Note on the mathematical formulation of the crossover design

     The linear model for a two treatment crossover design, including all the elements that allow the computation of the final value yijk of a subject j in group i and in period k can be described as (Kuehl, 2000):

, where the range of the subindices is as follows: i (number of groups: A, B), k (number of periods: 1, 2), t (number of treatments: 1, 2), c (carryover of the first period into the second: 1, 2),  j = 1, ..., s (s is the number of subjects in the experiment, 18 for Test 1 and 16 for Test 2).

     The parameters that have to be estimated in an analysis of variance (ANOVA) are: mean (m); group or sequence (s); period (g); treatment (t) and carryover (l) (subject (group) is the random subject effect). The group or sequence effect s (GROUP) tests the residual difference between the two sequences of application: a) SEF then SE; b) SE then SEF.  Throughout the text we will always use the term “group” since it eliminates the ambiguity of the term “sequence”. There is a strict correspondence between each of the two sequences and the two groups. The period effect g (PERIOD) tests for the existence of differences due to the two periods of treatment (due to learning, maturation or other causes). The paramenter treatment t (TREAT) is the direct effect of applying SEF or SE. The carryover effect l (CARRYOVER) is the residual effect of applying the treatment after the first period. This effect has to be included in the equations modeling the second period. In clinical studies it represents the residual effects of the treatment of the first period which persist into the second period, contaminating the effects of the second treatment. For the sake of clarity, g effects will not be considered as CARRYOVER effects, although in many texts g and l are both named as residual or carryover effects.

Table 1. Four observed means.

     The equations for each of the observed (measured) means are described in Table 1. By the own nature of the crossover design we assume that the sum of the two treatment effects is 0, since the effects are cancelled, so t2 = -t1 and similarly happens for sB = -sA, g2 = -g1 and l2 = -l1. Therefore the equations can be reduced to four equations with five parameters.

     With only four observed means , , ,  for estimating these five parameters (m, sA, t1, g1 and l1) we have to eliminate one the parameters (Kuehl, 2000, pp. 537-538) in order to estimate the remaining ones. This means that this 2x2 crossover design has the problem that some effects may be confounded with interactions or with other effects. Therefore the null hypothesis corresponding to those effects cannot be computed from the ANOVA model. For example, one or more parameters of the model should be eliminated for detecting the l1 effect or for estimating others. If the s, g and l effects are not statistically significant, the design is appropriate for estimating the null hypothesis for the treatment variable (t). 

     There are discussions about the circumstances in which some of these parameters can be discarded and how the analysis should be carried out. There are several procedures for dealing with this issue, of which three widely used are those of (Grizzle, 1965) (Hills and Armitage, 1979) (Milliken and Johnson, 1984, pp. 435-438).

   Crossover designs are not full-factorial designs (in factors GROUP, TREAT and PERIOD) as not all combinations of the levels are found in the design. Thus, the main effects are also estimates of the two-factor interactions (Dallal, 2000).

For example, the estimates of the GROUP effect are also the estimates of the interaction PERIOD x TREAT and of the CARRYOVER effects. The estimates of TREAT are the estimates of GROUP x PERIOD, and similarly for PERIOD and GROUP x TREAT. These considerations must be taken into account when analysing the data.

References

G.E. Dallal, "The Computer-Aided Analysis of Crossover Studies," http://www.tufts.edu/~gdallal/crossovr.htm, 2000.

J.E. Grizzle, “The Two-Period Change-Over Design and its Use in Clinical Trials”, Biometrics, Vol. 21, pp. 467-480, 1965.

M. Hills, and P. Armitage, "The Two-Period Cross-Over Clinical Trial," British Journal of Clinical Pharmacology, Vol. 8, pp. 7-20, 1979. R.O. Kuehl. Design of Experiments: Statistical Principles of Research Design and Analysis, 2nd edition, Duxbury Thomson Learning, California, USA, 2000.

G.A. Milliken and D.E. Johnson, Analysis of messy data. Vol I: Designed experiments, Cambridge University Press,  Cambridge (UK), 1984.