequences, so there is no bias caused by the order the treatments are administered. There is no significant difference between periods, so there is no time related bias. There is no significant interaction, so there is no bias due to any combination of treatment and time. Having thus been assured, the significant difference between the treatments can be accepted.

Model 2 : Latin Square The data in a crossover trial resembles that of the Latin Square model and can be similarly analysed. The same data table can be used, except that the subjects are not partitioned into between and within groups, and no interaction is calculated.

Source	df	SSq	MSq	F	p
Between Subjects	11	256.6	23.3	0.40	0.94
Periods	2	36.0	18.0	0.31	0.74
Treatments	2	960.5	480.3	8.3	0.002
Error	20	1153.1	57.7
Total	35	2406.2

The results suggest that subjects and periods are not statistically significant, giving reassurance that there is no bias, so that the significant treatment effect can be accepted.

Model 3 : Assuming crossover (residual) effects exist

The same data as that used in model 1 is used. However, data from groups (sequence) and interactions are usually not calculated or presented, as the results are adjusted in a different approach.

Source	df	SSq	MSq	F	p
Between Subjects	11	256.6	23.3
Period Within Squares	8	36.0	4.5
Unadjusted Direct Effect	2	960.5	480.3
Adjusted Direct Effect	2	792.9	396.4	4.1	0.04
Unadjusted Residual Effect	2	169.9	84.9
Adjusted Residual Effect	2	2.2	1.1	0.01	0.99
Error	12	1150.8	95.9
Total	35	2406.2

Please note that the data is analysed and presented in a completely different manner, in particular, the following.

• The subjects are not partitioned into between and within groups (sequences), as the interest here is not to see if there are difference between sequences, but merely to improve the efficiency of the model by identifying those variance related to individual cases.
• The periods and treatments are readjusted. The periods becomes periods within squares, and the treatments partitioned to the direct effects and the indirect (carry over) effects.
• The sum of unadjusted direct and adjusted indirect effects is the same as the sum of adjusted direct and unadjusted indirect effects. The calculations for both are usually presented for checking, but the thing to note are the adjusted direct and adjusted indirect effects.
• The error sums of squares and degrees of freedom are altered by these adjustments

Although the sums of squares and degrees of freedoms are presented for checking, the statistics to note is whether the adjusted direct and indirect effects are statistically significant. The former represents the effects of treatment, the latter any carry over effects.

Treatment	Adjusted mean	Residual Effect
1	4.03	-0.03
2	8.78	-0.44
3	16.75	0.47

Finally, as the mean values for any treatment is a combination of the effect of treatment and the carry over effect of the previous treatment, these needs to be adjusted, and the table of adjusted mean (representing the true effects of treatment), and residual (carry over) effects (how the treatment affect subsequent results) are presented.

Grp	Subj	Perd	Tmt	Res
4	1	1	2	**
4	1	2	3	**
4	1	3	1	**
2	2	1	1	**
2	2	2	3	**
2	2	3	2	**
6	3	1	3	**
6	3	2	2	**
6	3	3	1	**
1	4	1	1	**
1	4	2	2	**
1	4	3	3	**
3	5	1	2	**
3	5	2	1	**
3	5	3	3	**
5	6	1	3	**
5	6	2	1	**
5	6	3	2	**
4	7	1	2	**
4	7	2	3	**
4	7	3	1	**
1	8	1	1	**
1	8	2	2	**
1	8	3	3	**
5	9	1	3	**
5	9	2	1	**
5	9	3	2	**
3	10	1	2	**
3	10	2	1	**
3	10	3	3	**
6	11	1	3	**
6	11	2	2	**
6	11	3	1	**
2	12	1	1	**
2	12	2	3	**
2	12	3	2	**

As explained in the previous sections, crossover studies are powerful and precise, requiring smaller sample sizes. The risk however is its vulnerability to effects of time and the sequence of treatment.

In order to control possible bias, a full and balance allocation is necessary. This means that all the possible sequence of treatments must be randomised and made available.

This means that, a two treatment model must have multiples of 2 subjects (AB,BA), 3 treatment means a multiple of 6 subjects (ABC,ACB,BAC,BCA,CAB,CBA), 4 treatment multiple of 24 subjects, and so on. To provide a full and balance model, the number of subjects must be multiples of the Factorial of the number of treatments.

To ensure that all possible sequence are included, to have 2 or more replications for each sequence (group), having them all randomised, and creating a database for the results is a fairly tedious task. The data template program in the Cross Over Trials Program Page provides a facility to do this.

The program accepts two parameters, the number of treatments, and the number of replications. The number of subjects required will then be Factorial(treatment) x replication.

Once the table is created, the sequences (groups) is randomized. The program uses the date/time provided by the computer clock as the random seed, so that the orders are different at every run. The user can copy the template, as paste it into a data file (such as in an Excel file) to use.

The table to the right is an example of the template generated with the parameters of 3 treatment and 2 replications. The abbreviations are Grp=groups or sequence, Subj=subject, Perd=period, Tmt=treatment, and Res=result observed

Data for each subject occupies 3 consecutive rows (3 periods for 3 treatments). There are 6 groups (Factorial 3 for 3 treatments), and each group occurs twice (replication=2), resulting in 12 subjects being used, so the database has 3 x 12 = 36 rows.

• Subjects 4 and 8 (col 2) belongs to group 1 (col 1), and receive treatments 1,2,3 (col 4) in that order (col 3 and 4).
• Subjects 2 and 12 belongs to group 2 and receive treatments 1,3,2 in that order.
• Subjects 5 and 10 belongs to group 3 and receive treatments 2,1,3 in that order.
• Subjects 1 and 7 belongs to group 4 and receive treatments 2,3,1 in that order.
• Subjects 6 and 9 belongs to group 5 and receive treatments 3,1,2 in that order.
• Subjects 3 and 11 belongs to group 6 and receive treatments 3,2,1 in that order.

If more than one outcome measurement is made, additional columns can be added to the right, as the first 4 columns are constant for each subject.

The whole table can be copied and pasted into a spreadsheet like excel, the subjects are assigned in order of the second column as they are recruited, and the results entered to replace the ** as they are collected. When data collection is completed, the first 4 columns, plus the result column (without the header row) can be copied and pasted into the program for analysis of the results.

Crossover studies are highly efficient, but conceptually complex, and those with a full understanding of Analysis of Variance can adapt the basic concepts, producing designs to suit the research problems at hand. Although StatTools has provided three simple models in these pages, the intents are to introduce the concept and explain the potential used of these models.

Developing the programs began after receiving a request for help. Initial search led to STATA and its crossover procedure, which model 1 of these pages are based on. Following STATA's references the text book by Senn was consulted, and after accepting its argument to ignore carry over effects, the algorithm was presented on the web.

Criticisms soon followed, that the model presented was insufficient, and that algorithms that include carry over effects should be provided for those who want them. The algorithms from the original text book on experimental design by Cochran and Cox (see reference) were then added to the programs.

Cochran and Cox provided 2 models.

• when carry over effect is not considered, the book suggested a model even simpler than model 1 presented in these pages, a simple 3 way analysis of variance based on the Latin Square, partitioning rows (subjects), columns (periods), and treatments, and testing the statistical significance of each. This model is presented in these pages as model 2.

  Although historically, model 2 was proposed before model 1, they are placed in this order because model 2 can be re-assembled using the results from model 1. There is no partitioning subjects into those between groups and within groups, and no calculation for interactions.

• Cochran and Cox also provided in details a calculation developed by Williams, which took into consideration the presence of carry over effects. Model 3 of these pages is based on those calculations. This model should be used when carry over effect cannot be ruled out, but it has the disadvantage of being complex, and not intuitively easy to understand by those not familiar with statistical concepts and terminology.

The crossover model is a generic type, and there are many versions of how to analyse the data, each constructed to suit the particular needs of a research project. Users should therefore not embarked upon a crossover research project without a full understanding of its basis, and they are urged to consult professional statisticians in the planning and analysis.

Some terms commonly used in crossover studies

• Treatment is the different interventions under trial. In a two treatment trial, the treatments may be designated A and B (or 1 and 2)
• Period is the time or order the treatment is given. The number of periods is the same as the number of treatments.
• Group or Sequence is the order that the treatments are given. In a 2 treatment trial there are two groups (AB and BA). In a 3 treatment trial there are 6 groups (ABC,ACB,BAC,BCA,CAB,CBA), in a 4 treatment trial there are 24 groups, and so on. The number of groups is the Factorial of the number of treatments.
• Replication in the number of times each group is used. In a two treatment trial replicate 3 time means 3 x (AB,BA) = 6 subjects, with two observations per subject, a total of 12 observations.

Cochran WG and Cox GW (1957) Experimental Design Second Edition. John Wiley and Son New York. ISBN 0 471 16203 5. p. 127-139

Senn S (2002) Cross-over Trials in clinical Research (Second Edition) John Wiley and Son, Chichester. ISBN 0-471-49653-7

STATA 9.2. (1984) Statacorp, Texas, USA - pkcross procedure