# Tracking Changes in Cardiac Output: Statistical Considerations on the 4-Quadrant Plot and the Polar Plot Methodology

When comparing 2 technologies for measuring hemodynamic parameters with regard to their ability to track changes, 2 graphical tools are omnipresent in the literature: the 4-quadrant plot and the polar plot recently proposed by Critchley et al. The polar plot is thought to be the more advanced statistical tool, but care should be taken when it comes to its interpretation. The polar plot excludes possibly important measurements from the data. The polar plot transforms the data nonlinearily, which may prevent it from being seen clearly. In this article, we compare the 4-quadrant and the polar plot in detail and thoroughly describe advantages and limitations of each. We also discuss pitfalls concerning the methods to prepare the researcher for the sound use of both methods. Finally, we briefly revisit the Bland-Altman plot for the use in this context.

From the ^{*}Department of Anesthesiology, Center of Anesthesiology and Intensive Care Medicine, University Medical Center Hamburg-Eppendorf, Hamburg, Germany; and ^{â€ }Department of Economic and Social Statistics, University of Cologne, Cologne, Germany and Institute of Operations Research, Karlsruhe Institute of Technology, Karlsruhe, Germany.

Accepted for publication January 15, 2015.

Funding: None.

The authors declare no conflicts of interest.

Address correspondence to Julia Y. Wagner, MD, Department of Anesthesiology, Center of Anesthesiology and Intensive Care Medicine, University Medical Center Hamburg-Eppendorf, Martinistrasse 52, 20246 Hamburg, Germany. Address e-mail to jwagner.science@gmail.com.

Because cardiac output (CO) is an important hemodynamic parameter in caring for hemodynamically unstable patients, studies describing novel technologies for CO assessment are of high interest in the fields of perioperative and critical care medicine. In these studies, an innovative method for CO determination (i.e., studied technology) is usually compared with an established reference technology using different statistical methodologies. Methods for the assessment of the accuracy and precision of a studied technology (e.g., Bland-Altman analysis^{1},^{2} and calculation of the percentage error^{3}) have been described and discussed in detail elsewhere.^{4},^{5}

Besides describing its absolute accuracy and precision, it is important to assess the ability of a novel technology designed to measure CO to adequately track changes in CO in comparison with the gold standard method. This means that the technologies detect changes in the same direction. Several methods for the evaluation of this trending ability have been described.^{6}

Although the Bland-Altman analysis can provide insights within a trending analysis, 2 of the most frequently used graphical statistical methods in trending analysis are the 4-quadrant plot and the polar plot. We therefore restrict ourselves to the analysis of these 2 methods in this article. (For readers interested in the Bland-Altman analysis, we provide additional treatments in Appendix 1.) The 4-quadrant plot was first used for the description of trending capabilities in studies comparing one CO measurement technology with another by Perrino et al.^{7},^{8} The polar plot was proposed by Critchley et al.^{6} in 2010 as a new alternative method. In their review article published in 2010 and in another article published later,^{9} Critchley et al. described the derivation of the polar plot from the 4-quadrant plot. In their important articles, Critchley et al. demonstrated the importance, and also the complexity, of CO-trending analysis compared with precision analysis and in doing so drew attention to the problem of quantifying the ability of a technology to track CO changes. Since its introduction, numerous studies have used the polar plot analysis to describe the ability of a CO-monitoring device to follow changes in the true CO measured with a gold standard technique.

We argue that a profound understanding of the statistical methods used is a prerequisite for the correct assessment of the trending ability of a CO measurement technology. Whereas the 4-quadrant plot provides a relatively intuitive picture of the analyzed data at hand, the more sophisticated polar plot demands a higher level of insight into its construction to adequately interpret the characteristics of the analyzed data. Therefore, the primary scope of the present article is to describe the computation of the 4-quadrant plot and the polar plot in detail and to derive the relation between these statistical methods. Furthermore, we describe the basic properties of both plots and, in particular, cite possibly dangerous pitfalls when analyzing polar plots. We briefly review the problem of measuring CO and assessing the trending ability of measurement technologies. We then discuss the 4-quadrant plot and the polar plot as proposed by Critchley et al. in detail. Finally, we summarize the advantages and disadvantages of both methods.

## MEASURING COâ€”THE PROBLEM OF TRACKING CHANGES

Pulmonary artery thermodilution,^{10},^{11} single-indicator transpulmonary thermodilution,^{12â€“14} and lithium indicator dilution^{15â€“17} are thought to represent clinical gold standard methods for CO determination and are therefore used as reference technologies in method comparison studies.

Various novel, less invasive, and noninvasive technologies have been described in recent years including pulse contour analysis (both calibrated and autocalibrated),^{18â€“25} esophageal Doppler,^{25},^{26} thoracic electrical bioimpedance and bioreactance,^{27â€“32} and technologies based on the vascular unloading technique,^{33â€“36} pulse wave transit time, and radial artery applanation tonometry.^{37}

When applying different CO measurement technologies, one has to keep in mind that CO can be measured intermittently (e.g., intermittent pulmonary artery thermodilution), continuously (e.g., pulse contour analysis providing a real-time beat-to-beat report), or semicontinuously (e.g., bioreactance-derived CO readings averaged over 60 seconds).

CO is a hemo*dynamic* variable that changes over time and is modified by a variety of factors closely related to oxygen supply and consumption, such as cardiac preload, cardiac afterload, and cardiac contractility. When performing validation studies for CO-monitoring technologies, it has therefore to be kept in mind that both the studied technology and the reference technology are aiming to hit a moving target.

Considering the dynamic nature of CO, evaluating the ability of a technology for CO assessment to trend changes, in addition to assessing its accuracy and precision, is essential for the sound interpretation of the measurement performance of a novel CO-monitoring device. However, adequately describing the ability of a CO-monitoring method to timely track decreases and increases in CO is statistically complex. Several statistical approaches have been described previously.^{6}

A direction of change analysis can be performed by calculating the concordance rate, that is, the ratio (percentage) of CO measurements assessed by the studied technology and the reference technology that change correctly in the same direction (decrease or increase) to the sum of all changes. However, although this direction of change analysis provides information whether the studied technology qualitatively follows CO changes assessed by the reference technology, it does not provide information on the magnitude of the changes in CO or the degree of agreement between the studied technology and the reference technology.^{6}

Therefore, alternative and more sophisticated methods for trend analysis in clinical studies have been described. In recent years, the most widely used methods to illustrate the trending ability of CO-monitoring devices are the 4-quadrant plot and the polar plot.

## FOUR-QUADRANT PLOT ANALYSIS

For the computation of a 4-quadrant plot, Î”CO values (i.e., differences between consecutively obtained CO values) for both the studied technology and the reference technology are calculated and plotted in a scatter plot. Figure 1 shows an example for a 4-quadrant plot with 9 artificial data points. The values on the horizontal axis (usually called the x-axis) refer to Î”CO values of the reference technology, whereas the vertical axis (the y-axis) refers to the Î”CO values of the studied technology. From visual inspection of the resulting scatter plot, one can see the distribution of data points lying within 1 of the 4 quadrants. When both the studied technology and the reference technology indicate an increase in CO, the respective data point will appear in the upper right quadrant of the 4-quadrant plot. Similarly, the lower left quadrant contains data points resulting from concordant CO measurements indicating a decrease in CO. Therefore, the upper right and the lower left quadrants of the 4-quadrant plot represent concordant measurements of the studied technology and the reference technology with regard to direction of changes. In Figure 1, these quadrants are therefore marked by green areas. From the coordinates of 1 data point within the quadrant, the magnitude of change in CO measured by the studied technology and the reference technology can directly be read off. This is an appealing property of the 4-quadrant plot. Data point 8, for example, means that the reference device detected a CO change by 0.5 L/min, whereas the studied device showed a change by 2 L/min. Although these measurements are concordant, in the sense that both devices indicated a positive change in the CO, the numerical values are not equal. Points with equal numerical values are located on the 45Â° diagonal within the quadrant (the dotted line in the green quadrants in Fig. 1). In data point 5, for example, both devices detect a CO change by 1 L/min.

When measurements of Î”CO obtained with the devices disagree with regard to the direction of change (i.e., the studied technology indicates an increase in CO while the reference technology indicates a decrease in CO or vice versa), the respective data points will appear in the upper left or lower right quadrant of the plot. These quadrants are therefore marked by red areas in Figure 1. Again, values of the points on the horizontal axis refer to changes indicated by the reference device, and the values on the vertical axis refer to changes indicated by the studied device. Data point 9 reflects a detected change by the reference device of âˆ’0.3 and of 2 L/min by the studied device. Situations in which both devices show changes of the same absolute values but in opposite directions are reflected by points on the decreasing 45Â° diagonal within the red quadrants (the dotted line in the red quadrants in Fig. 1). In data point 1, for example, the reference device measured a change of âˆ’2.5 L/min, whereas the studied device showed a positive change by 2.5 L/min.

The higher the number of data points in the green quadrants compared with the number of data points in the red quadrants, the higher is the concordance between the measurement devices. The simplest way to further quantify the level of concordance is to calculate the proportion of data points in the quadrants representing direction of change agreement (green quadrants) in all data points. However, many other concordance measures have been proposed. (We refer to Nelsen^{38} for an overview.)

Because no clinically applicable CO measurement system is perfectly accurate and precise, very small changes in CO readings may be attributed to noise and are supposed to not contribute sufficiently to or even disturb trending analysis. Therefore, it was suggested that an exclusion zone be defined at the center of the 4-quadrant plot to remove measurements driven by noise and increase the signal-to-noise ratio. Points of this zone that are also considered to represent clinically insignificant changes are excluded from further analysis. In Figure 1, an exclusion zone for absolute changes below 0.5 L/min is marked by the gray area. Data points 6 and 7 fall in this area and should therefore not be used in the assessment of the trending ability of CO-monitoring technologies. In this example, both points would indicate nonconcordant measurements. However, because of their small absolute values, it is not clear whether they represent real changes in the measurements or are mainly driven by noise.

In summary, the 4-quadrant plot is an intuitive tool to illustrate the trending ability of measurement devices that allows for fast visual assessment of the characteristics of the studied technology and the reference technology. It is important to note that not only the quadrant of a point is important. From the x and y coordinates of a data point, we also obtain information about the magnitude and direction of CO changes of both technologies.

For example, Figure 2 shows 4-quadrant plots for 4 different situations. Clearly, Figure 2A shows a situation with low trending ability. The measurements in this example are, in fact, completely independent of each other. Figure 2B shows a better trending ability, whereas Figure 2D shows a quite good trending ability with only very few discordant measurements. Figure 2C shows a large number of discordant measurements. Here, the studied device tends to indicate changes in the opposite direction of the reference device.

A limitation of the 4-quadrant plot and concordance analysis is the lack of clearly defined cutoff values for the definition of good, acceptable, and poor agreement. Many such values have been suggested previously, but there are no generally accepted thresholds to describe the trending ability of CO measurement technologies. Also, because the results of the 4-quadrant plot analysis depend on the time interval between consecutive measurements, the plot can be influenced by choosing different time intervals for the analysis.

As described above, very small Î”CO values should not be included in the trending analysis, and, thus, a central exclusion zone should be applied. Authors normally use a Î”CO exclusion zone of 0.5 L/min or 10%. However, the exclusion zone should be adapted considering the range of Î”CO values observed in the study population and the time interval between CO readings used for the calculation of Î”CO.

Further, it should be remembered that, in addition to small Î”CO values, very large Î”CO values also might limit the validity of trending analysis. Whether zones excluding very high Î”CO values should be used in the 4-quadrant plot analysis is still a matter of debate.^{6}

## POLAR PLOT ANALYSIS

Basically, the polar plot by Critchley et al. is methodologically derived from a 4-quadrant plot and is supposed to be a more advanced statistical method for the description of the trending ability of a CO monitor. In the following section, we provide a detailed and critical analysis of this statistical approach by using worked examples. While explaining the individual steps of derivation of the polar plot from the 4-quadrant plot, we simultaneously point out some critical aspects of the polar plot methodology that have not yet been previously described.

In general, the polar plot is based on polar coordinates. This means that every point is addressed by (a) an angle and (b) a radius instead of horizontal and vertical coordinates (x, y). The angle (a) is the angle between the horizontal axis and the line from the point of interest to the central point (0, 0). The radius (b) is the distance of the point of interest to the central point (0, 0). In contrast, in the usual Cartesian coordinate system, points are addressed by their coordinates (x, y) on the horizontal and vertical axes. It is important to notice that both ways of addressing the points are mathematically equivalent and do not affect the position of the points. For example, for data point 5 in Figure 1, the angle between the horizontal axis and the line between (1, 1) and (0, 0) is 45Â°, and its distance to the point (0, 0) may be calculated by using the formula of Pythagoras to

. Thus, this point can either be described by the coordinates (x = 1, y = 1), where x and y refer to the horizontal and vertical coordinates, respectively, or by the tuple (angle = 45Â°, radius =

) of angle and radius. Given any of the two, one would find the same point in the plot.

The innovation of the polar plot by Critchley et al. is not only to use the angle and radius to address the points (which would not change the points), but also to transform the points: (a) the angle in the polar plot by Critchley et al. coincides with the angle between the (x, y) to (0,0) line with the 45Â° diagonal (instead of the horizontal axis) in the 4-quadrant plot, (b) the radius in the polar plot by Critchley et al. is calculated as

. The variables x and y refer to the horizontal and vertical coordinates in the 4-quadrant plot again. Figure 3 illustrates the transformation of data points from the 4-quadrant plot to the polar plot by Critchley et al. The left graph is a 4-quadrant plot with the same 9 data points as shown in Figure 1. We did not color the 4 quadrants of the plot as in Figure 1 but added some markings that will be explained later. The right graph of Figure 3 is a polar plot as proposed by Critchley et al.^{6} The numbers around the graph denote the values of angles measured from the horizontal axis. The dotted circles mark the radius measured from the center of the plot with values *r* = 1, 2, and 3 in the plot. Every data point of the 4-quadrant plot is also shown in the polar plot. For example, data point 5 with coordinate (x = 1, y = 1) in the 4-quadrant plot has an angle of 0 with the 45Â° diagonal line (it happens to lie on this line) and it holds

. The point is therefore drawn at angle 0 and radius *r* = 1 in the polar plot. To better illustrate how points transform between the plots, in addition to the blue data points, the 4-quadrant plot shows colored data points that lie in a circle. Transformed to the polar plot by Critchley et al., these points mark 2 circles. The yellow and dark blue points (which lie in the concordance quadrants in the 4-quadrant plot) are transformed to points near the horizontal axis in the polar plot. The red and light blue data points (which lie in the discordance quadrants in the 4-quadrant plot) are transformed to the center of the polar plot. Critchley et al. define an exclusion zone in the polar plot that is marked by the gray circle in the middle of the plot. Points within this area are removed from the analysis. The intention of the exclusion zone is to increase the signal-to-noise ratio. The gray area marked in the 4-quadrant plot on the left side denotes the set of points that would be transformed to the exclusion zone of the polar plot. It is very important to notice that this area not only contains points referring to small changes in the original measurements, but also, the contrary may be true. Points like data point 1, which show a clear discordant behavior of the devices, are mapped to the exclusion zone and are therefore removed from the analysis (also data points 6 and 7 are transformed to this zone). Thus, the exclusion zone of the polar plot excludes all measurements from the data where either both changes are small (=the noisy measurements) or the changes are of similar absolute value but contrary in direction (=the most discordant measurements).

Critchley et al. define points under and above certain horizontal lines in the polar plot as reflecting a low degree of trending capacity and the points between these lines as reflecting a high degree of trending capacity. Such horizontal lines are shown in the polar plot in Figure 3. These lines are transformed to the 4-quadrant plot. The areas outside (i.e., above and under) the horizontal lines in the polar plot correspond to the areas in the North, East, South, and West of the black lines in the 4-quadrant plot, respectively. Data points 8 and 9, and 2 and 4 lie within these areas. From the 4-quadrant plot, we can understand what these areas stand for better than we can understand what they stand for from the polar plot: they refer to the cases when one of the devices shows a relatively large change, whereas the other device shows a rather small change. It is again important to notice that the cases of opposite or nearly opposite changes do not fall into these regions but, rather, into the exclusion zone of the polar plot and are therefore removed from the analysis instead of classified as data points reflecting low trending ability.

Critchley et al. propose further refinements of the polar plot methodology. First, they propose not to use horizontal lines as separators between points standing for high and low trending ability, but, rather, lines with angles of +30Â° and âˆ’30Â° to the horizontal axis. These are shown in Figure 4 together with transformations of these lines into the 4-quadrant plot. It can be seen that these straight lines also correspond to straight lines now and that the angle between them is also 2 Ã— 30Â° = 60Â°. Points that are classified to indicate low trending ability are those lying in the North or the South of the polar plot (corresponding to North-West and to South-East in the 4-quadrant plot). Note, however, that the points in the gray area of the 4-quadrant plot fall again into the exclusion zone of the polar plot and are therefore removed from the analysis. Similar to the bias and limits of agreement that are used in the Bland-Altman method,^{39} Critchley et al. propose the angular bias and the radial limits of agreement to assess trending ability.^{9} The angular bias is the mean of all polar angles from a set of data points and the radial limits of agreement is described as the radial sector that contains 95% of the data points.

To set guidelines for good trending ability, clinical data from different CO measurement method comparison studies were analyzed. Based on these analyses, Critchley et al. defined an angular bias <Â±5Â° and radial limits of agreement <Â±30Â° for good trending ability. The idea of radial limits may lead to the misunderstanding that all points with the same angle in a polar plot reflect the same trending ability. Note, however, that this is not the case and that points with the same angle in the polar plot but different radii may indeed represent very different levels of trending abilities.

A further refinement of the polar plot is to turn all data points that lie on the left-hand side of the graph by 180Â° (half-circle polar plot). This is shown in Figure 5, where the circle of colored points in the 4-quadrant plot is transformed to 2 circles lying on the right-hand side of the polar plot (for convenience, we use 2 half-circles with different radii to avoid overlapping of the circles in the polar plot). Again, data points between the 30Â° lines in the East of the polar plot are classified to denote high trending ability, whereas data points outside this area are classified to reflect low trending ability. Again, all points of the gray area in the 4-quadrant plot are transformed to the exclusion zone of the polar plot and are therefore removed from further analysis.

## FOUR-QUADRANT PLOT OR POLAR PLOT ANALYSISâ€”WHICH METHOD SHOULD BE USED?

The aim of this article is to contribute to a better understanding of the 4-quadrant plot and polar plot as statistical methods referring to tracking changes in CO. For a better overview, the advantages and drawbacks of the 2 techniques are summarized in Table 1. Therefore, the awareness of the major critical aspects of the newer polar plot method is crucial. To sum up these aspects, the polar plot transforms the original data points in a nonlinear way, which makes identification of the actual situations leading to single data points difficult. In addition, the nonlinear transformation leads to the exclusion of parts of the data by mapping them into the so-called exclusion zone: the exclusion zone of the polar plot excludes all measurements from the data not only where both changes are small (=the noisy measurements), but also where the changes are of similar absolute value but contrary in direction, which means the most discordant measurements that correspond to the low trending ability of the 2 technologies.

To further illustrate, 2 simultaneously measured Î”CO data points with the same absolute value (or a close absolute value) but changing in opposite directions (e.g., device 1: +1 L/min and device 2: âˆ’1 L/min) are set to 0 (or fall in the exclusion zone) and are therefore ignored in the polar plot analysis, although they might be of high relevance for evaluating the trending ability of a CO monitor. This becomes obvious because there are no data points on or close to the 90Â° or 270Â° line, respectively, in the polar plot. (To further illustrate these properties and limitations, a number of studies are available to the interested reader that evaluate the trending ability of 2 CO measurement technologies and apply both the 4-quadrant plot and the polar plot methodology.^{40}) Furthermore, adequate interpretation of the trending ability of a CO monitor by using a polar plot is highly demanding. With the refinement to turn all data points on the left-hand side of the polar plot by 180Â°, interpretability becomes even more difficult. With regard to the relatively simple applicability and interpretability of the 4-quadrant plot, the question arises as to whether the complexity of the polar plot is justifiable.

## EXTENSIONS AND PERSPECTIVES

In this section, we address the challenge of trending analysis in patient groups with inhomogeneous levels of CO values.

An important consideration when evaluating the trending ability of the studied CO measurement technology is the fact that the levels of CO values within the studied patient population might lie in a broad range (e.g., between 2 and 15 L/min). If the range of CO values of the different patients within the studied patient group is small, it is perfectly appropriate to use absolute values of CO changes to compare the reference and the studied CO measurement technologies. However, if the range of CO values within the studied patient population is broad, it might be more appropriate to use relative CO changes in the comparison analysis. For example, an absolute decrease in CO of 2 L/min has more importance for a patient with CO values around 3 L/min than for a patient with CO values around 12 L/min. Therefore, if the range of CO values of the different patients within the studied patient group is broad, the CO changes are more comparable if one uses relative CO changes.

Such proportional analyses have also been applied on different topics dealing with inhomogeneous patient data.^{41â€“43}

Next, we would like to address 2 possible extensions of the 4-quadrant plot method. First, we show that measures usually derived from the polar plot (the angular bias and the radial limits of agreement) may as well be derived from the 4-quadrant plot. Second, we apply the 4-quadrant plot method for the case of delayed trending ability, that is, when one of the CO measurement devices reacts faster to CO changes than the other.

The angular bias (<Â±5Â°) and radial limits of agreement (<Â±30Â°) are useful measures to rate and compare the trending ability of different CO measurement technologies and different studies. However, we do not necessarily need the complexity and the drawbacks of the polar plot methodology to calculate the 2 measures. Angular bias is the mean of the angles in the polar plot. Because the angles in the polar plot correspond basically to the angles of the (0, 0; x, y) line to the 45Â° line in the 4-quadrant plot, this number can also be calculated from the points of the 4-quadrant plot. The radial limit is the symmetric angle around the 45Â° line in which 95% of the data points fall and can also be calculated from the points of the 4-quadrant plot. A detailed description of the calculation of the 2 measures is provided in Appendix 2.

The second extension of the 4-quadrant plot methodology considers the following scenario. Both the reference and the studied technologies detect the same changes in CO. However, the studied technology detects these changes with a time delay. If the time delay is small enough, it could be neglected with regard to clinical relevance. However, this clinically irrelevant time delay might have a serious impact on the concordance analysis and thus the measurement performance of the studied technology might be misjudged. Similar to cross-correlation analysis with time delay that is used, say, by engineers in signal analysis, the 4-quadrant plot methodology can be adapted to this case. Therefore, we shift the time series of measurements derived from the studied technology by a certain time lag and again perform our analyses. That is, we calculate the corresponding concordance rate and create the 4-quadrant plot for the shifted studied technologyâ€™s time series. Figure 6 illustrates this for artificial data. Figure 6A shows a 4-quadrant plot for the original data, that is, simultaneously recorded data pairs of CO values. Figure 6, B and C, show analyses in which the studied device is delayed by 2 and 4 time units, respectively. Clearly, in the second case (Fig. 6B), trending capacity is highest. The concordance coefficients are 0.65, 0.97, and 0.65.

Therefore, we suggest extending the trending analysis in CO measurement by accounting for a potential time delay. That is, we suggest defining a clinically acceptable range of time delays before the beginning of the clinical study comparing the trending ability of 2 CO measurement devices. When analyzing the obtained CO data, separate analyses must be performed for each time lag within the previously defined range of time delays. The final evaluation of the measurement performance of the studied technology should then be based on the time lag with the highest agreement.

## CONCLUSIONS

In comparison with the 4-quadrant plot analysis, there is an absence of definite advantages combined with a complex interpretability of the polar plot method. Therefore, the polar plot analysis can currently not be recommended as the superior method for the statistical evaluation of trending ability of a CO monitor compared with the 4-quadrant plot. Furthermore, accounting for a potential time delay between 2 CO measurement technologies is an important aspect in the field of trending analysis that needs to be addressed.

## APPENDIX 1

### Bland-Altman Analysis in the Context of Trending Analysis

The Bland-Altman analysis is the leading method to assess the accuracy and precision between a reference and a studied technology in cardiac output (CO) measurement comparison studies. In its simplest form, the Bland-Altman plot quantifies how much the reference (e.g., the gold standard) and studied technology may deviate from each other. To this end, it provides boundaries such that 95% of the nonsystematic differences between measurements from reference and studied technology lie within these boundaries (the exact value of 95% holds only if the differences are normally distributed random variables, otherwise it holds only for approximately 95% of the data). If the calculated boundaries of possible differences between the methods are too large to be clinically negligible, the researcher decides that the 2 methods are not interchangeable. However, if the boundaries refer to differences between the methods that are clinically negligible, the Bland-Altman analysis rates them as equally good and thus interchangeable.

To address this mathematically, let

, be the values of the gold standard method (g) and the studied technology (s), respectively, where *N*^{p} denotes the number of patients and *N*_{i} denotes the number of measurements per patient *i*. The Bland-Altman plot displays

on the x-axis and

on the y-axis for

. It is common in the scientific literature to use

for the x-axis values in the Bland-Altman plot rather than only the gold standard

. In their work published in 1995, Bland and Altman illustrate with an example why plotting the difference between

and

against the standard method instead of the average of

and

is misleading.^{44}

One then considers the mean and SD of the differences

on the y-axis for

.

Now it follows that if the differences are normally distributed random variables (which is a reasonable assumption for measurement errors) and independent, 95% of the differences lie between mean (

) minus 1.96 multiplied by SD and mean (

) plus 1.96 multiplied by SD, which are the so-called limits of agreement (see Bland and Altman 1986).^{39} Here, the mean refers to the systematic difference (if we know the mean, we could simply correct for this difference by subtracting it from the measurements of the studied technology) and 2 Ã— 1.96 = 3.92 multiplied by the SD to the boundaries for nonsystematic differences, respectively. These numbers hold if the data are independent and the true SD is known. Having to estimate the SD from the data, however, one should provide confidence bands for the estimates, that is, for the estimated limits of agreement. To this end, Bland and Altman estimate the variance of the limits of agreement by

, where

is the estimated variance of the differences above. Confidence bands for the limits of agreement will be *t* multiplied by

to each side, where *t* is the 95% quantile of the *t* distribution with n â€“ 1 degrees of freedom. Note that this formula holds only for independent data.

Often the data are not independent and identically distributed, for example, when >1 measurement in 1 individual is performed (see Bland and Altman 2012 for an overview).^{2} Extensions and improvements of the Bland-Altman analysis therefore deal in particular with the calculation of the SD and the limits above in such cases.^{45},^{46} Usually, one distinguishes between repeated and nonrepeated measurements. The term repeated measurements refers to a scenario in which we perform >1 measurement in the same patient. In this scenario, we have to further distinguish between cases in which the true value is constant and cases in which the true value varies over time (e.g., when measuring a patientâ€™s CO). In the latter case, we are using the repeated measurements to monitor, the patientâ€™s CO, for instance. If the true value is constant, repeating the measurement results in a higher precision of the measurements because the measurement errors average out across the measurements. If the true value varies over time, we do not increase the precision with each measurement. Furthermore, it is important whether we perform measurements in only 1 patient or in several patients. The variability of the measurements may be different from 1 patient to another (so called fixed effect on the patient). Again, denote the measurements with the 2 methods by g (gold standard) and by s (studied technology) and note that we are interested in the variance of D = g âˆ’ s. Partitioning of the variances of each method leads to

and

, where

is the variance due to changes in true value over time,

and

are variances due to different variations from patient to patient (fixed effects), and

and

are the variances of the measurement errors. The estimation of the variances of g and s is based on estimating all these single terms.

For the sake of simplicity, we now focus on the case where we only monitor 1 patient (so

). The true value, however, varies over time (so

is not equal to 0). Because we do not know the true value, we cannot estimate

without further information. But, if we look at the time series of the differences D = g âˆ’ s instead, the variation of the true value cancels out, and we find

, which is the sum of the variances of the measurement errors without further effects. Bland and Altman assume for this case (see, e.g., Bland and Altman, 1999, section 5.3^{47}) that the single measurements are independent, so that we can estimate Var(g âˆ’ s) by calculating the SD between the differences D = g âˆ’ s of the measurements. Furthermore, we can calculate the confidence bands for the limits of agreement as stated above for the independence case.

Before applying the Bland-Altman plot for trending analysis, we should recall the essence of trending analysis. Two methodologies show a good trending if they react on changes in the same direction. The idea is that both methods could be used to monitor and stabilize a patient.

When applying Bland-Altman analysis in this context, there are generally 2 different possibilities. First, one could apply it directly to the measurements as summarized above, and second one could apply the Bland-Altman plot to

for

, that is, to the changes measured by each methodology instead of the actual measurements.

In Figure 7, we visualize both ways for a simulation example. The data points are deliberately chosen to illustrate the challenges of the Bland-Altman plot in the context of trending analysis. Multiple measurements in 1 subject are shown. Figure 7A shows CO measurements done every 5 minutes while monitoring a patient who has been stable in CO for longer. Suddenly, the patientâ€™s CO decreases. The CO then increases again after some intervention. It can be seen from the graph that both CO measurement devices detect the CO change but react with different sensitivity. However, both devices are suitable for detecting a patientâ€™s decrease in CO. This is illustrated in the 4-quadrant plot (Fig. 7B). All data points lie on a straight line in the first and third quadrant of the plot, indicating an almost perfect linear relation of the reference technology and the studied technology, which, therefore, always detect the same direction of CO change. We additionally plotted a polar plot by Critchley et al. (Fig. 7D) and a 4-quadrant plot based on relative numbers (Fig. 7F) as discussed in Extensions and Perspectives. Figure 7C shows a Bland-Altman plot. Figure 7E shows a Bland-Altman plot applied to changes of the measurement. Now, the x-axis refers to the average of the detected changes of both devices at the same time, whereas the y-axis refers to the difference of the detected changes. The directions of the changes cannot be seen from the Bland-Altman plot (Fig. 7C) and the 4-quadrant plot should be consulted.

Figure 8 emphasizes this point. In the figure, the patientâ€™s decrease in CO is detected with the reference technology (gold standard method), whereas the studied technology always detects the changes in the opposite direction (Fig. 8A). The 2 Bland-Altman plots (Fig. 8, C and E) report similar mean differences and 95% limits of agreement as before (and are thus not able to distinguish the current scenario from the one before), whereas the 4-quadrant plot clearly shows the poor trending ability of the devices (Fig. 8, B and F).

Overall, Bland-Altman analysis is an appealing concept for assessing the absolute agreement and precision of technologies. However, if the technologies show a good trending ability but deviate in absolute measures, Bland-Altman analysis cannot confirm trending ability. In these cases, the methods as discussed in this article should additionally be applied.

## APPENDIX 2

### Calculation of the Measures Angular Bias and Radial Limits of Agreement (Usually Derived from the Polar Plot) from the 4-Quadrant Plot

Let *x*_{i} and *y*_{i} denote the Î”CO values of the reference and studied device as before. Then the angle between the (0,0)-(xi,yi)-line and the 45Â° axis is

where atan2 is a common variation of the arctangent function to cope with the different orthants of the data.

The mean of all these thetas corresponds to the angular bias calculated from the polar plot. It may be depicted in the 4-quadrant plot by a line.

Note that the numbers calculated this way do not necessarily match the numbers as calculated from the polar plot. The reason is the exclusion zone of the polar plot, which excludes also most of the discordant pairs from the data. The drawbacks of the exclusion zone in the polar plot are described in detail in Polar Plot Analysis. However, by excluding the same data points from the analysis, both the angular bias when applying the polar plot methodology and the angular bias derived from the 4-quadrant plot methodology coincide.

The radial limit of agreement is the symmetric angle around the 45Â° line in which 95% of the data points fall. It may be calculated from the theta values above. Therefore let

denote an ordered set of absolute values of the

calculated above beginning with the smallest value. Let

, that is, the 0.95 multiplied by the total number of points rounded to the next natural number. The angle of the radial limit is then the m-th smallest value of the absolute values of

as calculated above. Again, by excluding the same data points from the analysis both the radial limit of agreement when applying the polar plot methodology and the radial limit of agreement derived from the 4-quadrant plot methodology coincide.

## DISCLOSURES

**Name:** Bernd Saugel, MD.

**Contribution:** This author was responsible for the intellectual content of the article and manuscript preparation.

**Attestation:** Bernd Saugel approved the final manuscript.

**Name:** Oliver Grothe, PhD.

**Contribution:** This author was responsible for the intellectual content of the article and manuscript preparation.

**Attestation:** Oliver Grothe approved the final manuscript.

**Name:** Julia Y. Wagner, MD.

**Contribution:** This author was responsible for the intellectual content of the article and manuscript preparation.

**Attestation:** Julia Y. Wagner approved the final manuscript.

**This manuscript was handled by:** Franklin Dexter, MD, PhD.

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