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A Bland–Altman plot (difference plot) in analytical chemistry or biomedicine is a method of data plotting used in analyzing the agreement between two different assays. It is identical to a Tukey mean-difference plot,^[1] the name by which it is known in other fields, but was popularised in medical statistics by J. Martin Bland and Douglas G. Altman.^[2][3]

Agreement versus correlation[edit]

Bland and Altman make the point that any two methods that are designed to measure the same parameter (or property) should have good correlation when a set of samples are chosen such that the property to be determined varies considerably. A high correlation for any two methods designed to measure the same property could thus in itself just be a sign that one has chosen a widespread sample. A high correlation does not necessarily imply that there is good agreement between the two methods.

Construction[edit]

Consider a sample consisting of ${\displaystyle n}$ observations (for example, objects of unknown volume). Both assays (for example, different methods of volume measurement) are performed on each sample, resulting in ${\displaystyle 2n}$ data points. Each of the ${\displaystyle n}$ samples is then represented on the graph by assigning the mean of the two measurements as the ${\displaystyle x}$-value, and the difference between the two values as the ${\displaystyle y}$-value.

The Cartesian coordinates of a given sample ${\displaystyle S}$ with values of ${\displaystyle S_1}$ and ${\displaystyle S_2}$ determined by the two assays is

${\displaystyle S(x,y)=\left(\frac{S_1+S_2}{2},S_1-S_2\right).}$

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.^[4] 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:

${\displaystyle S(x,y)=\left(\frac{{\log }_2{S}_1+{\log }_2{S}_2}{2},{\log }_2{S}_1-{\log }_2{S}_2\right).}$

This version of the plot is used in MA plot.

Application[edit]

One primary application of the Bland–Altman plot is to compare two clinical measurements each of which produced some error in their measures.^[5] It can also be used to compare a new measurement technique or method with a gold standard, as even a gold standard does not—and should not—imply it to be without error.^[4] See Analyse-it, MedCalc, NCSS, GraphPad Prism, R, or StatsDirect for software providing Bland–Altman plots.

Bland–Altman plots are extensively used to evaluate the agreement among two different instruments or two measurements techniques. Bland–Altman plots allow identification of any systematic difference between the measurements (i.e., fixed bias) or possible outliers. The mean difference is the estimated bias, and the SD of the differences measures the random fluctuations around this mean. If the mean value of the difference differs significantly from 0 on the basis of a 1-sample t-test, this indicates the presence of fixed bias. If there is a consistent bias, it can be adjusted for by subtracting the mean difference from the new method. It is common to compute 95% limits of agreement for each comparison (average difference ± 1.96 standard deviation of the difference), which tells us how far apart measurements by two methods were more likely to be for most individuals. If the differences within mean ± 1.96 SD are not clinically important, the two methods may be used interchangeably. 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 ^[3] or by more precise methods.^[6]

Bland–Altman plots were also used to investigate any possible relationship of the discrepancies between the measurements and the true value (i.e., proportional bias). The existence of proportional bias indicates that the methods do not agree equally through the range of measurements (i.e., the limits of agreement will depend on the actual measurement). To evaluate this relationship formally, the difference between the methods should be regressed on the average of the 2 methods. When a relationship between the differences and the true value was identified (i.e., a significant slope of the regression line), regression-based 95% limits of agreement should be provided.^[4]

Notes[edit]

A similar method was proposed in 1981 by Eksborg.^[7] This method was based on Deming regression—a method introduced by Adcock in 1878.

Bland and Altman's Lancet paper ^[3] was number 29 in a list of the top 100 most-cited papers of all time with over 23,000 citations.^[8]

See also[edit]

References[edit]

1. ^Cleveland, William S. (1993). Visualizing data. Murray Hill, N.J.: At & T Bell Laboratories. pp. 22–23. ISBN978-0963488404. OCLC29456028.
2. ^Altman DG, Bland JM (1983). 'Measurement in medicine: the analysis of method comparison studies'. The Statistician. 32 (3): 307–317. doi:10.2307/2987937. JSTOR2987937.
3. ^ ^abcBland JM, Altman DG (1986). 'Statistical methods for assessing agreement between two methods of clinical measurement'(PDF). Lancet. 327 (8476): 307–10. CiteSeerX10.1.1.587.8931. doi:10.1016/S0140-6736(86)90837-8. PMID2868172. S2CID2844897.
4. ^ ^abcBland JM, Altman DG (1999). 'Measuring agreement in method comparison studies'. Statistical Methods in Medical Research. 8 (2): 135–60. doi:10.1177/096228029900800204. PMID10501650. S2CID9851097.
5. ^Hanneman SK (2008). 'Design, analysis, and interpretation of method-comparison studies'. AACN Advanced Critical Care. 19 (2): 223–234. doi:10.1097/01.AACN.0000318125.41512.a3. PMC2944826. PMID18560291.
6. ^Carkeet A (2015). 'Exact parametric confidence intervals for Bland–Altman Limits of Agreement'(PDF). Optometry and Vision Science. 92 (3): e71–e80. doi:10.1097/OPX.0000000000000513. PMID25650900. S2CID11643889.
7. ^Eksborg S (1981) Evaluation of method-comparison data. Clin Chem 27:1311–1312
8. ^Van Noorden, Richard; Maher, Brendan; Nuzzo, Regina (2014). 'The top 100 papers'. Nature. 514 (7524): 550–553. doi:10.1038/514550a. ISSN0028-0836. PMID25355343.

Plot y versus x as lines and/or markers.

Call signatures:

The coordinates of the points or line nodes are given by x, y.

The optional parameter fmt is a convenient way for defining basicformatting like color, marker and linestyle. It's a shortcut stringnotation described in the Notes section below.

You can use Line2D properties as keyword arguments for morecontrol on the appearance. Line properties and fmt can be mixed.The following two calls yield identical results:

When conflicting with fmt, keyword arguments take precedence.

Plotting labelled data

There's a convenient way for plotting objects with labelled data (i.e.data that can be accessed by index obj['y']). Instead of givingthe data in x and y, you can provide the object in the dataparameter and just give the labels for x and y:

All indexable objects are supported. This could e.g. be a dict, apandas.DataFame or a structured numpy array.

Plotting multiple sets of data

There are various ways to plot multiple sets of data.

• The most straight forward way is just to call plot multiple times.Example:

• Alternatively, if your data is already a 2d array, you can pass itdirectly to x, y. A separate data set will be drawn for everycolumn.

  Example: an array a where the first column represents the xvalues and the other columns are the y columns:

• The third way is to specify multiple sets of [x], y, [fmt]groups:

  In this case, any additional keyword argument applies to alldatasets. Also this syntax cannot be combined with the dataparameter.

By default, each line is assigned a different style specified by a'style cycle'. The fmt and line property parameters are onlynecessary if you want explicit deviations from these defaults.Alternatively, you can also change the style cycle using the'axes.prop_cycle' rcParam.

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See also

scatter

XY scatter plot with markers of varying size and/or color ( sometimes also called bubble chart).

Notes

Format Strings

A format string consists of a part for color, marker and line:

Each of them is optional. If not provided, the value from the stylecycle is used. Exception: If line is given, but no marker,the data will be a line without markers.

Other combinations such as [color][marker][line] are alsosupported, but note that their parsing may be ambiguous.

Markers

Line Styles

Example format strings:

Colors

Youtube Plot Diagram

The supported color abbreviations are the single letter codes

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and the 'CN' colors that index into the default property cycle.

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If the color is the only part of the format string, you canadditionally use any matplotlib.colors spec, e.g. full names('green') or hex strings ('#008000').