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7
UDC 616-037; 57.087.1; 519. DOI: 10.26565/2313-6693-2020-40-01
STATISTICAL ANALYSIS OF MEDICAL TIME SERIES
Martynenko A., Raimondi G., Sotnikova-Meleshkina Zh., Danylenko H., Budreiko N._
Statistical analysis of data sets is a necessary component of any medical research. Modern methods of mathematical statistics and statistical application suites provide extensive capabilities for analysis of random values. However, when a data set is represented by a series of data ordered by time, or when structure and order of data are essential components of research, special approaches to statistical analysis become necessary.Presented in this article are special statistical methods developed by the authors for analysis of a time series: Time Series Mann-Whitney M-test is an analogue of the known nonparametric Mann-Whitney U-test for two Time Series with an equal number of elements; Nominal Time Series Measure is a statistical estimator of dynamics of a nominal series consisting of «0» (no) and «1» (yes); Time Series Entropy EnRE is a specially developed robust formula for a Time Series, intended for calculation of nonlinear stochastic measure of order or disorder, popular in various researches. Presented methods are accompanied by a detailed demonstration of capacity for statistical analysis of medical Time Series: Analysis of growth dynamics of boys and girls aged 6-7-8 years (data by World Health Organization); analysis of the number of seizures and choice of anti-epileptic drugs (data by The National Society for Epilepsy); Time series entropy EnRE for Detecting Congestive Heart Failure by standard 5-minutesHeart Rate Variability samples (data by Massachusetts Institute of Technology - Boston's Beth Israel Hospital RR database). It has been noted that, in every case, using the named special methods for statistical analysis of medical Time Series allows one to avoid errors in interpreting results received through statistical methods and substantially increases the accuracy of statistical analysis of medical Time Series.
KEY WORDS: medical time series, nonparametric test, nominal measure, entropy INFORMATION ABOUT AUTHORS
Martynenko Alexander, D. Sc., Professor, Department of Hygiene and Social Medicine, V. N. Karazin Kharkiv National University, 6, Svobody sq., Kharkiv, Ukraine, 61022, e-mail: Alexander.v.martynenko@karazin.ua, ORCID ID: https://orcid.org/0000-0002-0609-2220
Gianfranco Raimondi, MD, PhD, Prof., Sapienza University of Rome (Italy), Piazzale Aldo Moro 5, 00185, Rome, Italy, e-mail: gianfrancoraimondi@uniroma1 .it
Sotnikova-Meleshkina Zhanna, PhD in Medicine, Head of Department of hygiene and social medicine V. N. Karazin Kharkiv National University School of Medicine, 6, Svobody Sq., Kharkiv, Ukraine, 61022, e-mail: zhanna.v.sotnikova@karazin.ua, ORCID ID: https://orcid.org/0000-0001-5534-8264
Danylenko Heorhii., MD, PhD, Full Prof., State Enterprise «Institute for the Protection of Children and Adolescent Health of the National Academy of Medical Sciences of Ukraine», 52-a, Yubileiny Avenue, Kharkiv, Ukraine, 61153, e-mail: danylenkog@ukr.net, ORCID ID: https://orcid.org/0000-0001-7086-2720
Budreiko Nikita, Assistant, Department of hygiene and social medicine, V. N. Karazin Kharkiv National University School of Medicine, 6, Svobody sq., Kharkiv, Ukraine, 61022, e-mail: nbudreiko@protonmail.com
Fundamental researches
OBJECTIVE
Statistical analysis of data sets is a necessary component of any medical research. Modern methods of mathematical statistics and statistical application suites provide extensive capabilities for analysis of random values. However, when a data set is represented by a series of data ordered by time, or when structure and order of data are essential components of research, special approaches to statistical analysis become necessary. Presented in this article are the following special statistical methods
developed by the authors for analysis of a Time Series:
1. Time Series Mann-Whitney M-test -an analogue of the known nonparametric Mann-Whitney U-test for two Time Series with an equal number of elements;
2. Nominal Time Series Measure - a statistical estimator of dynamics of a nominal series consisting of «0» (no) and «1» (yes). Such dichotomous Time Series are often used when there is a need for describing qualitative events in development or course of a disease;
© Martynenko A., Raimondi G., Sotnikova-Meleshkina Zh., Danylenko H., Budreiko N., 2020
3. Time Series Entropy EnRE - specially developed robust formula for a Time Series, intended for calculation of nonlinear stochastic measure of order or disorder, popular in various researches. In medicine, such nonlinear methods have, to the greatest degree, proved their worth for analysis and prognosis of sudden changes in medical condition, such as atrial fibrillation, epileptic seizures, etc.
All presented methods are accompanied by a detailed demonstration of capacity for statistical analysis of medical Time Series.
MATERIALS AND METHODS
Used for statistical analysis of a time series have been methods, algorithms and programs developed by the authors, these being compared with known statistical methods presented in software suites «IBM SPSS Statistics» and «Statistica» by StatSoft.
In order to illustrate presented methods for statistical analysis of time series, developed by the authors, the following cases have been used:
Case 1: height-for-age data by World Health Organization (WHO) for school-aged children and adolescents [1]. The growth curves for ages 5 to 19 years were thus constructed using data from 18 months to 24 years. The final sample used for fitting the growth curves included 30 907 observations (15 537 boys, 15 370 girls) for the height-forage curves;
Case 2: Epileptic patient case and example of anti-epileptic drugs (AEDs) therapy choice by The National Society for Epilepsy https://www.epilepsysociety.org.uk/;
Case 3: long-term HRV records by Massachusetts Institute of Technology -Boston'sBethlsraelHospital (MIT-BIH) from [28] (http://www.physionet.org) for Normal Sinus Rhythm (NSR) RR Interval Database includes beat annotation files for 54 long-term ECG recordings of subjects in normal sinus rhythm (30 men, aged 28.5 to 76, and 24 women, aged 58 to 73). Congestive Heart Failure (CHF) RR Interval Database includes beat annotation files for 29 long-term ECG recordings of subjects aged 34 to 79, with congestive heart failure (NYHA classes I, II, and III).
RESULTS AND DISCUSSION
1. Time Series Mann-Whitney M-test
The most popular statistical tests for comparison of two or more data sets are Student's t-test and Mann-Whitney U-test. The first one is used for data with normal distribution, and the second is a nonparametric alternative to the Student's t-test. Both tests identify differences of just the mean values in data sets and are insensitive to the order of data in a time series. If the named tests are used for a Time Series, this causes substantial errors. Specifically for comparison of Time Series with an equal number of elements we have developed a modification -Time Series Mann-Whitney M-test (MW M-test). Mann-Whitney U-test for data sets with number of elements N and sums of ranks Ri, R2 is represented by a known formula for U
[3]: +
Acquired value is compared with table Critical Values for chosen level of significance p (usually p = 0.05). There are numerous known modifications of the original method. In case of large samples, for one, U is approximately distributed normally, so standardized value use
Additional correction of o is performed in case of a need to take into consideration a small number of associated ranks. However, such approach is not applicable if all elements in data sets are interconnected as a single sequence of a time series.
In order to use the test for comparison of Time Series with an equal number of elements N, we have proposed a modification - Time Series Mann-Whitney M-test with a formula for M:
In this case, r2 are ranks of the elements in original time series, RitRi are ranks of time series in a general series after merging. In the proposed modification, we have taken into account the changed positions of elements of a time series before and after merging of data into a single sequence by calculating their total distances D1 ,D2. It should be noted that Critical Values of Time Series MannWhitney M-test are the same for MannWhitney U-test.
Case 1: Analysis of growth dynamics of boys and girls (6-7 years) and (6-8 years), according to data of the WHO [1].
In order to illustrate the use of developed Time Series MW M-test, we shall analyze the growth dynamics of boys and girls (67 years) and (6-8 years), according to data of the WHO [1]. The growth dynamics will be estimated by monthly increment of growth median in children. In table 1.a, it is shown that in boys and girls aged 6-7 years, growth dynamics differ at the mean level, which is confirmed, with significance level of p < 0.05, by all three conducted tests: Student's t-test, Mann-Whitney U-test and Time Series MW M-test. Should similar analysis be
conducted in children aged 6-8 years, it becomes apparent that mean values AL have come closer, due to nonlinearity of curve to monthly increment of growth median in girls (Table 1.b). In this case, Student's t-test and Mann-Whitney U-test are not applicable, because while there is actually a greater change of growth dynamics than in children aged 6-7 years, these tests are insensitive to substantial changes in time series. Time Series MW M-test correctly takes into account veracious difference in time series at the level of p < 0.05, as well as decreasing value of correlation coefficient, from 0.986 (6-7 years) to 0.8 (6-8 years).
Table 1.a
Growth dynamics in children (6-7 years) and (6-8 years) based on monthly increment of growth median
Age 61-84 m. (6-7 years) Age 61-96 m. (6-8 years)
Boys Girls Boys Girls
AL, sm (Mean ± Standard deviation) 0.499 ± 0.020 0.487 ± 0.017 0.485 ± 0.025 0.484 ± 0.015
Valid AL number, N 23 23 35 35
Student's t-test p < 0.05 p = 0.78
Mann-Whitney U-test p < 0.05 p = 0.84
Time Series MW M-test p < 0.05 p < 0.05
Pearson correlation 0.986 0.800
Table 1.b
Chart of growth dynamics in children (6-7 years) and (6-8 years) based on monthly increment of growth median
Boys
ñL, sm
Girls
Linear regression, y = -0.002x + 0.673, R2 = 0.957
Quadratic regression, y = 0.001x2 - 0.019x + 1.266, R2 = 0.979
2. Nominal Time Series Measure Using nominal data in medical practice is very common: everywhere the response options are represented by a dichotomy of «Yes» or «No», or either presence — «0» or
absence - «1» of any negative trait is important, using special statistical methods is necessary. When comparing two or more data sets, the most prevalent is Cochran Q test, which enables one to estimate the
significance of their difference. In case sets of nominal data are represented by time series, a modification of aforesaid Time Series MW M-test is applicable. However, if there is only one nominal time series for which it is needed to assess the nature of changes occurring in time, using a measure developed specially for that purpose becomes necessary. We have proposed an appropriate Nominal Time Series Measure:
a. If positive events are designated as «1»
yiv yJV
Mn = 2-:=iAi + 2-i=i!Ai
Ai + 0.5*.\'*(JV+ 1)
b. If negative events are designated as «1»
lj
Let us mark peculiarities of interpretation of a Nominal Time Series Measure Mn:
• Mn = 1, only when all the Xn = 1;
• Mn = 0, only when all the Xn = 0;
• Mn = 0.5, threshold value;
Seizures before and
• Mn < 0.5, from the position of an observer, in the end of a series, negative events or responses are prevalent;
• Mn > 0.5, from the position of an observer, in the end of a series, positive events or responses are prevalent.
Case 2: Analysis of number of seizures and choice of anti-epileptic drugs (according to data ofThe National Society for Epilepsy (NSE) https://www.epilepsysociety.org.uk/)
Number of epileptic seizures is a distinctive indicator for estimation of severity of an epilepsy and when choosing anti-epileptic drugs (AED) therapy. With the right AED, up to 70 % of people with epilepsy could have their seizures controlled or stopped. Shown in Table 2 are cases of epileptic seizure 2 and 4 weeks before treatment and during AED treatment. Facts of registered epileptic seizure are marked as «1», and the days without epileptic seizure are marked as «0». A simple estimator of the
Table 2
ing AED therapy
Before treatment
......|—'
—i—i—i—i—i—i—^—i—i— 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
AED treatment
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14
| l
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Day
number of epileptic seizures 2 weeks before treatment and during AED treatment does not enable us to draw any conclusions as to the quality of applied AED therapy, because both before and during the therapy, equal number of epileptic seizures is observed. However, objective clinical picture has enabled a doctor to continue the existing AED therapy for another 2 weeks. The doctor's choice has proven correct, because in a 4-week timespan the number of epileptic seizures during AED treatment has decreased, when compared to the state of affairs 4 weeks before treatment. At the same time, if the doctor had an opportunity to quantitatively estimate the dynamics of change in a nominal series, the choice would be a statistically valid one:
• 2 weeks before treatment Mn = 0.866;
• 2 weeks during AED treatment Mn = 0.899;
• 4 weeks before treatment Mn = 0.880;
• 4 weeks during AED treatment Mn = 0.972.
Therefore, we can see that even during the first 2 weeks since the beginning of AED treatment, measure Mn has shown positive dynamics, and in a 4-week timespan since the beginning of AED treatment, objective situation of Mn estimates has greatly improved.
3. Time Series Entropy
Nonlinear statistical methods, such as entropy, have found widespread use and have shown great efficiency as part of analyzing various medical data [4, 5]. Various methods of implementing the entropy calculation were developed, some of which and use thereof in medicine even have special issues of magazines dedicated to them [6]. However, presented methods of entropy calculation share a common characteristic, which is their insensitivity to change in data structure, i.e. data could be either randomly shuffled or ordered, and entropy would not change. Also, known methods usually are very demanding to quantity of analyzed data - required number of those can exceed thousands, which is sometimes hardly achievable in medical research. We have developed and proposed a robust formula for calculating entropy of time series EnRE [7]:
where MD is median of time series; Dy -distance between observed data points X and Xj in time series; A, l, m, k - estimated coefficients. Search conditions for coefficients A, l, m, k are the following [7]:
1/ accurate approximation for known distributions of a random value;
2/ independence of EnRE from N for initial time series and for series after sorting;
3/ independence of EnRE from additive changes of mean.
In [7], it was established that, in case of time series represented by RR-intervals, the following coefficient values had been found:
A =J~z~, I = 3, m = 1, k = 2. Let us note
important characteristics of the found generalized form of EnRE and coefficients:
1/ EnRE [7] and found coefficients l, m, k provide independence from additive change of mean series and from magnitude of selection N for basic series and for series after sorting;
2/ value EnRE is sensitive to structural changes in series, such as, for example, sorting which increases the degree of order in series, decreasing the EnRE. This offers additional advantages in research, as shown below for a case of NSR and CHF groups classification;
3/ readjusting coefficient A alone may be required to find the best EnRE value in another range of change in parameters of various random distributions, which can always be done using the method of least squares.
Case 3: Time series entropy EnRE for Detecting Congestive Heart Failure by standard 5-minute HRV samples (MIT-BIH RR database) [2].
Let us demonstrate the usage of EnRE for Detecting Congestive Heart Failure in short segments (N = 500) by MIT-BIH RR database. In [8], it has been shown that the minimal length of an RR-segment, for which it is possible to classify NSR and CHF groups by way of Multiscale Entropy Analysis, is N = 1000. The performance of such classification is: Se = 0.70; Sp = 0.76; Acc = 0.74. Given in Table 3 are Mean and Standard deviation of EnRE for NSR and CHF for basic RR-intervals and series after sorting (N = 500). In both cases, the differences between groups are reliable to the degree of p < 10-7.
Table 3
Mean and Standard deviation of EnRE for NSR and CHF (original and sorted data)
HRV Record (N = 500) Original RR Sorted Time Series
EnRE ^-value EnRE(sort) ^-value
NSR 1.72 ± 0.47 10-7 1.42 ± 0.51 10-7
CHF 0.65 ± 0.76 0.55 ± 0.81
Using discriminant analysis, let us assess the quality of division into NSR and CHF groups and build classification functions:
1. When using one parameter of classification EnRE, we receive the following indicators of quality of division into NSR and CHF groups
Se = 0.66; Sp = 0.93; Acc = 0.83 and classification function (1 variable EnRE; Wilks' Lambda: 0.56 approx. F (1,81) = 62.43;p < 10-7):
NSR = 4.96 * EnRE - 4.71
CHF = 1.87 * EnRE - 1.66
2. When using two parameters of classification, EnRE and EnRE(sort) for sorted series, we receive the following indicators of quality of division into NSR and CHF groups
Se = 0.76; Sp = 0.98; Acc = 0.90 and classification function (2 variables EnRE and EnRE(sort); Wilks' Lambda: 0.44; approx. F (2,80) = 50.54; p < 10-7):
NSR = 19.28 * EnRE - 13.85*EnRE(sort) - 7.22
CHF = 6.91 * EnRE - 4.87*EnRE(sort) -1.97
Therefore, proposed generalized form for Robust Entropy Estimator EnRE allows, with high accuracy, to divide NSR and CHF groups in short records (N = 500), which had remained unachieved in [8] by the way of Multiscale Entropy Analysis, and presents additional advantages provided by EnRE in case of structural changes in series (such as sorting). Quality of classification achieved by using two variables EnRE and EnRE(sort) is superior to results received in [8] by way of Multiscale Entropy Analysis for RR segments
with length of N = 1000, N = 2000 and N = 5000.
CONCLUSIONS
Developed by the authors and presented in this article are special methods for statistical analysis of Time Series:
1. Time Series Mann-Whitney M-test - an analogue of the known nonparametric MannWhitney U-test for two Time Series with an equal number of elements;
2. Nominal Time Series Measure - a statistical estimator of dynamics of a nominal series consisting of «0» (no) and «1» (yes);
3. Time Series Entropy EnRE — specially developed robust formula for a Time Series, intended for calculation of nonlinear stochastic measure of order or disorder, popular in various researches.
All presented methods are accompanied by a detailed demonstration of capacity for statistical analysis of medical Time Series:
1. Analysis of growth dynamics of boys and girls aged 6-7-8 years (WHO);
2. Analysis of number of seizures and choice of anti-epileptic drugs (NSE);
3. Time series entropy EnRE for Detecting Congestive Heart Failure by standard 5-minute HRV samples (MIT-BIH RR database).
It has been noted that, in every case, using the named special methods for statistical analysis of medical Time Series allows one to avoid errors in interpreting results received through statistical methods and substantially increases the accuracy of statistical analysis of medical Time Series.
REFERENCES
1. Mercedes de Onis et al. (2007). Development of a WHO growth reference for school-aged children and adolescents.Bulletin of the World Health Organization, 85, 660-667.
2. Goldberger, A.L. et al. (2000).Physiobank, physiotoolkit, and physionet: Components of a new research resource for complex physiologic signals. Circulation, 101, 215-220.
3. Mann, H.B., Whitney, D.R. (1947). On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other. Annals of Mathematical Statistics, 18(1),50-60. DOI:10.1214/aoms/1177730491.
4. Neal, A., Parra-Henao, G. (2007).Uses of entropy in medical research.Revista CES MEDICINA, 21(1), 65-75.
5. Boregowda, S., Handy, R., Sleeth, D., Merryweather, A. (2016). Measuring Entropy Change in a Human Physiological SystemJournal of Thermodynamics, 1, 1-8. DOI:10.1155/2016/4932710.
6. Humeau-Heurtier, A. (ed.) (2016).Special Issue «Multiscale Entropy and Its Applications in Medicine and Biology».Entropy, 17-18, 88.
7. Martynenko, A., Raimondi, G., Budreiko, N. (2019). Robust Entropy Estimator for Heart Rate Variability.Journal of Clinical Informatics and Telemedicine, 14 (15), 67-73.
8. Chengyu, L., Rui, G. (2017).Multiscale Entropy Analysis of the Differential RR Interval Time Series Signal and Its Application in Detecting Congestive Heart Failure.Entropy, 19, 251. DOI:10.3390/e19060251.
СТАТИСТИЧНИЙ АНАЛ1З МЕДИЧНИХ ЧАСОВИХ РЯД1В
Мартиненко О., PaÜMOHdi Д., Сотткова-Мелешкша Ж., Даниленко Г., Будрейко М.
Статистичний аналiз даних е необхвдним компонентом будь-якого медичного дослiдження. CyHacHÍ методи математично! статистики i пакети прикладних комп'ютерних програм надають широк можливостi для статистичного анал!зу.Однак, коли наб!р даних представлений сер!ею даних, упорядкованих за часом, або коли структура i порядок даних е важливими компонентами дослвдження, стають необхвдш спещальт тдходи до статистичного анал!зу.У статл представлен! спещальт статистичш методи, що розроблеш авторами для анал!зу медичних часових ряд!в: Мана-Уиш M-тест часових ряд!в - аналог ведомого непараметричного методу Мана-Уиш U-тесту для двох часових ряд!в
з однаковим числом елеменпв; номтальна м!ра часових ряд!в - статистична ощнка динам!ки номтального ряду,що складаеться з «0» (m) i «1» (так); ентротя часових ряд!в EnRE - спещально розроблена для часових ряд!в робастна формула обчислення популярного в наукових дослщженнях нелншна стохастична м!ра порядку. Представлен! методи супроводжуються докладною демонстращею можливостей для статистичного анал!зу медичних часових ряд!в: Анал!з динашки ростухлопчишв и д!вчат 6-7-8 рошв (за даними Всесвггаьо! Оргашзацд Охорони здоров'я); Анал!з числа епшептичних напащв i виб!р протиепшептично! терапи (за даними Национально! епшептично! стлки, Великобритания); ентротя часових ряд!в EnRE для д!агностики серцево! недостатносп на щдгрунп стандартних 5-хвилинних запиав вар!абельносп серцевого ритму (за даними Массачусетського шституту технологт - Бостонського гостталю Бет-1зраель RR база даних). Ввдзначено, що у вах випадках застосуваннярозроблених спещальних метод!в для статистичного анал!зу медичнихчасових ряд!в дозволяеуникнутипомилок в штерпретацд отриманих результатй суттево щдвищують точшсть статистичного анал!зу медичних часових ряд!в.
КЛЮЧОВ1 СЛОВА: едичш часов! ряди, непараметричний тест, номтальна м!ра, ентротя ИНФОРМАЦ1Я ПРО АВТОР1В
Мартиненко Олександр, д.ф!з-мат.н., професор кафедри гтени та сощально! медицини Харкгвського национального ушверситету шен1 В. Н. Каразгна, пл. Свободи, 6, Харкгв, Укра!на, 61022, e-mail: Alexander.v.martynenko@karazin.ua, ORCID ID: https://orcid.org/0000-0002-0609-2220
Ж. Раймонди, д.мед.н., проф., Римський ушверситет Ла Сатенца (1ташя), Piazzale Aldo Moro 5, 00185, Рим, 1талш, e-mail: gianfrancoraimondi@uniroma1.it
Сотнжова-Мелешкша Жанна, к.мед.н., зав. кафедри плени та сощально! медицини Харкгвського национального ушверситету шен1 В. Н. Каразша, майдан Свободи, 6, Харюв, Укра!на, 61022, e-mail: zhanna.v.sotnikova@karazin.ua, ORCID ID: https://orcid.org/0000-0001-5534-8264
Даниленко Георгш, д.мед.н., проф., ДУ «1нститут охорони здоров'я дней та тдшткгв НАМН Укра!ни», пр. Ювшейний, 52-а, Харкгв, Укра!на, 61153, e-mail: danylenkog@ukr.net, ORCID ID: https://orcid.org/0000-0001-7086-2720
Будрейко Микита, асистент кафедри гтени та сощально! медицини Харювського национального ушверситету шен1 В. Н. Каразша майдан Свободи, 6, Харкгв, Укра!на, 61022, e-mail: nbudreiko@protonmail.com
СТАТИСТИЧЕСКИЙ АНАЛИЗ МЕДИЦИНСКИХ ВРЕМЕННЫХ РЯДОВ
Мартыненко A., Раймонди Д., Сотникова-Meлешкинa Ж., Даниленко Г., Будрейко Н.
Статистический анализ данных является необходимым компонентом любого медицинского исследования. Современные методы математической статистики и пакеты прикладных компьютерных программ предоставляют широкие возможности для статистического анализа. Однако, когда набор данных представлен серией данных, упорядоченных по времени, или когда структура и порядок данных являются важными компонентами исследования, становятся необходимы специальные подходы к статистическому анализу. В статье представлены специальные статистические методы, разработанные авторами для анализа медицинских временных рядов: Манна--Уитни M-тест временных рядов - аналог известного непараметрического метода Манна-Уитни U-теста для двух временных рядов с одинаковым числом элементов; номинальная мера временных рядов - статистическая оценка динамики номинального ряда, состоящего из «0» (нет) и «1» (да); энтропия временных рядов EnRE -специально разработанная для временных рядов робастная формула вычисления популярного в научных исследованиях нелинейная стохастическая мера порядка. Представленные методы сопровождаются подробной демонстрацией возможностью для статистического анализа медицинских временных рядов: Анализ динамики роста мальчиков и девочек 6-7-8лет (по данным Всемирной Организации Здравоохранения); Анализ числа эпилептических приступов и выбор антиэпилептической терапии (по данным Национального эпилептического общества, Великобритания); энтропия временных рядов EnRE для диагностики сердечной недостаточности на основании стандартных 5-минутных записей вариабельности сердечного ритма (Массачусетского института технологий - Бостонского госпиталя Бет-Израиль база данных RR).Отмечено, что во всех случаях использование указанных специальных методов для статистического анализа медицинских временных рядов позволяет избежать ошибок в интерпретации полученных результатов и существенно повышают точность статистического анализа медицинских временных рядов.
КЛЮЧЕВЫЕ СЛОВА: медицинские временные ряды, непараметрический тест, номинальная мера, энтропия
ИНФОРМАЦИЯ ОБ АВТОРАХ
Мартыненко Александр, д.физ-мат.н., профессор кафедры гигиены и социальной медицины Харьковского национального университета имени В. Н. Каразина, пл. Свободи, 6, Харьков, Украина, 61022, e-mail: Alexander.v.martynenko@karazin.ua, ORCID ID: https://orcid.org/0000-0002-0609-2220
Ж. Раймонди, д.мед.н., проф., Университет Рима «Сапиенза», Рим, Италия, Piazzale Aldo Moro 5, 00185, Рим, Италия, e-mail: gianfrancoraimondi@uniroma1 .it
Сотникова-Мелешкина Жанна, к.мед.н., зав. кафедры гигиены и социальной медицины Харьковского национального университета имени В. Н. Каразина, пл. Свободи, 6, Харьков, Украина, 61022, e-mail: zhanna.v.sotnikova@karazin.ua, ORCID ID: https://orcid.org/0000-0001-5534-8264
Даниленко Георгий, д. мед. н., проф., ГУ «Институт охраны здоровья детей и подростков НАМН Украины», проспект Юбилейный, 52-а, Харьков, Украина, 61153, e-mail: danylenkog@ukr.net, ORCID ID: https://orcid.org/0000-0001 -7086-2720
Будрейко Никита, асистент кафедры гигиены и социальной медицины Харьковского национального университета имени В. Н. Каразина, пл. Свободи, 6, Харьков, Украина, 61022, e-mail: nbudreiko@protonmail.com
