Mean Amplitude of Glycemic Excursions (MAGE) is a glucose variability metric that can be applied to continuous glucose monitoring (CGM) data. From iglu version 3 onward, the package includes two algorithms for MAGE calculation:

`iglu::mage(., version = 'ma')`

uses short and long moving averages to computationally emulate the original definition,`iglu::mage(., version = 'naive')`

is a naive approximation**solely included**for backwards compatibility with earlier versions of iglu.

The moving average algorithm was published in the Journal of Diabetes
Science and Technology: Open-source Algorithm
to Calculate Mean Amplitude of Glycemic Excursions Using Short and Long
Moving Averages, and the code to reproduce all tests can be found at
the accompanying repository: Nathaniel-Fernandes/mage_algorithm_data.
Below is a supplementary vignette describing the algorithm in more
detail, our design choices, and the methods we used to test it. Some
example code to get started using MAGE in iglu is included in the *Implementation* section.

**Background**: Glucose variability is increasingly
regarded as a risk factor for diabetes-related complications. The mean
amplitude of glycemic excursions (MAGE) is a common measure of glucose
variability. While its original definition requires manual calculations,
there exist several software programs for automated calculations which
have shown varying degrees of agreement. The objective of this study was
to develop an automated algorithm for MAGE that would achieve high
accuracy against the reference standard of manual calculations and
benchmark it against existing alternatives.

**Methods**: Manual calculations of MAGE were performed
on 45 publicly available CGM profiles from a diverse set of patients
(i.e., Type 1 diabetes, Type 2 diabetes, and those without diabetes)
following the definition described in Service et al. [3]. An automated
algorithm for MAGE calculation was developed to identify peaks and
nadirs of glycemic excursions based on the crosses of a short and long
moving average of the glucose profile, and implemented in version 3+ of
R package iglu. The accuracy of the algorithm was evaluated relative to
manual calculations using a five-fold cross-validation and compared to
other publicly available MAGE calculators, as listed in Table 2.

**Results**: The newly developed algorithm had the
median error of 1.4% relative to manual calculations (iglu v4.1+). The
median errors of the six other approaches including AGATA, EasyGV,
cgmanalysis, naive iglu algorithm, rGV, and cgmquantify are 9.8%, 11.3%,
20.3%, 42.0%, 78.4%, and 78.1%, respectively.

**Conclusions**: The newly developed algorithm
eliminates the need for tedious manual MAGE calculations and
approximates the manual derivation better than existing approaches.

Glucose variability is increasingly regarded as a risk factor for diabetes-related complications beyond HbA1c [1, 2]. The mean amplitude of glycemic excursions (MAGE) is a commonly used measure of glucose variability. As defined by Service et al. [3], MAGE is the arithmetic mean of the amplitude (i.e. height) of glucose excursions that are greater than the standard deviation of the glucose values. MAGE is calculated manually by visually inspecting the glucose profiles. However, this manual approach to calculating MAGE is error-prone and impractical for large data sets of continuous glucose monitors (CGMs). Thus, an automated algorithm for MAGE calculation that approximates the manual reference is needed.

Several software programs for automated MAGE calculations have been developed, and these implementations can be divided into two categories: those that only return the numeric MAGE values, and those that additionally provide a visual display of the calculations [4]. Examples in the first category include the Excel®-based workbook EasyGV [5], online GlyCulator2 [6], R package cgmanalysis [7], early versions of R package iglu [8], and Python package cgmquantify [9]. Because these software programs do not output a visual display of the performed calculations, the user has no means to inspect the calculations and confirm accuracy. Examples in the second category include programs developed by Fritzsche [10] and Baghurst [11]. However, we found the software developed by Fritzsche are not compatible with newer operating systems (e.g., Windows 10) to allow importing of CGM data. Additionally, Baghurst does not provide a downloadable implementation of his algorithm. However, according to documentation for cgmanalysis [7] and cgmquantify [9], these packages provide free implementations of Baghurst’s algorithm, albeit they may not fully reflect the original implementation.

The comparison of GlyCulator, EasyGV, Fritzsche, and Baghurst algorithms demonstrates varying degrees of agreement across MAGE algorithms [12]. While the aforementioned algorithms were compared to each other, they were not compared with the manual reference standard.

Thus, the objective of current study is to establish an approach of calculating MAGE that is user-friendly and approximates well the results obtained manually. Therefore, these goals are defined as follows.

To computationally emulate the manual method in Service et al. [3] with high accuracy based on the automated detection of peaks and nadirs tailored specifically towards CGM data

To provide a freely-available implementation with a visual display

To evaluate accuracy, the newly developed algorithm was compared with the manual derivations along with the other computational approaches.

*CGM Data and Manual MAGE Calculation*

Publicly-available CGM data were obtained from [13], from which 45 CGM traces were selected to represent a diverse set of patients (Type 1 diabetes, Type 2 diabetes, and those without diabetes), see Table 1. Each CGM trace corresponds to one day of measurements (24 h), where a day is defined from midnight to midnight as it is customary to calculate MAGE daily. Manual MAGE calculations were performed on each of the 45 CGM traces following the original description of MAGE calculations by Service et al. [3] by two of the authors, who were unaware of the outcome of the proposed algorithm’s outputs.

**Table 1:** Summary of the 45 CGM traces used for
manual calculations. The first three datasets have been obtained as
described in [13]. The last dataset is available within R package iglu
[8].

Dataset | Diabetes Type | Age group | Subject ID | Number of days used (CGM traces) |
---|---|---|---|---|

Hall et al. (2018) | None | Adults (>18) | 16 36-69-001 | 5 |

16 36-69-026 | 6 | |||

Tsalikian et al. (2005) | Type 1 | Children (10 - 18) | 2 | 1 |

7 | 2 | |||

8 | 1 | |||

11 | 2 | |||

15 | 1 | |||

43 | 1 | |||

44 | 1 | |||

Dubosson et al. (2005) | Type 1 | Adults (>18) | 1 | 5 |

2 | 2 | |||

Broll et al. (2021) | Type 2 | Adults (>18) | Subject 1 | 1 |

Subject 2 | 5 | |||

Subject 3 | 1 | |||

Subject 4 | 5 | |||

Subject 5 | 6 |

*Existing MAGE Calculators*

We benchmarked the proposed algorithm against a variety of open and closed-source tools. To the best of our knowledge, the list in Table 2 comprises a relatively exhaustive list of existing packages to compute MAGE on CGM data as of May 15, 2024.

**Table 2:** An exhaustive list (to the best of our
knowledge) of existing packages and algorithms to compute MAGE as of May
15, 2024. We also specify the version and release date if we benchmark
the proposed MAGE algorithm against it, or the reason we were unable
to.

Tool or Algorithm Name | Has MAGE | Benchmarked? | Version Used | Release Date |
---|---|---|---|---|

iglu::mage_ma (Proposed) | Yes | Yes | iglu v4.1 | Present |

iglu::mage_naive | Yes | Yes | iglu v2.0 | Feb 27, 2020 |

EasyGV [5] | Yes | Yes | EasyGV v9.0 (The download link for Version 10 is broken & only downloads Version 9.0) |
Oct 6, 2020 |

cgmanalysis [7] | Yes | Yes | cgmanalysis v2.7.7 | Nov 20, 2023 |

rGV [18] | Yes | Yes | rGV v0.0.4 | Jan 17, 2023 |

cgmquantify [9] | Yes | Yes | Commit: d4a788c | Feb 8, 2021 |

AGATA [19] | Yes | Yes | agata -2.1.1.zip | Jan 27, 2023 |

CGDA [20] | Yes | No: skips computing MAGE for some CGM traces without warning | N/A | N/A |

CGM Shiny [21] | Yes | No: Unable to load CSV files with sensor data | N/A | N/A |

“Group of Signs” Method [22] | Yes | No: no open-source code provided. | N/A | N/A |

“Spline Interpolation” Method [23] | Yes | No: no open-source code provided. Coded from scratch as per paper | N/A | N/A |

“Linear Interpolation” Method [24] | Yes | No: no open-source code provided. Coded from scratch as per paper | N/A | N/A |

GVAP Tool [25] | Yes | No: Requires CGM traces to have >100 points, which fails to work with some manual traces with large gaps | N/A | N/A |

CGMTSA [26] | Yes | No: Cannot process some CGM traces when less than 24 hours worth of data | N/A | N/A |

CGMStatsAnalyser [27] | Yes | No: Unable to load CSV files with sensor data | N/A | N/A |

Fritzche [10] | Yes | No: Software not compatible with modern operating systems (Windows 10 / MacOS) | N/A | N/A |

GlyCulator2 [6] GlyCulator3 [28] |
Yes | No: program requires input of exact # of glucose values which fails to work with variable- length MAGE computation or CGM traces with gaps | N/A | N/A |

MAGECAA v1.0 [4] | Yes | No: Closed -source - copy requested from the author | N/A | N/A |

Baghurst [11] | Yes | No: Closed source. Unable to acquire a copy. | N/A | N/A |

CGM-Guide [29] | Yes | No: Closed source. Unable to acquire a copy. | N/A | N/A |

Tidepool [30] | No | N/A | N/A | N/A |

GLU [31] | No | N/A | N/A | N/A |

CGMAnalyzer [32] | No | N/A | N/A | N/A |

*The Algorithm*

The original MAGE description in Service et al. [3] requires (i) identification of turning points (e.g., peaks and nadirs) for calculation of glucose excursions; (ii) filtering those turning points to only keep relevant glucose excursions and calculating the mean of those excursion amplitudes (i.e., heights).

To the best of our knowledge, our implementations for both parts (i) and (ii) are unique.

**Identification of peaks and nadirs.**The proposed algorithm uses the crosses of a short and long moving average to identify time intervals where a peak or nadir may exist. Let \(\alpha\) be the window size for the short moving average and 𝛽 be the window size for the long moving average, such that \(\alpha < \beta\). Since the shorter moving average is more affected by local variation in the glucose values than the longer moving average, a peak or nadir (i.e., a turning point, \({tp}_i\)) must exist on the intervals that are bounded by the crosses of the two moving averages \(cp_i\) and \(cp_{i+1}\) as demonstrated in Figure 1, with the exception of “whiplash”. (Note: the first and last glucose values are always treated as crossing points [Appendix: Why are the first and last data points always treated as crossing points?].) To solve the issue of “whiplash”, where the true peak/nadir falls outside the interval bounded by \(cp_i\) and \(cp_{i+1}\) for back-to-back sinusoidal excursions [Appendix: What is whiplash?], we make the assumption that peaks and nadirs alternate (peak, nadir, peak, nadir, etc.) or vice versa and calculate subsequent peaks or nadirs sequentially as the maximum or minimum glucose value, respectively, between the index of \(tp_{i-1}\) (left boundary, exclusive) and \(cp_{i+1}\) (right boundary, inclusive) for \(i=2\). The left boundary is \(cp_{i}\) for the base case of \(i=1\).A distinct advantage of using moving averages to identify the turning points is that they inherently smooth out local fluctuations and noise within larger trends in a glucose profile. The amount of smoothing can be tuned by varying window size parameters \(\alpha\) and \(\beta\). By default, the algorithm uses \(\alpha = 5\) and \(\beta = 32\) as these values maximized the algorithm’s accuracy on the manual MAGE calculations, where \(\alpha \in [1, 7]\) and \(\beta \in [16, 38]\).

**Eliminating Excursions and Calculation of Mean Amplitudes.**An excursion is defined by three turning points \(E_{i,j,k}=(tp_i, tp_j, tp_k) \text{ s.t. } i < j < k\) , where the amplitude of the excursion is calculated as follows: \(a(j, i) = |tp_j - tp_i|\) and \(|E_{i,j,k}| = (|E_{left}|,|E_{right}|)=(a(j, i), a(k, j))\). An “acceptable” excursion, as defined by Service et al. [3], requires the amplitudes of both the left-side and right-side half-excursions to exceed the standard deviation of the entire CGM trace. Computationally, we devised the procedure below to calculate acceptable excursions, given a set of alternating turning points. To prevent double counting excursions, only ascending (MAGE+) or descending (MAGE–) half-excursion amplitudes are averaged (i.e., \(|E_{left}|\) or \(|E_{right}|\), respectively), where ascending means \(tp_j\) is a peak and vice versa. When excursions are symmetric, MAGE+ = MAGE–, however in general they may disagree [11]. By default, our implementation follows the description in Service et al. [3], and returns MAGE+ or MAGE– depending on which type of excursion first crosses the one SD threshold; however, the user can specify either if desired.*Procedure:*Starting from \(i=1, j=2\) (note: R is 1-based), check whether the amplitude exceeds the one standard deviation threshold, \(a(j, i) = a(2, 1) > sd\). If false, the algorithm will calculate \(\max\; \{a(3,1), \; a(3,2)\} > sd\) and so on until a large enough amplitude is found or the CGM trace ends. Note: we use the maximum to enable “accumulation”, an optimization that allows an earlier larger half-excursion to absorb a later smaller half-excursion. Once an acceptable left-side half-excursion is found, the boundaries \(i\) and \(j\) are recorded and the same procedure is used to find an acceptable right-side excursion, where \(j\) is updated if \(a(k, i) >= a(j, i)\). We record \(E_{left}\) once \(E_{right} > sd\) or the CGM trace ends.

*Handling Gaps*

The algorithm outlined above assumes that the input CGM trace should be treated contiguously, with MAGE calculated over the entire trace and turning points alternate (peak, nadir, peak, nadir, etc.) or vice versa. However, this can lead to suboptimal results if large gaps are present in the trace. While small gaps can be interpolated before MAGE calculation (starting in iglu v3.3.0), CGM values separated by large gaps should be treated as separate segments, with MAGE calculated on each segment independently. By default, MAGE released in iglu v3.5.1+ segments the CGM trace by gaps larger than \(max\_gap=180 \text{ minutes}\) and calculates MAGE via the above algorithm on each segment [Appendix: [Why set a default of 180 minutes for max_gap?]]. The user can then return the MAGE values for each segment (\(MAGE_i\)) or a one-number summary for the variability of the entire CGM trace (\(MAGE_{overall}\)). MAGE for the entire trace is reported as an average of the segment MAGE values, weighted by segment length (\(MAGE_{overall} = \Sigma \frac{n_i}{N}MAGE_i\), where \(n_i\) is the length of segment \(i\) and \(N = \Sigma n_i\)). The plot displays segments with a solid, black vertical line for the left-boundary and a dashed, black vertical line for the right boundary.

*Implementation*

The newly developed algorithm for MAGE calculation can be found from
version 3 onward in R package iglu [17] as well as the accompanying GUI
via Shiny App [8] - both are free and open-source. Implementation in a
script-based programming language like R allows the creation of
reproducible scripts for all data processing and MAGE calculation steps,
and the availability of a point-and-click GUI meets the needs of
researchers with limited programming experience. By default, the
algorithm uses \(\alpha = 5\) and \(\beta = 32\), however these choices can be
adjusted by the user.

```
# load package and example data
library(iglu)
# mage calculation using default parameters
mage(example_data_5_subject)
#> # A tibble: 5 × 2
#> # Rowwise:
#> id MAGE
#> <fct> <dbl>
#> 1 Subject 1 72.4
#> 2 Subject 2 118.
#> 3 Subject 3 116.
#> 4 Subject 4 70.9
#> 5 Subject 5 142.
# Adjust default window sizes in mage calculation
mage(example_data_5_subject, short_ma = 3, long_ma = 30)
#> # A tibble: 5 × 2
#> # Rowwise:
#> id MAGE
#> <fct> <dbl>
#> 1 Subject 1 72.3
#> 2 Subject 2 118.
#> 3 Subject 3 116.
#> 4 Subject 4 70.9
#> 5 Subject 5 142.
```

In addition to providing a numerical MAGE value, the software allows the creation of glucose trace plots that highlight the selected peaks and nadirs used for MAGE calculation.

```
# subset data to better see identified peaks and nadirs
fig1data <- example_data_1_subject[1:200, ]
# plot using mage function with plot = TRUE and show_ma = TRUE
mage(fig1data, plot = TRUE, show_ma = TRUE, title = "Glucose Trace - Subject 1")
```

**Figure 1:** Glucose trace of Subject 2 with Type 2
diabetes from Broll et al. [8], the solid green line is a short MA with
\(\alpha\) = 5 and the solid orange
line is a long MA with \(\beta\) = 32.
The peaks and nadirs occur between the crosses of short and long moving
averages. By default, the algorithm calculates MAGE+ as in this profile
the first segment to exceed one SD threshold is ascending. The provided
graphical display in iglu software [17] automatically highlights only
those peaks (red) and nadirs (blue) that are used for final
calculation.

Additionally, the plot will highlight any gaps in the data due to missing readings. If these gaps are shorter than \(max\_gap = 180\), they will be part of the same segment.

```
# subset data to better show gaps
fig2data <- example_data_1_subject[200:400, ]
# plot using mage function with plot = TRUE
mage(fig2data, plot = TRUE, title = "Glucose Trace with Gaps - Subject 1")
```

**Figure 2**. Glucose trace of Subject 5 with Type 2
diabetes from Broll et al. [8] with regions of missing glucose readings
automatically highlighted using iglu software [17].

*Assessment of Accuracy*

The accuracy of the proposed algorithm was evaluated against 45 manual
calculations. For each CGM trace, the accuracy was measured as the
relative percent difference in the automatically calculated MAGE value
and the manual MAGE value, that is: \(\frac{|MAGE_{algorithm} -
MAGE_{manual}|}{MAGE_{manual}}\cdot 100\%\) .

To find optimal combination of window sizes, we considered \(\alpha \in [1, 7]\) and \(\beta \in [16, 38]\), and recorded the mean errors across 45 CGM traces for each pair of (\(\alpha, \beta\)) values. The optimal (\(\alpha, \beta\)) pair was determined as the one that leads to the minimal in-sample error. Since the obtained minimal error is a biased assessment of accuracy as the optimal (\(\alpha, \beta\)) were selected based on the same 45 traces, five-fold cross-validation was also performed to gain an unbiased assessment of accuracy. The 45 CGM traces were randomly split into five groups of nine samples each. The best (\(\alpha, \beta\)) combination was then chosen using four out of the five groups, and used to evaluate the accuracy on the remaining group. The best combination was defined as the one that gave the lowest percent error compared to manual calculations. This out-of-sample evaluation was performed on all five groups.

The accuracy of EasyGV [5], cgmanalysis [7], cgmquantify [9], rGV, AGATA, and the MAGE algorithm implemented in the earlier version of iglu [8] was also determined against the manual reference derivation. The latter is a naive simplification of original MAGE definition, as it calculates MAGE as the mean of absolute differences between glucose values and the glucose mean that are larger than one standard deviations (and thus does not attempt to explicitly identify peaks and nadirs). We use iglu_ma to denote the proposed algorithm based on moving averages, and iglu_naive to denote the naive algorithm used in earlier versions of the software. We were unable to evaluate GlyCulator2 [6] because the software requires CGM trace to have at least 576 time points, and thus returns NA when applied to 24-hour CGM data. We were also unable to evaluate Fritzsche [10] as importing the CGM data into the provided user interface on Windows 10 lead to errors.

*Assessment of Parameter Frequency Robustness*

We also test if the optimal long and short window parameters chosen
on the 5-minute CGM data - i.e., (\(\alpha_{best}, \beta_{best}\)) - are robust
to the frequency of the meter / CGM data (e.g., 1-, 5-, 10-, and
15-minute). To do this, we use the `iglu::CGMS2DayByDay`

function to change the raw data’s frequency to 1-, 10-, and 15-minute
intervals, and automatically imputes glucose values for gaps less than
45 minutes. We then rerun the hyperparameter tuning code to discover the
best (\(\alpha, \beta\)) selected on
the new data and report the mean, median, and IQR of errors. Note that
this is not a perfect comparison: information is lost when downsampling
the 5-minute CGM trace to 10- and 15-minutes but the manual calculations
compared against are performed on 5-minute data. Thus, we expect to see
a slightly degraded performance on the 10- and 15-minute data simply due
to this design limitation and not necessarily an algorithm
limitation.

The optimal choice of window sizes (\(\alpha, \beta\)) for the proposed algorithm on the 5-minute CGM data was determined to be (5, 32) as this combination minimized the average error relative to 45 manual calculations. Figure 3 shows a heatmap of the percent average error (across 45 traces) as a function of \(\alpha\) and \(\beta\), where the darkest regions represent the most favorable combinations of \(\alpha\) and \(\beta\). (Note: while there was a tie, we chose hyperparameters in the middle of the dark region.) The minimum corresponds to a mean average error of 8.9% with a much smaller median percent error of 1.4% and [Q1, Q3] = [0.5%, 7.1%]. As expected, the unbiased out-of-sample cross-validated errors are slightly larger, but still relatively low: 9.1% mean with the median of 10.0% and inter-quartile range of [9.4%, 11.8%].

**Figure 3.** Heatmap of average errors over 45 CGM
traces of automatically calculated MAGE values via proposed algorithm
relative to manual MAGE values. The short moving average window size
\(\alpha \in [1, 7]\) and the long
moving average window size \(\beta \in [16,
38]\). The darker regions correspond to smaller errors, with the
minimal error of % achieved at \(\alpha =
5\), \(\beta = 32\).

Compared to the other available automated algorithms, the next best results were achieved by EasyGV, which had an average error of 23.4%, the median error of 11.3% and the IQR of [6.0%, 22.3%]. In constrast, cgmanalysis, iglu_naive, and cgmquantify all had significantly higher errors, with median errors (and interquartile range) of 20.3% [8.6%, 32.1%], 42.0% [32.6%, 48.3%], and 78.4% [30.8%, 131.9%], respectively. Figure 4 shows boxplots of relative errors of all algorithms on 45 CGM traces. Figure 5 and Table 3 show a correlation matrix of all algorithms on the 45 CGM traces. (Note: see the “Important Notes on rGV” in the Discussion below.)

**Table 3.** Correlation matrix of various MAGE
calculators’ outputs on the 45 CGM traces.

Manual Calc | Iglu_ma | Iglu_naive | CGM_Analysis | Easy_GV | agata_err | rGV | CGM_Quantify | |
---|---|---|---|---|---|---|---|---|

Manual Calc | 1.00 | 0.93 | 0.90 | 0.82 | 0.91 | 0.92 | 0 | 0.66 |

Iglu_ma | 0.93 | 1.00 | 0.96 | 0.85 | 0.98 | 0.92 | 0 | 0.70 |

Iglu_naive | 0.90 | 0.96 | 1.00 | 0.92 | 0.98 | 0.88 | 0 | 0.69 |

CGM_Analysis | 0.82 | 0.85 | 0.92 | 1.00 | 0.87 | 0.80 | 0 | 0.66 |

Easy_GV | 0.91 | 0.98 | 0.98 | 0.87 | 1.00 | 0.89 | 0 | 0.66 |

agata_err | 0.92 | 0.92 | 0.88 | 0.80 | 0.89 | 1.00 | 0 | 0.66 |

rGV | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1 | 0.00 |

CGM_Quantify | 0.66 | 0.70 | 0.69 | 0.66 | 0.66 | 0.66 | 0 | 1.00 |

**Figure 4.** Comparison of relative errors of different
automatic algorithms for MAGE calculation relative to manual MAGE values
on 45 CGM traces. The proposed method is implemented as ma (moving
average) algorithm in R package iglu [17]; best corresponds to the
errors with \(\alpha = 5\), \(\beta = 32\); cv corresponds to the
out-of-sample errors based on 5- fold-cross-validation for \(\alpha, \beta\).

**Figure 5.** Heatmap of correlation matrix of various
MAGE calculators’ outputs on the 45 CGM traces.

**Table 4.** Correlation matrix of various MAGE
calculators’ outputs on the 45 CGM traces and the mean and standard
deviations of the 45 CGM traces.

mean | sd | |
---|---|---|

Manual Calc | 0.68 | 0.88 |

Iglu_ma | 0.70 | 0.93 |

Iglu_naive | 0.67 | 0.98 |

CGM_Analysis | 0.61 | 0.94 |

Easy_GV | 0.66 | 0.95 |

agata_err | 0.72 | 0.86 |

rGV | 0.00 | 0.00 |

CGM_Quantify | 0.90 | 0.64 |

mean | 1.00 | 0.66 |

sd | 0.66 | 1.00 |

For the proposed algorithm, both in-sample errors with the best choice of window sizes (\(\alpha, \beta\)) and out-of-sample cross-validated errors are displayed. The corresponding numerical summaries are summarized in Table 2.

**Table 5:** Numerical summaries of relative errors of
automatic MAGE calculators on 45 CGM traces compared to manual
calculations. The proposed method is implemented as ma (moving average)
algorithm in R package iglu [17]; best corresponds to the errors with
\(\alpha\) = 5, \(\beta\) = 32; cv corresponds to the
out-of-sample errors based on 5-fold-cross-validation for \(\alpha\), \(\beta\).

Algorithm | Mean | Median | IQR |
---|---|---|---|

iglu (ma, best) | 8.9% | 1.4% | [0.5%, 7.1%] |

iglu (ma, cv) | 9.1% | 10.0% | [9.4%, 11.8%] |

iglu (naive) | 39.0% | 42.0% | [32.6%, 48.3%] |

EasyGV | 23.4% | 11.3% | [6.0%, 22.3%] |

cgmanalysis | 23.9% | 20.3% | [8.6%, 32.1%] |

cgmquantify | 91.9% | 78.4% | [30.8%, 131.9%] |

rGV | 74.7% | 78.1% | [64.9%, 86.7%] |

AGATA | 16.7% | 9.8% | [4.4%, 26.3%] |

**Table 6:** Numerical summaries of relative errors of
the proposed MAGE algorithm on 45 CGM traces compared to manual
calculations at varying frequencies of the underlying data. The
relatively constant mean error rate demonstrates the proposed method and
selected hyperparameters (\(\alpha\) =
5, \(\beta\) = 32) are robust to the
frequency of the meter.

CGM Frequency | Mean | Median | IQR |
---|---|---|---|

1-min | 6.5% | 0.6% | [0.3%, 3.6%] |

5-min | 8.9% | 1.4% | [0.5%, 7.1%] |

10-min | 9.7% | 2.5% | [1.2%, 8.4%] |

15-min | 10.7% | 5.3% | [2.1%, 9.0%] |

There are several software programs for automatic MAGE calculation, however, they show varying agreement [12]. In the current study, a new automated algorithm for MAGE was developed based on moving averages which demonstrated the least amount of error relative to manual reference when compared to other automated algorithms.

The newly developed algorithm approximated the manual results better than EasyGV, cgmanalysis, iglu_naive, and cgmquantify. The largest discrepancy was noted for cgmanalysis and cgmquantify both of which implemented Baghurst’s algorithm [11]. It is unclear whether the deviation from manual reference is due to the underlying algorithm or the specific of implementations of the algorithm. The differences in accuracy among the MAGE algorithms points to the necessity of a visual display of the underlying calculations with identified peaks and nadirs. While such visual display is implemented for the proposed MAGE algorithm in version 3 of R package iglu [17], this functionality is absent from EasyGV, cgmanalysis, and cgmquantify, making it more difficult for the user to verify the accuracy.

The correlation between our proposed algorithm (with Service definition) and manual calculations per Service is 0.93 when compared across the same 45 CGM traces. This is the largest correlation value across the implementations we have compared (0.92 for AGATA, 0.91 for EasyGV, 0.90 for older iglu MAGE algorithm, 0.82 for CGManalysis and 0.66 for CGMQuantify), which is consistent with our proposed approach showing the lowest error rate relative to manual calculations. When comparing correlations between our approach and existing methods, we find correlation of 0.98 with EasyGV, 0.96 with older iglu MAGE algorithm, 0.92 with AGATA, 0.85 with CGManalysis and 0.7 with CGMquantify.

As can be seen in Table 4, there is a relatively high correlation between MAGE and standard deviation of the CGM trace. This has been noted before [33] and has led some to question the importance of using MAGE over the simpler standard deviation. Nevertheless, further research is needed to see if MAGE is better correlated with clinical outcomes (e.g., morbidity or mortality) than the simple standard deviation.

A major challenge for any automated MAGE algorithm is separation of true peaks and nadirs in a larger trend from the smaller glucose fluctuations. A particular advantage of the proposed algorithm is the use of moving averages which inherently smooth out local fluctuations within a larger trend. The amount of smoothing can be tuned by varying the window sizes of the short and long moving averages, and we found that the algorithm is quite robust to the choice of window sizes, with high accuracy in a large range (Figure 3). Additionally, the proposed optimal short and long moving averages are robust to the frequency of the CGM meter, as shown by the similar mean errors of the MAGE algorithm despite the meter frequency in Table 4. One disadvantage is that the moving average is a lagging indicator, and thus is undefined in the beginning of the time period (e.g., a moving average of window size 20 is undefined for the first 19 time points). This could skew the calculations of MAGE if the first times points contain a lot of variation in subject’s glucose levels. To adjust for this, the algorithm replaces the undefined moving average values at the first time points with the first calculated moving average value (e.g., a moving average of window size 20 will be first calculated at 20th time point, and that same value will be assigned to the first 19 time point to be later used in determining the crosses of short and long moving averages). This adjustment can be seen for the long moving average in Figure 1, where its value is flat for the first 32 points. Given the high accuracy of the proposed algorithm compared to manual calculations, this adjustment is adequate, however it is possible that it may not be adequate for some CGM profiles. In practice, we recommend to inspect the visual display of MAGE calculations provided by iglu [17] as in Figures 1 and 2 for additional accuracy validation. Another challenge for MAGE calculations is the presence of asymmetric excursions. To prevent double counting, typically only ascending (MAGE+) or descending (MAGE–) excursion amplitudes are averaged. When excursions are symmetric, MAGE+ = MAGE–, and the accuracy is not affected irrespective of the choice of direction. However, when excursions are asymmetric, the direction matters as the two disagree. By default, our algorithm follows the description in Service et al. [3], and calculates MAGE+ or MAGE– depending on which type of excursion first crosses the one SD threshold. The same decision is made in [12]. However, Baghurst [11] argues that it is more appropriate to use \(MAGE_{avg} = \frac{1}{2}(MAGE_{plus}+MAGE_{minus})\). Our implementation allows the user to change the default settings, and explicitly calculate MAGE+, MAGE–, MAGEavg, or MAGEmax, which makes our algorithm’s implementation more flexible compared to existing MAGE software.

As seen in Figure 4 and Table 3 above, the errors for rGV and cgmquantify are quite high and much different than the other calculators. After further analysis, we believe the following reasons explain the high errors.

- The MAGE function in rGV outputs 20 regardless of the input data - this behavior is reproducible even with the example given in the rGV function.
- cgmquantify may incorrectly index a dataframe during the mage computation, as identified in https://github.com/brinnaebent/cgmquantify/issues/10.

We present a new algorithm for MAGE calculation that significantly outperforms existing algorithms. The implementation of this algorithm is free and open-source, it is available in the R package iglu version 3 [17] as well as in the accompanying GUI via Shiny App at https://irinagain.shinyapps.io/shiny_iglu/. Furthermore, the visual displays of the CGM traces, exact values for manual MAGE and the code to reproduce all analyses, along with instructions on how to download each CGM trace, are publicly available at https://github.com/Nathaniel-Fernandes/mage_algorithm_data. The public nature of CGM traces, manual MAGE values, and the algorithm allow additional validation of presented results by a community at large.

**Funding Sources**: NSF CAREER Award DMS-2044823 to
IG

**Acknowledgements**: The source of subset of the data
is the T1D Exchange, but the analyses, content and conclusions presented
herein are solely the responsibility of the authors and have not been
reviewed or approved by the T1D Exchange.

**Disclosures**: None

If we didn’t, we would throw away at the beginning or end of CGM trace because the MAGE algorithm calculates MAGE between the first and last crossing point. For example, assume a CGM trace of 100 values with the first and last crossing point at index 20 and 80, respectively. The turning points are looked for between index 20 and 80, which throws out the beginning and end. Thus, we automatically record the first and last points in the CGM trace as crossing points.

We coined the term whiplash to describe two back-to-back excursions that roughly form a sinusoid. These sinusoidal excursions pose a problem for identifying the true peaks and nadirs defining an excursion since the moving average is a lagging indicator. In other words, at the end of one period of a sinusoidal excursion, the long moving average will be approximately equal to the short moving average (i.e., \(MA_{long}(\vec x)\approx MA_{short}(\vec x)\approx \bar x\)). Since the moving average is a lagging indicator, by the time a cross between the short and long moving average occurs, the true peak or nadir indicating the start of an excursion may have already occurred. Thus, starting in MAGE iglu v3.5.1, we calculate turning points sequentially between \(tp_{i-1}\) and \(cp_{i+1}\) as compared to \(cp_i\) and \(cp_{i+1}\), previously. See the difference in the identified \(tp_{12}\) in the images below, the first being MAGE iglu v3.5.1 (whiplash corrected) and the second MAGE iglu v3.5.0 (whiplash not corrected).

**Figure 5.** Plot of Subject 1636-69-026 via MAGE iglu
v3.5.1 with correction for whiplash. Notice the true peak is properly
identified for \(tp_{12}\).

**Figure 6.** Plot of Subject 1636-69-026 via MAGE iglu
v3.5.0 without whiplash corrected for. Notice the true peak is not
properly identified for \(tp_{12}\).

`max_gap`

?The purpose of `max_gap`

is to split the CGM trace into
different logical segments - i.e., there are so many missing values
between segments 1 and 2 that it makes more sense to treat them as
separate than as one continuous trace. For example, say a patient wears
their CGM on Monday and Wednesday but forgets on Tuesday. There would be
a stretch of 24-h of missing data! Thus, it makes more sense to
calculate MAGE on Monday and Wednesday independently, as we cannot
extrapolate what occurred in the time in-between.

While it is fairly obvious in the above example that we should
segment the CGM trace, what about shorter stretches of missing data?
Since the purpose of MAGE is to calculate the mean amplitude of glycemic
*excursions*, the ideal `max_gap`

will split the trace
when there is a possibility of missing an excursion. Since post-prandial
excursions typically last a maximum of 180 minutes [citation needed], we
chose 180 minutes as an acceptable default threshold. To explain in
natural language: if we have >180 minutes of missing data, we likely
have missed either an entire excursion or multiple excursions, so we
treat the trace as two distinct segments.

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