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Bland–Altman plot

A Bland–Altman plot in analytical chemistry or biomedicine is a method of data plotting used in analyzing the agreement between two different assays. It was popularised in medical statistics by J. Martin Bland and Douglas G. Altman. In other fields, it is also called a Tukey mean-difference plot, named after John Tukey.

Construction
Consider a sample consisting of n observations (for example, objects of unknown volume). Both assays (for example, different methods of volume measurement) are performed on each sample, resulting in 2n data points. Each of the n samples is then represented on the graph by assigning the mean of the two measurements as the x-value, and the difference between the two values as the y-value. The Cartesian coordinates of a given sample S with values of S_1 and S_2 determined by the two assays is : S(x,y)=\left( \frac{S_1+S_2}{2}, S_1-S_2 \right). For comparing the dissimilarities between the two sets of samples independently from their mean values, it is more appropriate to look at the ratio of the pairs of measurements. Log transformation (base 2) of the measurements before the analysis will enable the standard approach to be used; so the plot will be given by the following equation: : S(x,y)=\left( \frac{\log_2 S_1 +\log_2 S_2}{2}, \log_2 S_1 -\log_2 S_2 \right). This version of the plot is used in MA plot. == Interpretation ==
Interpretation
Interpretation of a Bland-Altman plot is contingent on the construction of the plot and data at hand. Variations to the default plot have introduced throughout the years and each should be interpreted accordingly. Original construction The original plot displays a scatter plot of differences between individual data points. The differences should be of the new reference system minus a gold standard. The 95% limits of agreement can be unreliable estimates of the population parameters especially for small sample sizes so, when comparing methods or assessing repeatability, it is important to calculate confidence intervals for 95% limits of agreement. This can be done by Bland and Altman's approximate method Sample size and power estimation Determining an adequate sample size is a key consideration in Bland–Altman analysis, as it influences the precision of the estimated limits of agreement and the statistical power to detect clinically meaningful differences between measurement methods. Historically, there has been limited formal guidance on how to perform power or sample size calculations for Bland–Altman studies. Early recommendations by Martin Bland suggested estimating sample size from the expected width of the confidence interval for the limits of agreement, an approach that does not explicitly account for Type II error and may yield insufficient sample sizes for typical study designs. A more rigorous approach was later introduced by Lu et al. (2016), who proposed a statistical framework for assessing power and determining sample size based on the distribution of measurement differences and predefined limits of clinical agreement. Their method explicitly incorporates Type II error control and provides more accurate estimates of required sample sizes for studies targeting a given statistical power, typically 80%. Simulation studies in that work demonstrated good performance of the method under practical conditions; however, the authors did not provide publicly available software to implement the approach. Several software packages now include implementations of the Lu et al. methodology. The commercial MedCalc statistical software provides sample size and power estimation tools for Bland–Altman analyses. In addition, an open-source implementation is available in the R package blandPower, which provides functions to estimate power curves, determine required sample sizes, and visualize confidence interval widths as a function of sample size. The blandPower package was developed to promote reproducibility and accessibility of power and sample size calculations for method comparison studies using the Bland–Altman framework. Visualization variations In the case that the differences grow proportionally to the magnitude of the data, then the data is said to have a 'proportional bias'. There are many methods for visualizing the plot and subsequent analysis to accommodate for it. Firstly, a linear regression could illustrate any relevant trends. If the distribution of differences are equal at all points around the regression the data is said to be homoscedastic and the trend is a simple proportional bias. Inversely, if the data has wider spread at different magnitudes of the data, then the differences are said to be heteroscedastic, which has further implications. Statistical tests such as the Breusch–Pagan test or the White test can provide statistical indicators of heteroscedasticity.One typical example of a plot with heteroscedastic data is one whose variation of differences grows proportional to the magnitude of the data, visualized as an expanding 'v' shape. Similarly, the plot of differences could be visualized logarithmically. In either case, the relationship between the two systems illustrates a multiplicative relationship as opposed to linear one. This also indicates that the magnitude of the data correlates with variations of accuracy for the systems. == Application ==
Application
One primary application of the Bland-Altman plot is to compare two clinical measurements that produce continuous output. It can be used to compare a new reference system, technique, or method with a verified gold standard, but a gold standard does not imply it to be without error. Many recommendations and scholarly articles have also been published in efforts of polishing the technique, the underlying statistical construction, and validity of the plot. See Analyse-it, MedCalc, NCSS, GraphPad Prism, R, StatsDirect, or JASP for software providing Bland–Altman plots. == See also ==
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