First-order and Higher-order Carryover Effects Section
Within time period \(j, j = 2, \dots, p\), it is possible that there are carryover effects from treatments administered during periods \(1, \dots, j - 1\). Usually in period j we only consider first-order carryover effects (from period \(j - 1\)) because:
- if first-order carryover effects are negligible, then higher-order carryover effects usually are negligible;
- the designs needed for eliminating the aliasing between higher-order carryover effects and treatment effects are very cumbersome and not practical. Therefore, we usually assume that these higher-order carryover effects are negligible.
In actuality, the length of the washout periods between treatment administrations may be the determining factor as to whether higher-order carryover effects should be considered. We focus on designs for dealing with first-order carryover effects, but the development can be generalized if higher-order carryover effects need to be considered. We will focus on:
Uniformity Section
A crossover design is labeled as:
- uniform within sequences if each treatment appears the same number of times within each sequence, and
- uniform within periods if each treatment appears the same number of times within each period.
For example, AB/BA is uniform within sequences and period (each sequence and each period has 1 A and 1 B) while ABA/BAB is uniform within period but is not uniform within sequence because the sequences differ in the numbers of A and B.
If a design is uniform within sequences and uniform within periods, then it is said to be uniform. If the design is uniform across periods you will be able to remove the period effects. If the design is uniform across sequences then you will be also be able to remove the sequence effects. An example of a uniform crossover is ABC/BCA/CAB.
Latin Squares Section
Latin squares historically have provided the foundation for r-period, r-treatment crossover designs because they yield uniform crossover designs in that each treatment occurs only once within each sequence and once within each period. As will be demonstrated later, Latin squares also serve as building blocks for other types of crossover designs. Latin squares for 4-period, 4-treatment crossover designs are:
Period 1 | Period 2 | Period 3 | Period 4 | |
Sequence ABCD | A | B | C | D |
Sequence BCDA | B | C | D | A |
Sequence CDAB | C | D | A | B |
Sequence DABC | D | A | B | C |
and
Period 1 | Period 2 | Period 3 | Period 4 | |
Sequence ABCD | A | B | C | D |
Sequence BDAC | B | D | A | C |
Sequence CADB | C | A | D | B |
Sequence DCBA | D | C | B | A |
Latin squares are uniform crossover designs, uniform both within periods and within sequences. Although with 4 periods and 4 treatments there are \(4! = (4)(3)(2)(1) = 24\) possible sequences from which to choose, the Latin square only requires 4 sequences.
Balanced Designs Section
The Latin square in [Design 8] has an additional property that the Latin square in [Design 7] does not have. Each treatment precedes every other treatment the same number of times (once). For example, how many times is treatment A followed by treatment B? Only once. How many times do you have one treatment B followed by a second treatment? Only once. This is an advantageous property for Design 8. This same property does not occur in [Design 7]. When this occurs, as in [Design 8], the crossover design is said to be balanced with respect to first-order carryover effects.
Try it!
The designs that are balanced with respect to first order carryover effects are:
Designs 1, 2, 3, 5, 6, 8.
When r is an even number, only 1 Latin square is needed to achieve balance in the r-period, r-treatment crossover. When r is an odd number, 2 Latin squares are required. For example, the design in [Design 5] is a 6-sequence, 3-period, 3-treatment crossover design that is balanced with respect to first-order carryover effects because each treatment precedes every other treatment twice.
Strongly Balanced Designs Section
A crossover design is said to be strongly balanced with respect to first-order carryover effects if each treatment precedes every other treatment, including itself, the same number of times. A strongly balanced design can be constructed by repeating the last period in a balanced design.
Here is an example:
Period 1 | Period 2 | Period 3 | Period 4 | Period 5 | |
Sequence ABCDD | A | B | C | D | D |
Sequence BDACC | B | D | A | C | C |
Sequence CADBB | C | A | D | B | B |
Sequence DCBAA | D | C | B | A | A |
This is a 4-sequence, 5-period, 4-treatment crossover design that is strongly balanced with respect to first-order carryover effects because each treatment precedes every other treatment, including itself, once. Obviously, the uniformity of the Latin square design disappears because the design in [Design 9] is no longer is uniform within sequences.
Uniform and Strongly Balanced Design Section
Latin squares yield uniform crossover designs, but strongly balanced designs constructed by replicating the last period of a balanced design are not uniform crossover designs. The following 4-sequence, 4-period, 2-treatment crossover design is an example of a strongly balanced and uniform design.
Period 1 | Period 2 | Period 3 | Period 4 | |
Sequence ABBA | A | B | B | A |
Sequence BAAB | B | A | A | B |
Sequence AABB | A | A | B | B |
Sequence BBAA | B | B | A | A |