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Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory
A crossover study, also referred to as a crossover trial, is a longitudinal study in which subjects receive a sequence of different treatments (or exposures). While crossover studies can be observational studies, many important crossover studies are controlled experiments, which are discussed in this article. Crossover designs are common for experiments in many scientific disciplines, for example psychology, education, pharmaceutical science, and medicine.
Randomized, controlled crossover experiments are especially important in health care. In a randomized clinical trial, the subjects are randomly assigned to different arms of the study which receive different treatments. When the randomized clinical trial is a repeated measures design, the same measures are collected multiple times for each subject. A crossover clinical trial is a repeated measures design in which each patient is randomly assigned to a sequence of treatments, including at least two treatments (of which one "treatment" may be a standard treatment or a placebo).
Nearly all crossover designs have "balance", which means that all subjects should receive the same number of treatments and that all subjects participate for the same number of periods. In most crossover trials, in fact, each subject receives all treatments.
Statisticians suggest that designs have four periods, a design which allows studies to be truncated to three periods while still enjoying greater efficiency than the twoperiod design.^{[1]}^{[2]} However, the twoperiod design is often taught in nonstatistical textbooks, partly because of its simplicity.
Contents
Clinical trial protocol specifies the statistical analysis
The data are analyzed using the statistical method that was specified in the clinical trial protocol, which needs to have been approved by the appropriate institutional review boards and regulatory agencies before the trial could begin. Again, the clinical trial protocols specify the method of statistical analysis. Most clinical trials are analyzed using repeatedmeasurements anova (analysis of variance) or mixed models that include random effects.
In most longitudinal studies of human subjects, patients may withdraw from the trial or become "lost to followup" (due e.g. to moving abroad or to dying from another disease). There are statistical methods for dealing with such missingdata and "censoring" problems. An important method analyzes the data according to the principle of the intention to treat.
Advantages
A crossover study has two advantages over a noncrossover longitudinal study. First, the influence of confounding covariates is reduced because each crossover patient serves as his or her own control. In a noncrossover study, even randomized, it is often the case that different treatmentgroups are found to be unbalanced on some covariates. In a controlled, randomized crossover designs, such imbalances are implausible (unless covariates were to change systematically during the study).
Second, optimal crossover designs are statistically efficient and so require fewer subjects than do noncrossover designs (even other repeated measures designs).
Optimal crossover designs are discussed in the graduate textbook by Jones and Kenward and in the review article by Stufken. Crossover designs are discussed along with more general repeatedmeasurements designs in the graduate textbook by Vonesh and Chinchilli.
Limitations and disadvantages
These studies are often done to improve the symptoms of patients with chronic conditions; for curative treatments or rapidly changing conditions, crossover trials may be infeasible or unethical.
Crossover studies often have two problems:
First is the issue of "order" effects, because it is possible that the order in which treatments are administered may affect the outcome. An example might be a drug with many adverse effects given first, making patients taking a second, less harmful medicine, more sensitive to any adverse effect.
Second is the issue of "carryover" between treatments, which confounds the estimates of the treatment effects. In practice, "carryover" effects can be avoided with a sufficiently long "washout" period between treatments. However, the planning for sufficiently long washout periods does require expert knowledge of the dynamics of the treatment, which often is unknown, of course.
Also, there might be a "learning" effect. This is important where you have controls who are naive to the intended therapy. In such a case e.g. you cannot make a group (typically the group which learned the skill first) unlearn a skill such as yoga and then act as a control in the second phase of the study.
See also


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Notes
References
 Jones, Byron; Kenward, Michael G. (2003). Design and Analysis of CrossOver Trials, Second, London: Chapman and Hall.
 Najafi Mehdi, (2004). Statistical Questions in Evidence Based Medicine. New York: Oxford University Press. ISBN 0192629921
 Senn, S. (2002). CrossOver Trials in Clinical Research, Second edition. Wiley. ISBN 9780471496533
 Stufken, J. (1996). "Optimal Crossover Designs" Ghosh, S. and Rao, C. R. Design and Analysis of Experiments, 63–90, NorthHolland.
 (1997) "Crossover Experiments" Linear and Nonlinear Models for the Analysis of Repeated Measurements, 111–202, London: Chapman and Hall.
Statistics  

Descriptive statistics 
Mean (Arithmetic, Geometric)  Median  Mode  Power  Variance  Standard deviation 
Inferential statistics 
Hypothesis testing  Significance  Null hypothesis/Alternate hypothesis  Error  Ztest  Student's ttest  Maximum likelihood  Standard score/Z score  Pvalue  Analysis of variance 
Survival analysis 
Survival function  KaplanMeier  Logrank test  Failure rate  Proportional hazards models 
Probability distributions  
Correlation 
Confounding variable  Pearson productmoment correlation coefficient  Rank correlation (Spearman's rank correlation coefficient, Kendall tau rank correlation coefficient) 
Regression analysis 
Linear regression  Nonlinear regression  Logistic regression 
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