Crossover study

Short description: Research study in medicine

"Crossover experiment" redirects here. For the chemical reaction/mechanism analysis, see crossover experiment (chemistry).

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In medicine, a crossover study or 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, 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 trial has a repeated measures design, the same measures are collected multiple times for each subject. A crossover trial has a repeated measures design in which each patient is assigned to a sequence of two or more treatments, of which one may be a standard treatment or a placebo.

Nearly all crossover are designed to have "balance", whereby all subjects receive the same number of treatments and participate for the same number of periods. In most crossover trials each subject receives all treatments, in a random order.

Statisticians suggest that designs should have four periods, which is more efficient than the two-period design, even if the study must be truncated to three periods.[1][2] However, the two-period design is often taught in non-statistical textbooks, partly because of its simplicity.

## Contents

* 1 Analysis
* 2 Advantages
* 3 Limitations and disadvantages
* 4 See also
* 5 Notes
* 6 References

## Analysis

The data is analyzed using the statistical method that was specified in the clinical trial protocol, which must have been approved by the appropriate institutional review boards and regulatory agencies before the trial can begin. Most clinical trials are analyzed using repeated-measurements 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 follow-up". There are statistical methods for dealing with such missing-data 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 both a parallel study and a non-crossover longitudinal study. First, the influence of confounding covariates is reduced because each crossover patient serves as their own control.[3] In a randomized non-crossover study it is often the case that different treatment-groups 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 non-crossover 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 repeated-measurements 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, cross-over 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 "carry-over" between treatments, which confounds the estimates of the treatment effects. In practice, "carry-over" effects can be avoided with a sufficiently long "wash-out" period between treatments. However, planning for sufficiently long wash-out periods requires expert knowledge of the dynamics of the treatment, which is often unknown.

## See also

* Design of experiments
* Glossary of experimental design
* Randomized controlled trial
* Survival analysis
* N of 1 trial
* Single-subject design

## Notes

1. ↑ Vonesh & Chinchilli (1997)
2. ↑ Jones & Kenward (2003)
3. ↑ Jones, B., & Kenward, M. G. (2003). Design and analysis of cross-over trials (2nd ed.). Boca Raton, Fla.: Chapman & Hall/CRC.

## References

* M. Bose and A. Dey (2009). Optimal Crossover Designs. World Scientific. ISBN 978-9812818423
* D. E. Johnson (2010). Crossover experiments. WIREs Comp Stat, 2: 620–625. [1]
* Jones, Byron; Kenward, Michael G. (2014). Design and Analysis of Cross-Over Trials (Third ed.). London: Chapman and Hall. ISBN 978-0412606403.
* K.-J. Lui, (2016). Crossover Designs: Testing, Estimation, and Sample Size. Wiley.
* Najafi Mehdi, (2004). Statistical Questions in Evidence Based Medicine. New York: Oxford University Press. ISBN 0-19-262992-1
* D. Raghavarao and L. Padgett (2014). Repeated Measurements and Cross-Over Designs. Wiley. ISBN 978-1-118-70925-2
* D. A. Ratkowsky, M. A. Evans, and J. R. Alldredge (1992). Cross-Over Experiments: Design, Analysis, and Application. Marcel Dekker. ISBN 978-0824788926
* Senn, S. (2002). Cross-Over Trials in Clinical Research, Second edition. Wiley. ISBN 978-0-471-49653-3
* Stufken, J. (1996). "Optimal Crossover Designs". Design and Analysis of Experiments. Handbook of Statistics. 13. North-Holland. pp. 63–90. ISBN 978-0-444-82061-7.
* Vonesh, Edward F.; Chinchilli, Vernon G. (1997). "Crossover Experiments". Linear and Nonlinear Models for the Analysis of Repeated Measurements. London: Chapman and Hall. pp. 111–202. ISBN 978-0824782481.

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Design of experiments

Scientific
method|

* Scientific experiment
* Statistical design
* Control
* Internal and external validity
* Experimental unit
* Blinding

* Optimal design: Bayesian
* Random assignment
* Randomization
* Restricted randomization
* Replication versus subsampling
* Sample size

Treatment
and blocking|

* Treatment
* Effect size
* Contrast
* Interaction
* Confounding
* Orthogonality
* Blocking
* Covariate
* Nuisance variable

Models
and inference|

* Linear regression
* Ordinary least squares
* Bayesian

* Random effect
* Mixed model
* Hierarchical model: Bayesian

* Analysis of variance (Anova)
* Cochran's theorem
* Manova (multivariate)
* Ancova (covariance)

* Compare means
* Multiple comparison

Designs

Completely
randomized|

* Factorial
* Fractional factorial
* Plackett-Burman
* Taguchi

* Response surface methodology
* Polynomial and rational modeling
* Box-Behnken
* Central composite

* Block
* Generalized randomized block design (GRBD)
* Latin square
* Graeco-Latin square
* Orthogonal array
* Latin hypercube
Repeated measures design
* Crossover study

* Randomized controlled trial
* Sequential analysis
* Sequential probability ratio test

* Glossary
* Category
*   * Statistical outline
* Statistical topics

* v
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* e

Statistics

* Outline
* Index

Descriptive statistics

| Continuous data|

| Center|

* Mean
* arithmetic
* geometric
* harmonic
* Median
* Mode

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Dispersion|

* Variance
* Standard deviation
* Coefficient of variation
* Percentile
* Range
* Interquartile range

Shape|

* Central limit theorem
* Moments
* Skewness
* Kurtosis
* L-moments

Count data|

* Index of dispersion

Summary tables|

* Grouped data
* Frequency distribution
* Contingency table

Dependence|

* Pearson product-moment correlation
* Rank correlation
* Spearman's rho
* Kendall's tau
* Partial correlation
* Scatter plot

Graphics|

* Bar chart
* Biplot
* Box plot
* Control chart
* Correlogram
* Fan chart
* Forest plot
* Histogram
* Pie chart
* Q–Q plot
* Run chart
* Scatter plot
* Stem-and-leaf display
* Radar chart
* Violin plot

Data collection

| Study design|

* Population
* Statistic
* Effect size
* Statistical power
* Optimal design
* Sample size determination
* Replication
* Missing data

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Survey methodology|

* Sampling
* stratified
* cluster
* Standard error
* Opinion poll
* Questionnaire

Controlled experiments|

* Scientific control
* Randomized experiment
* Randomized controlled trial
* Random assignment
* Blocking
* Interaction
* Factorial experiment

Adaptive Designs|

* Adaptive clinical trial
* Up-and-Down Designs
* Stochastic approximation

Observational Studies|

* Cross-sectional study
* Cohort study
* Natural experiment
* Quasi-experiment

Statistical inference

| Statistical theory|

* Population
* Statistic
* Probability distribution
* Sampling distribution
* Order statistic
* Empirical distribution
* Density estimation
* Statistical model
* Model specification
* Lp space
* Parameter
* location
* scale
* shape
* Parametric family
* Likelihood (monotone)
* Location–scale family
* Exponential family
* Completeness
* Sufficiency
* Statistical functional
* Bootstrap
* U
* V
* Optimal decision
* loss function
* Efficiency
* Statistical distance
* divergence
* Asymptotics
* Robustness

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Frequentist inference|

| Point estimation|

* Estimating equations
* Maximum likelihood
* Method of moments
* M-estimator
* Minimum distance
* Unbiased estimators
* Mean-unbiased minimum-variance
* Rao–Blackwellization
* Lehmann–Scheffé theorem
* Median unbiased
* Plug-in

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Interval estimation|

* Confidence interval
* Pivot
* Likelihood interval
* Prediction interval
* Tolerance interval
* Resampling
* Bootstrap
* Jackknife

Testing hypotheses|

* 1- & 2-tails
* Power
* Uniformly most powerful test
* Permutation test
* Randomization test
* Multiple comparisons

Parametric tests|

* Likelihood-ratio
* Score/Lagrange multiplier
* Wald

Specific tests|

* Z-test (normal)
* Student's t-test
* F-test

Goodness of fit|

* Chi-squared
* G-test
* Kolmogorov–Smirnov
* Anderson–Darling
* Lilliefors
* Jarque–Bera
* Normality (Shapiro–Wilk)
* Likelihood-ratio test
* Model selection
* Cross validation
* AIC
* BIC

Rank statistics|

* Sign
* Sample median
* Signed rank (Wilcoxon)
* Hodges–Lehmann estimator
* Rank sum (Mann–Whitney)
* Nonparametric anova
* 1-way (Kruskal–Wallis)
* 2-way (Friedman)
* Ordered alternative (Jonckheere–Terpstra)

Bayesian inference|

* Bayesian probability
* prior
* posterior
* Credible interval
* Bayes factor
* Bayesian estimator
* Maximum posterior estimator

* Correlation
* Regression analysis

| Correlation|

* Pearson product-moment
* Partial correlation
* Confounding variable
* Coefficient of determination

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Regression analysis|

* Errors and residuals
* Regression validation
* Mixed effects models
* Simultaneous equations models
* Multivariate adaptive regression splines (MARS)

Linear regression|

* Simple linear regression
* Ordinary least squares
* General linear model
* Bayesian regression

Non-standard predictors|

* Nonlinear regression
* Nonparametric
* Semiparametric
* Isotonic
* Robust
* Heteroscedasticity
* Homoscedasticity

Generalized linear model|

* Exponential families
* Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance|

* Analysis of variance (ANOVA, anova)
* Analysis of covariance
* Multivariate ANOVA
* Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

| Categorical|

* Cohen's kappa
* Contingency table
* Graphical model
* Log-linear model
* McNemar's test

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Multivariate|

* Regression
* Manova
* Principal components
* Canonical correlation
* Discriminant analysis
* Cluster analysis
* Classification
* Structural equation model
* Factor analysis
* Multivariate distributions
* Elliptical distributions
* Normal

Time-series|

| General|

* Decomposition
* Trend
* Stationarity
* Seasonal adjustment
* Exponential smoothing
* Cointegration
* Structural break
* Granger causality

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Specific tests|

* Dickey–Fuller
* Johansen
* Q-statistic (Ljung–Box)
* Durbin–Watson
* Breusch–Godfrey

Time domain|

* Autocorrelation (ACF)
* partial (PACF)
* Cross-correlation (XCF)
* ARMA model
* ARIMA model (Box–Jenkins)
* Autoregressive conditional heteroskedasticity (ARCH)
* Vector autoregression (VAR)

Frequency domain|

* Spectral density estimation
* Fourier analysis
* Wavelet
* Whittle likelihood

Survival|

| Survival function|

* Kaplan–Meier estimator (product limit)
* Proportional hazards models
* Accelerated failure time (AFT) model
* First hitting time

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Hazard function|

* Nelson–Aalen estimator

Test|

* Log-rank test

Applications

| Biostatistics|

* Bioinformatics
* Clinical trials / studies
* Epidemiology
* Medical statistics

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Engineering statistics|

* Chemometrics
* Methods engineering
* Probabilistic design
* Process / quality control
* Reliability
* System identification

Social statistics|

* Actuarial science
* Census
* Crime statistics
* Demography
* Econometrics
* Jurimetrics
* National accounts
* Official statistics
* Population statistics
* Psychometrics

Spatial statistics|

* Cartography
* Environmental statistics
* Geographic information system
* Geostatistics
* Kriging

* Category
* '
* Commons
* WikiProject

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