Method comparison & evaluation
The Bland-Altman plot (Bland & Altman, 1986 and 1999), or difference plot, is a graphical method to compare two measurements techniques. In this graphical method the differences (or alternatively the ratios) between the two techniques are plotted against the averages of the two techniques. Alternatively (Krouwer, 2008) the differences can be plotted against one of the two methods, if this method is a reference or "gold standard" method.
Horizontal lines are drawn at the mean difference, and at the limits of agreement, which are defined as the mean difference plus and minus 1.96 times the standard deviation of the differences.
If you have duplicate or multiple measurements per subject for each method, see Bland-Altman plot with multiple measurements per subject.
You can select the following variations of the Bland-Altman plot (see Bland & Altman, 1995; Bland & Altman, 1999; Krouwer, 2008):
- Plot against (X-axis)
In the original Bland-Altman plot (Bland & Altman, 1986) the differences* between the two methods are plotted against the averages of the two methods.
Alternatively, you can choose to plot the differences*
- against one of the two methods, if this is a reference or "gold standard" method (Krouwer, 2008)
- against the geometric mean of both methods
- against the sample rank (where the samples are ranked by the average of the two methods), the rank of the first method, or the rank of the second method; these options will give a "Ranked order difference plot" according to CLSI, 2013.
(*) or ratios when this option is selected (see below).
- Plot differences
This is the default option corresponding to the methodology of Bland & Altman, 1986.
- Plot differences as %
When selecting this option the differences will be expressed as percentages of the observations represented on the X-axis (i.e. proportionally to the magnitude of measurements). This option is useful when there is an increase in variability of the differences as the magnitude of the measurement increases.
- Plot ratios
When this option is selected then the ratios of the measurements will be plotted instead of the differences (MedCalc performs the calculations on log-transformed data in the background). This option as well is useful when there is an increase in variability of the differences as the magnitude of the measurement increases. However, the program will give a warning when either one of the two techniques includes zero values.
- Maximum allowed difference between methods: (optionally) the pre-defined clinical agreement limit D. Depending on the option (Plot differences or ratios) selected above, a difference, a difference expressed as a percentage, or a ratio must be entered.
The value D must be chosen so that differences in the range −D to D (for ratios 1/D to D) are clinically irrelevant or neglectable.
- Draw line of equality: useful for detecting a systematic difference.
- 95% CI of mean difference*: the 95% Confidence Interval of the mean difference illustrates the magnitude of the systematic difference. If the line of equality is not in the interval, there is a significant systematic difference.
- 95% CI of limits of agreement: shows error bars representing the 95% confidence interval for both the upper and lower limits of agreement (Bland & Altman, 1999).
- Draw regression line of differences*: this regression line may help to detect a proportional difference. The regression parameters are shown in the graph's info panel. Optionally, you can select to show the 95% confidence interval of this regression line.
- Click if you want to identify subgroups in the Bland-Altman plot. A new dialog box is displayed in which you can select a categorical variable. The graph will use different markers for the different categories in this variable.
(*) or ratios when this option is selected.
It is recommended (Stöckl et al., 2004; Abu-Arafeh et al., 2016) to enter a value for the "Maximum allowed difference between methods", and select the option "95% CI of limits of agreement".
How to define the Maximum allowed difference
Jensen & Kjelgaard-Hansen (2010) give two approaches to define acceptable differences between two methods.
- In the first approach the combined inherent imprecision of both methods is calculated (CV2method1 + CV2method2)1/2, or in case of duplicate measurements [(CV2method1 /2)+ (CV2method2)/2)]1/2.
- In the second approach acceptance limits are based on analytical quality specifications such as for example reported by the Clinical Laboratory Improvement Amendments (CLIA).
A third approach might be to base acceptance limits on clinical requirements. If the observed random differences are too small to influence diagnosis and treatment, these differences may be acceptable and the two laboratory methods can be considered to be in agreement.
After you click OK you obtain the following graph:
The graph displays a scatter diagram of the differences plotted against the averages of the two measurements. Horizontal lines are drawn at the mean difference, and at the limits of agreement.
The limits of agreement (LoA) are defined as the mean difference ± 1.96 SD of differences. If these limits do not exceed the maximum allowed difference between methods Δ (the differences within mean ± 1.96 SD are not clinically important), the two methods are considered to be in agreement and may be used interchangeably.
Proper interpretation (Stöckl et al., 2004) takes into account the 95% confidence interval of the LoA, and to be 95% certain that the methods do not disagree, Δ must be higher than the upper 95 CI limit of the higher LoA and −Δ must be less than the lower %95 CI limit of the lower LoA:
To get more statistical information, right-click in the graph window and click Info on the context menu:
The Bland-Altman plot is useful to reveal a relationship between the differences and the magnitude of measurements (examples 1 & 2), to look for any systematic bias (example 3) and to identify possible outliers. If there is a consistent bias, it can be adjusted for by subtracting the mean difference from the new method.
Optionally, confidence intervals may be displayed for the average difference and for the limits of agreement. These confidence intervals can be represented as error bars or horizontal lines. Right-click on the error bar to set formatting options.
Some typical situations are shown in the following examples.
Example 1: Case of a proportional error.
Example 2: Case where the variation of at least one method depends strongly on the magnitude of measurements.
Example 3: Case of an absolute systematic error.
The Bland-Altman plot may also be used to assess the repeatability of a method by comparing repeated measurements using one single method on a series of subjects. The graph can then also be used to check whether the variability or precision of a method is related to the size of the characteristic being measured.
Since for the repeated measurements the same method is used, the mean difference should be zero. Therefore the Coefficient of Repeatability (CR) can be calculated as 1.96 (or 2) times the standard deviation of the differences between the two measurements (d2 and d1) (Bland & Altman, 1986):
The 95% confidence interval for the Coefficient of Repeatability is calculated according to Barnhart & Barborial, 2009.
To obtain this coefficient in MedCalc you
- create a Bland & Altman plot for the two measurements
- right-click in the display and click Info on the context menu
- in the info panel, click Coefficient of Repeatability
The coefficient of repeatability is not reported when you have selected the "Plot ratios" method.
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- Barnhart HX, Barborial DP (2009) Applications of the repeatability of quantitative imaging biomarkers: a review of statistical analysis of repeat data sets. Translational Oncology 2:231-235.
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