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This article introduces a crossover design that is often used in clinical studies, with the advantage of comparing treatment effects within one study subject. In particular, the advantages and disadvantages of the two-period, two-sequence crossover design (2 × 2 or AB/BA crossover design), which is widely used in clinical practice, are identified, and the elements necessary for analysis are introduced. This article explains the carryover effect, period effect, sequence effect, and period-by-treatment interaction in a crossover design and examines the analysis commands of SAS along with example data. After confirming the carryover effect using a general linear model, the treatment effect is analyzed using a linear mixed effect model.

In a crossover study design, two or more treatments (e.g., drugs, procedures) are provided to subjects at different time periods, and the sequence of treatments is randomized for each subject. This design is applied to several fields such as bioequivalence clinical trials, and the simplest study design is a two-period, two-sequence crossover design (AB/BA design)^{1)}^{2)}

A crossover design has the following advantages. First, the treatment effect is compared within a subject since each subject serves as his or her own control. It removes the inter-subject variability from the comparison between groups and the effect of covariates^{3)}

However, there are also limitations in the crossover design. First, the conditions of the subjects must be stable throughout the study. In other words, a case is inappropriate for a study if the disease status changes over time, such as an acute cure, or when symptoms disappear or are cured by treatment in the first period. Second, a washout period may be necessary until the effect of the first treatment subsides. Therefore, if a treatment drug has an extended half-life, it may be difficult to conduct a study with a crossover design. Third, there is a burden that all treatments are carried out on one subject, which often causes ethical problems. Fourth, the processing of dropped or missing data is more problematic than in a parallel design, and the statistical analysis is complex [

Based on these advantages and disadvantages, it will be helpful in planning and proceeding with an appropriate crossover design to fully understand the various factors encountered in its design and analysis process.

The standard 2 × 2 crossover design is used to assess between two groups (test group A and control group B). Each subject is randomly allocated to either an AB sequence or a BA sequence. Subjects in the AB sequence receive treatment A at the first period and treatment B at the second period.

A model can be used to describe the standard 2 × 2 crossover design as follows:

Where

_{ijk}^{th} subject in the ^{th} sequence at the ^{th} period,

subject _{k}^{th} subject in ^{th} sequence)

period

sequence

The subject (_{ik}_{ijk}^{4)}_{j}_{j,k}_{j-1,k}_{1,2} represents the direct fixed effect of the treatment at period 1 in sequence 2 (Here, at the first period of the BA sequence), and _{2-1,1} is the residual effect carried over from the (2-1)^{th} period to the second period in sequence 1 (Here, the AB sequence). The carryover effect in the standard 2 × 2 crossover design can occur at the second period. The fixed effects at each period in each sequence are summarized as follows.

Sequence | First period | Second period |
---|---|---|

AB sequence | ||

BA sequence |

Here,

_{jk}_{ijk}_{1} + _{2} = 0, _{A}_{B}_{A}_{B}

In studies using a crossover design, elements such as carryover effect, period effect, sequence effect, and period-by-treatment interaction should be evaluated before testing the treatment effect. Even if effects other than the treatment effect were carefully considered and excluded in the research planning stage, it is necessary to check the carryover effect or the period effect before analyzing the treatment effect. After checking the above-mentioned effects are not absent, it is common to analyze the treatment effect. It is important in the crossover study planning stage is to design an analysis that does not become complicated and allow effects other than the treatment effect to influence the interpretation of the results [

This refers to the direct effect of the treatment. In this article, it indicates the effects of A and B and _{j,k}

The period effect implies that the effect of the same treatment received at two different periods is different for each period and corresponds to _{j}^{5)}

The carryover effect, which corresponds to _{j-1,k}^{6)}

The fact that subjects are allocated to a particular sequence may affect the results. That is, when comparing the means of the dependent variables in the AB and BA sequences, there should be no difference if there is no sequence effect. This allows the assumption that there is no sequence effect by randomization in the AB/BA sequence. However, it should be noted that this assumption cannot be verified through statistical analysis [

Randomization is performed to eliminate selection bias and to provide statistical evidence for quantitative evaluation. In parallel design, randomization to different treatment groups (A or B) can ensure independence between groups. However, in the crossover design, randomization is performed in sequence, that is, AB or BA sequence, so that virtually no independence between treatment groups is guaranteed. Therefore, as discussed above, it is necessary to test whether the treatment effect shown in the first period remains in the second period. The randomization method may be reviewed in the article of Lim and In [

When planning the crossover design, to confirm the treatment effect, the carryover effect should not appear above all. The carryover effect can be found through data analysis, and when this effect is significant, it is difficult to interpret the treatment effect. Therefore, a protocol such as the above-mentioned washout period setting is an appropriate method to eliminate this effect. However, if the washout period is too long, the dropout rate may increase at the second period. In some cases, crossover studies may be impossible. For example, in the case of an acutely curable disease, if it is cured in the first period, there is no reason for the subjects to participate in the second period. Therefore, the crossover design is more suitable for chronic diseases than for acute diseases. These factors should be reflected in crossover design planning.

The example used below was published by Senn and Auclair [

The full data entry is presented in

title “PEF data”;

PROC PRINT DATA = PEF(OBS=5);

RUN;

PEF data

OBS sequence subject period treat pef

1 For-Sal 1 1 For 310

2 For-Sal 1 2 Sal 270

3 For-Sal 4 1 For 310

4 For-Sal 4 2 Sal 260

5 For-Sal 6 1 For 370

Determining whether various effects other than the treatment effect appearing in this data can be easily accomplished through the analysis of variance table. The following code is used for the construction and execution of a general linear model to which analysis of variance is applied. Here, sequence represents For→Sal or Sal→For, the subject is a pediatric patient, and period represents the first period (1) or the second period (2). treat means cure using For or Sal. CLASS defines the variables to be used in the model, and MODEL is a code that specifies the independent and dependent variables.

PROC GLM DATA=PEF;

CLASS sequence subject period treat;

MODEL pef = sequence subject(sequence) period treat;

RUN;

In the table of the sum of squares in

PROC MIXED DATA=PEF;

CLASS sequence subject period treat;

MODEL pef = sequence period treat;

RANDOM subject(sequence);

LSMEANS treat/PDIFF;

ESTIMATE 'Treatment' treat 1 -1 /CL ALPHA=0.05;

RUN;

The subject is included as a random effect through RANDOM subject (sequence). LSMEANS treat/PDIFF is a code that describes the P value for the difference between two treatments, and ESTIMATE 'Treatment' treat 1 -1 /CL ALPHA=0.05 represents the estimate of the treatment effect and 95% confidence interval.

After confirming that other effects do not appear through the analysis of variance table, the treatment effect can be analyzed using a linear mixed effect model (

The AB/BA design is a simplified form of the AA/AB/BA/BB crossover design introduced in clinical practice. It has many advantages, such as the ability of subjects themselves to reduce the influence of covariates by acting as a control group, and the higher power and statistical efficiency compared to those of the parallel design. However, there are also several limitations such as the difficulty in studying drugs with long half-lives, the requirement for subject stability even at different periods, administration of all treatments to one subject, and complicated processing of dropped or missing data. Therefore, it is necessary to understand the strengths and weaknesses of this study design and to apply an appropriate analysis method. It should also be noted that if a carryover effect occurs when using a crossover design, only the results of the first period can be explored.

A and B usually refer to test group A and control group B. It was introduced as a crossover design including the four orders AA/AB/BA/BB in consideration of the treatment type and treatment sequence; however, a simplified study design is mainly used in clinical studies, excluding AA and BB, which are the sequences in which the same drug is repeatedly administered.

Depending on the literature, it is also described as a sequence, a group, or an order. In this article, a group of subjects in the same order of receiving treatment is described as a sequence, and a set of data for subjects receiving the same treatment is described as a group.

In addition to the independent variables, a variable that the researcher wants to control as a factor that can affect the dependent variable is designated a covariate. Variables such as age and gender are often included.

Subjects are randomly assigned to AB or BA sequence so that the assumption of no sequence effect is established.

Both values are the result of treatment B minus the result of treatment A.

In the example of this article, the carryover effect is evaluated through sequence effect analysis.

No potential conflict of interest relevant to this article was reported.

Chi-Yeon Lim (Conceptualization; Resources; Supervision; Writing – review & editing)

Junyong In (Conceptualization; Software; Writing – original draft)

Two-period, two-sequence crossover design.

Part of the output of SAS PROC GLM (General linear model procedure). DF: degrees of freedom, SS: sum of squares.

Part of the output of SAS PROC MIXED (Mixed effect model procedure). DF: degrees of freedom, Den DF: DF for the denominator, Num DF: DF for the numerator.

Analysis of Variance Table for a Standard 2 × 2 Crossover Design (Example Data)

Source of variation | d.f. | Type III sum of squares | Mean square | F | P value |
---|---|---|---|---|---|

Between subjects | |||||

Carryover | 1 | 335 | 335 | 0.45 | 0.518 |

Residual | 11 | 114878 | 10443 | ||

Within-subjects | |||||

Treatment | 1 | 14036 | 14036 | 18.70 | 0.001 |

Period | 1 | 1632 | 1632 | 2.17 | 0.168 |

Residual | 11 | 8254 | 750 | ||

Total | 25 | 138488 |

Due to the limitation of the simplified 2 × 2 crossover design, all effects or interactions cannot be estimated separately. Therefore, depending on the model to be designed, the carryover effect is inherent in other effects or interactions. Here, the carryover effect is confirmed through the sequence. That is, in this crossover design model, the carryover effect is inherent in the sequence effect. Therefore, the result for the sequence is displayed as

DATA PEF;

INPUT sequence $ subject period treat $ pef;

DATALINES;

For-Sal 1 1 For 310

For-Sal 1 2 Sal 270

For-Sal 4 1 For 310

For-Sal 4 2 Sal 260

For-Sal 6 1 For 370

For-Sal 6 2 Sal 300

For-Sal 7 1 For 410

For-Sal 7 2 Sal 390

For-Sal 10 1 For 250

For-Sal 10 2 Sal 210

For-Sal 11 1 For 380

For-Sal 11 2 Sal 350

For-Sal 14 1 For 330

For-Sal 14 2 Sal 365

Sal-For 2 1 Sal 370

Sal-For 2 2 For 385

Sal-For 3 1 Sal 310

Sal-For 3 2 For 400

Sal-For 5 1 Sal 380

Sal-For 5 2 For 410

Sal-For 9 1 Sal 290

Sal-For 9 2 For 320

Sal-For 12 1 Sal 260

Sal-For 12 2 For 340

Sal-For 13 1 Sal 90

Sal-For 13 2 For 220

;

RUN;

Spaces are used to separate free formatted data. Data reported by Senn SJ and Auclair P (Statistics in Medicine 1990; 9: 1287-302) were permitted to reuse from the Wiley Publisher.