Grouping data and constructing a distribution series. Statistical study of variation series and calculation of average values How to find variation series in statistics

Condition:

There is data on the age composition of workers (years): 18, 38, 28, 29, 26, 38, 34, 22, 28, 30, 22, 23, 35, 33, 27, 24, 30, 32, 28, 25, 29, 26, 31, 24, 29, 27, 32, 25, 29, 29.

1. Construct an interval distribution series.
2. Construct a graphical representation of the series.
3. Graphically determine the mode and median.

Solution:

1) According to the Sturgess formula, the population must be divided into 1 + 3.322 lg 30 = 6 groups.

Maximum age - 38, minimum - 18.

Interval width Since the ends of the intervals must be integers, we divide the population into 5 groups. Interval width - 4.

To make calculations easier, we will arrange the data in ascending order: 18, 22, 22, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 29, 29, 29, 29, 30 , 30, 31, 32, 32, 33, 34, 35, 38, 38.

Age distribution of workers

Graphically, a series can be depicted as a histogram or polygon. Histogram - bar chart. The base of the column is the width of the interval. The height of the column is equal to the frequency.

Polygon (or distribution polygon) - frequency graph. To build it using a histogram, we connect the midpoints of the upper sides of the rectangles. We close the polygon on the Ox axis at distances equal to half the interval from the extreme values of x.

Mode (Mo) is the value of the characteristic being studied, which occurs most frequently in a given population.

To determine the mode from a histogram, you need to select the highest rectangle, draw a line from the right vertex of this rectangle to the upper right corner of the previous rectangle, and from the left vertex of the modal rectangle draw a line to the left vertex of the subsequent rectangle. From the intersection of these lines, draw a perpendicular to the x-axis. The abscissa will be fashion. Mo ≈ 27.5. This means that the most common age in this population is 27-28 years old.

Median (Me) is the value of the characteristic being studied, which is in the middle of the ordered variation series.

We find the median using the cumulate. Cumulates - a graph of accumulated frequencies. Abscissas are variants of a series. Ordinates are accumulated frequencies.

To determine the median over the cumulate, we find a point along the ordinate axis corresponding to 50% of the accumulated frequencies (in our case, 15), draw a straight line through it, parallel to the Ox axis, and from the point of its intersection with the cumulate, draw a perpendicular to the x axis. The abscissa is the median. Me ≈ 25.9. This means that half of the workers in this population are under 26 years of age.

Glossary of statistical terms

General statistics questions

WHAT ARE MEDICAL STATISTICS?

Statistics is the quantitative description and measurement of events, phenomena, objects. It is understood as a branch of practical activity (collection, processing and analysis of data on mass phenomena), as a branch of knowledge, i.e. special scientific discipline, and, as a set of summary, final digital indicators collected to characterize any area of social phenomena.

Statistics is a science that studies the patterns of mass phenomena using the method of generalizing indicators.

Medical statistics – independent social science, studying quantitative side of mass social phenomena inextricably linked with their qualitative side, allowing method of generalizing indicators study the patterns of these phenomena, the most important processes in economic, social life society, its health, the system of organizing medical care to the population.

Statistical methods are a set of techniques for processing mass observation materials, which include: grouping, summary, obtaining indicators, their statistical analysis, etc.

Statistical methods in medicine are used to:

condition study public health the population as a whole and its main groups by collecting and analyzing statistical data on the size and composition of the population, its reproduction, physical development, prevalence and duration of various diseases, etc.;
identifying and establishing connections general level morbidity and mortality from any individual diseases with various factors environment;
collection and study of numerical data on the network of medical institutions, their activities and personnel for planning health care activities, monitoring the implementation of development plans for the network and the activities of health care institutions and assessing the quality of work of individual medical institutions;
assessing the effectiveness of measures to prevent and treat diseases;
determination of the statistical significance of research results in the clinic and experiment.

Sections of medical statistics:

general theoretical and methodological foundations statistics,
population health statistics,
health statistics.

CREATION OF A DATABASE IN MS EXCEL

In order for the database to be convenient for subsequent processing, simple principles should be followed:

1) The optimal program for creating a database is MS Excel. Data from Excel can subsequently be easily transferred to other specialized statistical packages, such as Statistica, SPSS, etc. for more complex manipulations. However, up to 80-90% of calculations can be conveniently performed in Excel itself using the Data Analysis add-in.

2) The top line of the table with the database is designed as a header, where the names of those indicators that are taken into account in this column are entered. It is undesirable to use cell merging (this requirement generally applies to the entire database), since this will make many operations invalid. Also, you should not create a “two-story” header, in which the top line indicates the name of a group of homogeneous indicators, and the bottom line indicates specific indicators. To group homogeneous indicators, it is better to mark them with a single-color fill or include a grouping feature in parentheses in their name.

For example, not this way:

GENERAL BLOOD ANALYSIS
ER	LEU	TR

ER(UAC)

LEU(UAC)

TR(UAC)

in the latter version, both the “single-story” header and the visual homogeneity of the data are ensured (all of them relate to UAC indicators).

3) The first column should contain the patient’s serial number in this database, without linking it to any of the indicators being studied. This will allow you to subsequently ensure an easy rollback to the original order of patients at any stage, even after numerous sortings of the list.

4) The second column is usually filled with the last names (or full names) of patients.

5) Quantitative indicators (those that are measured in numbers, for example - height, weight, blood pressure, heart rate, etc.) are entered into the table in numerical format. It would seem that this is already clear, but you should remember that in Excel, starting from version 2007, fractional values are denoted by a dot: 4.5. If you write a number separated by a comma, it will be perceived as text, and these columns will have to be rewritten.

6) It’s more difficult with qualitative indicators. Those of them that have two variants of meaning (the so-called binary values: Yes-No, Present-Absent, Male-Female) are better translated into binary system: 0 and 1. The value 1 is usually assigned to a positive value (Yes, Present), 0 to a negative value (No, Absent).

7) Qualitative indicators that have several values, differing in severity, level of phenomenon (Weak-Medium-Strong; Cold-Warm-Hot) can be ranked and, accordingly, also translated into numbers. The lowest level of the phenomenon is assigned the lowest rank - 0 or 1, the following degrees are indicated by the values of the ranks in order. For example: No disease - 0, mild degree - 1, moderate degree - 2, severe degree - 3.

8) Sometimes several values correspond to one quality indicator. For example, in the “Concomitant diagnosis” column, if there are several diseases, we want to indicate them separated by commas. This should not be done, since processing such data is very difficult and cannot be automated. Therefore, it is better to make several columns with specific groups of diseases ("diseases of the cardiovascular system", "diseases of the gastrointestinal tract", etc.) or certain nosologies ("chronic gastritis", "IHD", etc.), in which we enter the data into binary, binary form: 1 (which means “This disease exists”) - 0 (“This disease does not exist”).

9) To distinguish between individual groups of indicators, you can actively use color: for example, columns with UAC indicators are highlighted in red, OAM data in yellow, etc.

10) Each patient must correspond to one row of the table.

Such a design of the database allows not only to significantly simplify the process of its statistical processing, but also to facilitate its completion at the stage of collecting material.

WHICH METHOD TO CHOOSE FOR STATISTICAL ANALYSIS?

After all the data has been collected, each researcher is faced with the question of choosing the most appropriate method of statistical processing. And this is not surprising: modern statistics combines a huge number of various criteria and methods. They all have their own characteristics and may or may not be suitable for two seemingly similar situations. In this article we will try to systematize all the basic, most common methods statistical analysis according to their purpose.

However, first, a few words about what kind of statistical data there are, since this is what determines the choice of the most suitable analysis method.

Measurement scale

When conducting a study, the values of each observation unit are determined various signs. Depending on the scale on which they are measured, all signs are divided into quantitative And quality. Qualitative indicators in studies are distributed according to the so-called nominal scale. In addition, indicators can be presented according to rank scale.

For example, a comparison is made of cardiac performance in athletes and people leading a sedentary lifestyle.

In this case, the following signs were determined in the subjects:

floor- is nominal an indicator that takes two values - male or female.
age - quantitative index,
sports - nominal an indicator that takes two meanings: engaged or not engaged,
heart rate - quantitative index,
systolic blood pressure - quantitative index,
presence of complaints of chest pain- is high quality indicator, the values of which can be determined both by nominal(there are complaints - there are no complaints), and according to rank scale depending on frequency (for example, if pain occurs several times a day - the indicator is assigned rank 3, several times a month - rank 2, several times a year - rank 1, if there are no complaints of chest pain - rank 0) .

Number of compared populations

The next issue that needs to be addressed in choosing a statistical method is the number of populations to be compared within the study.

In most cases, in clinical trials we deal with two groups of patients - basic And control. Basic, or experienced, is generally considered to be the group in which the method of diagnosis or treatment being studied was applied, or in which patients suffer from the disease that is the subject of this study. Test the group, in contrast, consists of patients receiving usual care, placebo, or those who do not have the disease being studied. Such populations, represented by different patients, are called unrelated.
There are still related, or doubles, aggregates, when we are talking about the same people, but the values of some characteristic obtained are compared before and after research. The number of compared populations is also equal to 2, but different techniques are applied to them than to unrelated ones.
Another option is to describe one totality, which, it must be admitted, generally underlies any research. Even if the main purpose of the work is to compare two or more groups, each of them must first be characterized. Methods used for this descriptive statistics. In addition, for one population methods can be applied correlation analysis , used to find a relationship between two or more characteristics being studied (for example, the dependence of height on body weight or the dependence of heart rate on body temperature).
Finally, there may be several populations being compared. This is very common in medical research. Patients can be grouped depending on the use of various drugs (for example, when comparing the effectiveness of antihypertensive drugs: group 1 - ACE inhibitors, 2 - beta-blockers, 3 - centrally acting drugs), according to the severity of the disease (group 1 - mild, 2 - medium, 3 - heavy), etc.

It is also important to ask normality of distribution populations being studied. This determines whether the methods can be applied parametric analysis or just nonparametric. The conditions that must be met in normally distributed populations are:

maximum proximity or equality of the values of the arithmetic mean, mode and median;
compliance with the “three sigma” rule (at least 68.3% variants are in the M±1σ interval, at least 95.5% variants are in the M±2σ interval, at least 99.7% variants are in the M±3σ interval;
indicators are measured on a quantitative scale;
positive results of testing for normality of distribution using special criteria - Kolmogorov-Smirnov or Shapiro-Wilk.

After determining all the characteristics we have indicated for the populations under study, we suggest using the following table to select the most optimal method of statistical analysis.

Method	Indicator measurement scale	Number of compared populations	Purpose of processing	Data distribution
Student's t-test	quantitative	2		normal
Student's t-test with Bonferroni correction	quantitative	3 or more	no comparison related sets	normal
Paired Student's t-test	quantitative	2		normal
One-way analysis of variance (ANOVA)	quantitative	3 or more	comparison of unrelated populations	normal
One-way analysis of variance (ANOVA) with repeated measures	quantitative	3 or more	comparison of related populations	normal
Mann-Whitney U test	quantitative, ranking	2	comparison of unrelated populations	any
Rosenbaum's Q test	quantitative, ranking	2	comparison of unrelated populations	any
Kruskal-Wallis test	quantitative	3 or more	comparison of unrelated populations	any
Wilcoxon test	quantitative, ranking	2	comparison of related populations	any
G-sign test	quantitative, ranking	2	comparison of related populations	any
Friedman criterion	quantitative, ranking	3 or more	comparison of related populations	any
Pearson's χ2 test	nominal	2 or more	comparison of unrelated populations	any
Fisher's exact test	nominal	2	comparison of unrelated populations	any
McNemar test	nominal	2	comparison of related populations	any
Cochran's Q test	nominal	3 or more	comparison of related populations	any
Relative risk (Risk Ratio, RR)	nominal	2	comparison of unrelated populations in cohort studies	any
Odds Ratio (OR)	nominal	2	comparison of unrelated populations in case-control studies	any
Pearson correlation coefficient	quantitative	2 rows of measurements		normal
Spearman's rank correlation coefficient	quantitative, ranking	2 rows of measurements	identifying connections between signs	any
Kendall correlation coefficient	quantitative, ranking	2 rows of measurements	identifying connections between signs	any
Kendall's coefficient of concordance	quantitative, ranking	3 or more rows of measurements	identifying connections between signs	any
Calculation of average values (M) and average errors (m)	quantitative	1	descriptive statistics	any
Calculation of medians (Me) and percentiles (quartiles)	rank	1	descriptive statistics	any
Calculation of relative values (P) and average errors (m)	nominal	1	descriptive statistics	any
Shapiro-Wilk test	quantitative	1	distribution analysis	any
Kolmogorov-Smirnov criterion	quantitative	1	distribution analysis	any
Smirnov-Cramer-von Mises criterion ω 2	quantitative	1	distribution analysis	any
Kaplan-Meier method	any	1	survival analysis	any
Cox proportional hazards model	any	1	survival analysis	any

Great Statisticians

Karl Pearson (March 27, 1857 – April 27, 1936)

Karl Pearson, the great English mathematician, statistician, biologist and philosopher, was born on March 27, 1857; founder mathematical statistics, one of the founders of biometrics.

Having received the position of professor at the age of 27 applied mathematics At University College London, Karl Pearson began to study statistics, which he perceived as a general scientific tool, consistent with his not at all generally accepted thoughts about the need to provide students with a broad outlook.

Pearson's main achievements in the field of statistics include the development of the foundations of the theory of correlation and contingency of characteristics, the introduction of “Pearson curves” to describe empirical distributions and the extremely important chi-square criterion, as well as the compilation of a large number of statistical tables. Pearson applied the statistical method and especially the theory of correlation in many branches of science.

Here is one of his statements: “The first amateur introduction of modern statistical methods into established science is opposed by typical contempt. But I lived to see the time when many of them began to secretly apply the very methods that they initially condemned.”

And already in 1920, Pearson wrote a note in which he stated that the goal of the biometric school was “to transform statistics into a branch of applied mathematics, to generalize, discard or justify the meager methods of the old school of political and social statisticians, and, in general, to transform statistics from the playing field to amateurs and debaters into a serious branch of science. It was necessary to criticize imperfect and often erroneous methods in medicine, anthropology, craniometry, psychology, criminology, biology, sociology, in order to provide these sciences with new and more powerful means. The battle lasted almost twenty years, but many appeared signs that the old hostilities were left behind and new methods were universally accepted."

Karl Pearson had very diverse interests: he studied physics in Heidelberg, was interested in the social and economic role of religion, and even lectured on German history and literature in Cambridge and London.

A little-known fact is that at the age of 28, Karl Pearson lectured on the “women’s question” and even founded the Men and Women’s Club, which existed until 1889, in which everything related to women, including relationships between the sexes, was freely and unrestrictedly discussed.

The club was made up of an equal number of men and women, mostly middle-class liberals, socialists and feminists.

The subject of the club's discussions was a wide range of issues: from sexual relations in ancient Greek Athens to the situation of Buddhist nuns, from attitudes towards marriage to the problems of prostitution. In essence, the Men and Women Club challenged long-established norms of male-female interaction, as well as ideas about “proper” sexuality. In Victorian England, where sexuality was seen by many as “base” and “animal” and ignorance about sex education was widespread, discussion of such issues was truly radical.

In 1898, Pearson was awarded the Darwin Medal by the Royal Society, which he refused, believing that awards “should be given to young people to encourage them.”

Florence Nightingale (12 May 1820 – 13 August 1910)

Florence Nightingale (1820-1910) - nurse and public figure in Great Britain, on whose birthday we celebrate International Nurses Day today.

She was born in Florence into a wealthy aristocratic family, received an excellent education, and knew six languages. WITH youth dreamed of becoming a sister of mercy, in 1853 she received nursing education at the community of sisters of Pastor Flender in Kaiserwerth and became the manager of a small private hospital in London.

In October 1854, during Crimean War, Florence, along with 38 assistants, went to field hospitals in Crimea. While organizing care for the wounded, she consistently implemented the principles of sanitation and hygiene. As a result, in less than six months, mortality in hospitals decreased from 42 to 2.2%!

Having set herself the task of reforming the medical service in the army, Nightingale ensured that hospitals were equipped with ventilation and sewage systems; hospital staff were required to undergo necessary preparation. A military medical school was organized, and explanatory work was carried out among soldiers and officers about the importance of disease prevention.

Florence Nightingale's great contributions to medical statistics!

Her 800-page book Notes on the Factors Affecting the Health, Efficiency and Management of British Army Hospitals (1858) contained an entire section devoted to statistics and illustrated with diagrams.
Nightingale was an innovator in the use of graphical images in statistics. She invented pie charts, which she called "cockscomb" and used to describe the structure of mortality. Many of her charts were included in the report of the Commission on Health Problems in the Army, which led to the decision to reform army medicine.
She developed the first form for collecting statistics in hospitals, which is the predecessor of modern reporting forms on hospital activities.

In 1859 she was elected a fellow of the Royal Statistical Society and subsequently became an honorary member of the American Statistical Association.

⠀

Johann Carl Friedrich Gauss (April 30, 1777 – February 23, 1855)

On April 30, 1777, the great German mathematician, mechanic, physicist, astronomer, surveyor and statistician Johann Carl Friedrich Gauss was born in the city of Braunschweig.

He is considered one of the greatest mathematicians of all time, the "King of Mathematicians". Laureate of the Copley Medal (1838), foreign member of the Swedish (1821) and Russian (1824) Academies of Sciences, and the English Royal Society.

Already at the age of three, Karl could read and write, even correcting his father’s calculation mistakes. According to legend, a school mathematics teacher, in order to keep children busy for a long time, asked them to count the sum of numbers from 1 to 100. Young Gauss noticed that pairwise sums from opposite ends are the same: 1+100=101, 2+99=101, etc. etc., and instantly got the result: 50×101=5050. Until his old age, he was accustomed to doing most of his calculations in his head.

The main scientific achievements of Carl Gauss in statistics are the creation of the least squares method, which underlies regression analysis.

He also studied in detail the normal distribution law widespread in nature, the graph of which has since often been called the Gaussian. The “three sigma” rule (Gauss’s rule) describing the normal distribution has become widely known.

Lev Semyonovich Kaminsky (1889 – 1962)

On the 75th anniversary of the Victory in the Great Patriotic War I would like to remember and talk about a wonderful scientist, one of the founders of military medical and sanitary statistics in the USSR - Lev Semenovich Kaminsky (1889-1962).

He was born on May 27, 1889 in Kyiv. After graduating with honors from the Faculty of Medicine of Petrograd University in 1918, Kaminsky was in the ranks of the Red Army, from April 1919 to the end of 1920 he held the position of chief physician of the 136th consolidated evacuation hospital of the South-Eastern Front.

Since 1922, Lev Semyonovich was in charge of the sanitary and epidemiological department of the medical and sanitary service of the North-Western Railway. During these years it began scientific activity Kaminsky under the guidance of prof. S.A.Novoselsky. In their joint fundamental work, “Losses in Past Wars,” statistical material was analyzed on human losses in the wars of various armies of the world from 1756 to 1918. In subsequent works, Kaminsky developed and substantiated a new, more accurate classification of military losses.

The monograph “National Nutrition and Public Health” (1929) examined in detail the sanitary and hygienic aspects of the impact of wars on public health, as well as issues of organizing medical care for the population and the army during the war.

From 1935 to 1943, Lev Semenovich headed the department of sanitary (since 1942 - medical) statistics of the People's Commissariat of Health of the USSR. In October 1943, Prof. Kaminsky became head of the department of military medical statistics at the Military Medical Academy named after. S.M. Kirov, and since 1956 he has held the position of professor at the Department of Statistics and Accounting at Leningrad State University.

Lev Semyonovich advocated widespread implementation quantitative methods into the practice of sanitary and medical statistics. In 1959, under his authorship, it was published tutorial“Statistical processing of laboratory and clinical data: the application of statistics in the scientific and practical work of a doctor,” which for many years became one of the best domestic textbooks on medical statistics. In the preface, L.S. Kaminsky notes:
“... It seems important that treating physicians know how to get down to business and know how to collect and process the correct numbers, suitable for comparisons and comparisons.”

Criteria and methods

STUDENT'S t-CRITERION FOR INDEPENDENT POPULATIONS

Student's t-test is a general name for a class of methods for statistical testing of hypotheses (statistical tests) based on the Student distribution. The most common uses of the t-test involve testing the equality of means in two samples.

This criterion was developed William Seeley Gossett

2. What is the Student's t-test used for?

Student's t test is used to determine the statistical significance of differences in means. It can be used both in cases of comparison of independent samples (for example, a group of patients with diabetes and a group of healthy people) and when comparing related populations (for example, the average heart rate in the same patients before and after taking an antiarrhythmic drug). In the latter case, the paired Student t-test is calculated

3. In what cases can the Student’s t-test be used?

To apply the Student's t-test, it is necessary that the original data have a normal distribution. The equality of variances (distributions) of the compared groups (homoscedasticity) is also important. For unequal variances, the t-test as modified by Welch (Welch's t) is used.

With absence normal distribution compared samples, instead of Student's t-test, similar methods of nonparametric statistics are used, among which the most well-known is Mann-Whitney U test.

4. How to calculate Student's t-test?

To compare average values, Student's t-test is calculated using the following formula:

Where M 1- arithmetic mean of the first compared population (group), M 2- arithmetic mean of the second compared population (group), m 1- average error of the first arithmetic mean, m 2- average error of the second arithmetic mean.

The resulting Student's t-test value must be interpreted correctly. To do this, we need to know the number of subjects in each group (n 1 and n 2). Finding the number of degrees of freedom f according to the following formula:

F = (n 1 + n 2) - 2

After this, we determine the critical value of the Student’s t-test for the required level of significance (for example, p = 0.05) and for a given number of degrees of freedom f according to the table (see below).

If the calculated value of the Student's t-test is equal to or greater than the critical value found from the table, we conclude that the differences between the compared values are statistically significant.
If the value of the calculated Student's t-test is less than the table value, then the differences between the compared values are not statistically significant.

To study the effectiveness of a new iron preparation, two groups of patients with anemia were selected. In the first group, patients received a new drug for two weeks, and in the second group they received a placebo. After this, hemoglobin levels in peripheral blood were measured. In the first group, the average hemoglobin level was 115.4±1.2 g/l, and in the second group - 103.7±2.3 g/l (data presented in M±m format), the compared populations have a normal distribution. The number of the first group was 34, and the second - 40 patients. It is necessary to draw a conclusion about the statistical significance of the differences obtained and the effectiveness of the new iron preparation.

Solution: To assess the significance of differences, we use Student’s t-test, calculated as the difference in mean values divided by the sum of squared errors:

After performing the calculations, the t-test value turned out to be 4.51. We find the number of degrees of freedom as (34 + 40) - 2 = 72. We compare the resulting Student's t-test value of 4.51 with the critical value at p = 0.05 indicated in the table: 1.993. Since the calculated value of the criterion is greater than the critical value, we conclude that the observed differences are statistically significant (significance level p<0,05).

PAIRED STUDENT'S t-TEST

The paired Student's t-test is one of the modifications of the Student's method, used to determine the statistical significance of differences in paired (repeated) measurements.

1. History of the development of the t-test

t-test was developed William Gossett to assess the quality of beer in the Guinness company. Due to obligations to the company regarding non-disclosure of trade secrets, Gosset's article was published in 1908 in the journal Biometrics under the pseudonym "Student".

2. What is the paired Student's t-test used for?

The paired Student's t-test is used to compare two dependent (paired) samples. Dependent measurements are those taken in the same patients but at different times, for example, blood pressure in hypertensive patients before and after taking an antihypertensive drug. The null hypothesis states that there are no differences between the samples being compared, the alternative hypothesis states that there are statistically significant differences.

3. In what cases can you use the paired Student's t-test?

The main condition is the dependence of the samples, that is, the values being compared must be obtained from repeated measurements of one parameter in the same patients.

As in the case of comparisons of independent samples, to use a paired t-test, the original data must be normally distributed. If this condition is not met, nonparametric statistical methods should be used to compare sample means, such as G-sign test or Wilcoxon T-test.

The paired t test can only be used when comparing two samples. If you need to compare three or more repeated measurements, you should use one-way analysis of variance (ANOVA) for repeated measures.

4. How to calculate paired Student's t-test?

The paired Student's t-test is calculated using the following formula:

Where M d- arithmetic average of the differences between the indicators measured before and after, σ d- standard deviation of differences in indicators, n- number of subjects studied.

5. How to interpret the Student's t-test value?

The interpretation of the resulting paired Student's t-test value does not differ from the assessment of the t-test for unrelated populations. First of all, you need to find the number of degrees of freedom f according to the following formula:

F = n - 1

After this, we determine the critical value of the Student’s t-test for the required level of significance (for example, p<0,05) и при данном числе степеней свободы f according to the table (see below).

We compare the critical and calculated values of the criterion:

If the calculated value of the paired Student's t-test is equal to or greater than the critical value found from the table, we conclude that the differences between the compared values are statistically significant.
If the value of the calculated paired Student's t-test is less than the table value, then the differences between the compared values are not statistically significant.

6. Example of calculating Student's t-test

To evaluate the effectiveness of the new hypoglycemic agent, blood glucose levels were measured in patients with diabetes mellitus before and after taking the drug. As a result, the following data was obtained:

Solution:

1. Calculate the difference of each pair of values (d):

Patient N	Blood glucose level, mmol/l		Difference (d)
Patient N	before taking the drug	after taking the drug	Difference (d)
1	9.6	5.7	3.9
2	8.1	5.4	2.7
3	8.8	6.4	2.4
4	7.9	5.5	2.4
5	9.2	5.3	3.9
6	8.0	5.2	2.8
7	8.4	5.1	3.3
8	10.1	6.9	3.2
9	7.8	7.5	2.3
10	8.1	5.0	3.1

2. Find the arithmetic mean of the differences using the formula:

3. Find the standard deviation of the differences from the average using the formula:

4. Calculate the paired Student’s t-test:

5. Let us compare the obtained value of Student's t-test 8.6 with the table value, which, with the number of degrees of freedom f equal to 10 - 1 = 9 and the significance level p = 0.05, is 2.262. Since the obtained value is greater than the critical value, we conclude that there are statistically significant differences in blood glucose levels before and after taking the new drug.

MANN-WHITNEY U-CRITERION

The Mann-Whitney U test is a nonparametric statistical test used to compare two independent samples in terms of the level of a quantitatively measured trait. The method is based on determining whether the zone of intersecting values between two variation series (a ranked series of parameter values in the first sample and the same in the second sample) is small enough. The lower the criterion value, the more likely it is that the differences between the parameter values in the samples are reliable.

1. History of the development of the U-criterion

This method of identifying differences between samples was proposed in 1945 by an American chemist and statistician Frank Wilcoxon.
In 1947, it was significantly revised and expanded by mathematicians H.B. Mann(H.B. Mann) and D.R. Whitney(D.R. Whitney), by whose names today it is usually called.

2. What is the Mann-Whitney U test used for?

The Mann-Whitney U test is used to assess differences between two independent samples in terms of the level of any quantitative characteristic.

3. In what cases can the Mann-Whitney U test be used?

The Mann-Whitney U test is a nonparametric test, therefore, unlike Student's t-test

The U-test is suitable for comparing small samples: each sample must have at least 3 characteristic values. It is allowed that there are 2 values in one sample, but then the second must have at least five.

The condition for applying the Mann-Whitney U test is the absence of matching attribute values in the compared groups (all numbers are different) or a very small number of such matches.

An analogue of the Mann-Whitney U test for comparing three or more groups is Kruskal-Wallis test.

4. How to calculate the Mann-Whitney U test?

First, from both compared samples, a single ranked series, by arranging observation units according to the degree of increasing attribute and assigning a lower rank to a smaller value. In the case of equal values of a characteristic for several units, each of them is assigned the arithmetic mean of successive rank values.

For example, two units occupying 2nd and 3rd place (rank) in a single ranked row have the same values. Therefore, each of them is assigned a rank equal to (3 + 2) / 2 = 2.5.

In the compiled single ranked series, the total number of ranks will be equal to:

N = n 1 + n 2

where n 1 is the number of elements in the first sample, and n 2 is the number of elements in the second sample.

Next, we again divide the single ranked series into two, consisting respectively of units of the first and second samples, while remembering the rank values for each unit. We calculate separately the sum of ranks that fall on the share of the elements of the first sample, and separately - on the share of the elements of the second sample. We determine the larger of the two rank sums (T x) corresponding to a sample with n x elements.

Finally, we find the value of the Mann-Whitney U test using the formula:

5. How to interpret the value of the Mann-Whitney U test?

We compare the resulting value of the U-test using the table for the selected level of statistical significance (p=0.05 or p=0.01) with the critical value of U for a given number of compared samples:

If the resulting value U less tabular or equals him, then the statistical significance of the differences between the levels of the trait in the samples under consideration is recognized (the alternative hypothesis is accepted). The smaller the U value, the higher the reliability of the differences.
If the resulting value U more tabular, the null hypothesis is accepted.

WILCOxon CRITERION

Wilcoxon test for related samples (also called Wilcoxon T-test, Wilcoxon test, Wilcoxon signed rank test, Wilcoxon rank sum test) is a nonparametric statistical test used to compare two related (paired) samples in terms of the level of any quantitative characteristic measured on a continuous or ordinal scale.

The essence of the method is that the absolute values of the severity of shifts in one direction or another are compared. To do this, first all absolute values of shifts are ranked, and then the ranks are summed up. If shifts in one direction or another occur randomly, then the sums of their ranks will be approximately equal. If the intensity of shifts in one direction is greater, then the sum of the ranks of the absolute values of shifts in the opposite direction will be significantly lower than it could be with random changes.

1. History of the development of the Wilcoxon test for related samples

The test was first proposed in 1945 by American statistician and chemist Frank Wilcoxon (1892-1965). In the same scientific work, the author described another criterion used in the case of comparing independent samples.

2. What is the Wilcoxon test used for?

The Wilcoxon T test is used to evaluate differences between two sets of measurements taken on the same population but under different conditions or at different times. This test can reveal the direction and severity of changes - that is, whether indicators are more shifted in one direction than in another.

A classic example of a situation in which the Wilcoxon T-test for related populations can be used is a before-after study that compares scores before and after treatment. For example, when studying the effectiveness of an antihypertensive drug, blood pressure is compared before and after taking the drug.

3. Conditions and limitations of using the Wilcoxon T-test

The Wilcoxon test is a nonparametric test, therefore, unlike paired Student's t-test, does not require a normal distribution of the populations being compared.
The number of subjects when using the Wilcoxon T-test must be at least 5.
The studied trait can be measured both on a quantitative continuous scale (blood pressure, heart rate, leukocyte content in 1 ml of blood) and on an ordinal scale (number of points, severity of the disease, degree of contamination with microorganisms).
This criterion is used only when comparing two series of measurements. An analogue of the Wilcoxon T-test for comparing three or more related populations is Friedman criterion.

4. How to calculate the Wilcoxon T-test for related samples?

Calculate the difference between the values of paired measurements for each subject. Zero shifts are not taken into account further.
Determine which of the differences are typical, that is, correspond to the direction of change in the indicator that is dominant in frequency.
Rank the differences of the pairs according to their absolute values (that is, without taking into account the sign), in ascending order. The smaller absolute value of the difference is assigned a lower rank.
Calculate the sum of ranks corresponding to atypical shifts.

Thus, the Wilcoxon T-test for related samples is calculated using the following formula:

where ΣRr is the sum of ranks corresponding to atypical changes in the indicator.

5. How to interpret the value of the Wilcoxon test?

The resulting value of the Wilcoxon T-test is compared with the critical value according to the table for the selected level of statistical significance ( p=0.05 or p=0.01) for a given number of compared samples n:

If the calculated (empirical) value of T em. less than the tabulated T cr. or equal to it, then the statistical significance of changes in the indicator in the typical direction is recognized (the alternative hypothesis is accepted). The lower the T value, the higher the reliability of the differences.
If T emp. more T cr. , the null hypothesis about the absence of statistical significance of changes in the indicator is accepted.

Example of calculating the Wilcoxon test for related samples

A pharmaceutical company is researching a new drug from the group of non-steroidal anti-inflammatory drugs. For this purpose, a group of 10 volunteers suffering from ARVI with hyperthermia was selected. Their body temperature was measured before and 30 minutes after taking the new drug. It is necessary to draw a conclusion about the significance of the decrease in body temperature as a result of taking the drug.

The source data is presented in the following table:

To calculate the Wilcoxon T-test, we calculate the differences between paired indicators and rank their absolute values. In this case, we highlight atypical ranks in red:

N	Surname	body t before taking the drug	t body after taking the drug	Difference of indicators, d	\|d\|	Rank
1.	Ivanov	39.0	37.6	-1.4	1.4	7
2.	Petrov	39.5	38.7	-0.8	0.8	5
3.	Sidorov	38.6	38.7	0.1	0.1	1.5
4.	Popov	39.1	38.5	-0.6	0.6	4
5.	Nikolaev	40.1	38.6	-1.5	1.5	8
6.	Kozlov	39.3	37.5	-1.8	1.8	9
7.	Ignatiev	38.9	38.8	-0.1	0.1	1.5
8.	Semenov	39.2	38.0	-1.2	1.2	6
9.	Egorov	39.8	39.8	0	—	—
10.	Alekseev	38.8	39.3	0.5	0.5	3

As we see, typical shift the indicator is its decrease, noted in 7 cases out of 10. In one case (in patient Egorov), the temperature did not change after taking the drug, and therefore this case was not used in further analysis. In two cases (in patients Sidorov and Alekseev) it was noted atypical shift temperatures upward. The ranks corresponding to an atypical shift are 1.5 and 3.

Let's calculate the Wilcoxon T-test, which is equal to the sum of ranks corresponding to the atypical shift of the indicator:
T = ΣRr = 3 + 1.5 = 4.5
Let's compare T emp. with T cr. , which at the significance level p=0.05 and n=9 is equal to 8. Therefore, T emp.
We conclude: the decrease in body temperature in patients with ARVI as a result of taking a new drug is statistically significant (p<0.05).

PEARSON CHI-SQUARE CRITERION

Pearson's χ 2 test is a nonparametric method that allows us to assess the significance of differences between the actual (revealed) number of outcomes or qualitative characteristics of the sample that fall into each category and the theoretical number that would be expected in the studied groups if the null hypothesis is true. To put it simply, the method allows you to evaluate the statistical significance of differences between two or more relative indicators (frequencies, proportions).

1. History of the development of the χ 2 criterion

The chi-square test for analyzing contingency tables was developed and proposed in 1900 by an English mathematician, statistician, biologist and philosopher, the founder of mathematical statistics and one of the founders of biometrics Karl Pearson(1857-1936).

2. Why is Pearson's χ 2 test used?

The chi-square test can be used in the analysis contingency tables containing information on the frequency of outcomes depending on the presence of a risk factor. For example, a four-field contingency table looks like this:

	There is an outcome (1)	No outcome (0)	Total
There is a risk factor (1)	A	B	A+B
No risk factor (0)	C	D	C+D
Total	A+C	B+D	A+B+C+D

How to fill out such a contingency table? Let's look at a small example.

A study is being conducted on the effect of smoking on the risk of developing arterial hypertension. For this purpose, two groups of subjects were selected - the first included 70 people who smoke at least 1 pack of cigarettes daily, the second included 80 non-smokers of the same age. In the first group, 40 people had high blood pressure. In the second, arterial hypertension was observed in 32 people. Accordingly, normal blood pressure in the group of smokers was in 30 people (70 - 40 = 30) and in the group of non-smokers - in 48 (80 - 32 = 48).

We fill in the four-field contingency table with the initial data:

In the resulting contingency table, each line corresponds to a specific group of subjects. Columns show the number of people with arterial hypertension or normal blood pressure.

The task that is posed to the researcher is: are there statistically significant differences between the frequency of people with blood pressure among smokers and non-smokers? This question can be answered by calculating the Pearson chi-square test and comparing the resulting value with the critical one.

Comparable indicators should be measured on a nominal scale (for example, the patient's gender is male or female) or on an ordinal scale (for example, the degree of arterial hypertension, ranging from 0 to 3).
This method allows you to analyze not only four-field tables, when both the factor and the outcome are binary variables, that is, they have only two possible values (for example, male or female gender, the presence or absence of a certain disease in the anamnesis...). The Pearson chi-square test can also be used in the case of analyzing multifield tables, when a factor and (or) outcome takes three or more values.
The groups being compared must be independent, that is, the chi-square test should not be used when comparing before-after observations. McNemar test(when comparing two related populations) or calculated Cochran's Q test(in case of comparison of three or more groups).
When analyzing four-field tables expected values in each cell there must be at least 10. If in at least one cell the expected phenomenon takes a value from 5 to 9, the chi-square test must be calculated with Yates's amendment. If in at least one cell the expected phenomenon is less than 5, then the analysis should use Fisher's exact test.
When analyzing multifield tables, the expected number of observations should not be less than 5 in more than 20% of the cells.

4. How to calculate the Pearson chi-square test?

To calculate the chi-square test you need:

This algorithm is applicable for both four-field and multi-field tables.

5. How to interpret the value of the Pearson chi-square test?

If the obtained value of the χ 2 criterion is greater than the critical value, we conclude that there is a statistical relationship between the studied risk factor and the outcome at the appropriate level of significance.

6. Example of calculating the Pearson chi-square test

Let us determine the statistical significance of the influence of the smoking factor on the incidence of arterial hypertension using the table discussed above:

We calculate the expected values for each cell:
Find the value of the Pearson chi-square test:
χ 2 = (40-33.6) 2 /33.6 + (30-36.4) 2 /36.4 + (32-38.4) 2 /38.4 + (48-41.6) 2 /41.6 = 4.396.
The number of degrees of freedom f = (2-1)*(2-1) = 1. Using the table, we find the critical value of the Pearson chi-square test, which at the significance level p=0.05 and the number of degrees of freedom 1 is 3.841.
We compare the obtained value of the chi-square test with the critical one: 4.396 > 3.841, therefore, the dependence of the incidence of arterial hypertension on the presence of smoking is statistically significant. The significance level of this relationship corresponds to p<0.05.

FISCHER'S EXACT CRITERION

Fisher's exact test is a test that is used to compare two relative indicators that characterize the frequency of a particular characteristic that has two values. The initial data for calculating Fisher's exact test are usually grouped in the form of a four-field table.

1. History of the development of the criterion

The criterion was first proposed Ronald Fisher in his book Design of Experiments. This happened in 1935. Fischer himself claimed that Muriel Bristol prompted him to this idea. In the early 1920s, Ronald, Muriel and William Roach were stationed in England at an agricultural experimental station. Muriel claimed that she could determine the order in which tea and milk were poured into her cup. At that time, it was not possible to verify the correctness of her statement.

This gave rise to Fisher's idea of the "null hypothesis". The goal was not to prove that Muriel could tell the difference between differently prepared cups of tea. It was decided to refute the hypothesis that a woman makes a choice at random. It was determined that the null hypothesis could neither be proven nor justified. But it can be refuted during experiments.

8 cups were prepared. The first four are filled with milk first, the other four with tea. The cups were mixed. Bristol offered to taste the tea and divide the cups according to the method of preparing the tea. The result should have been two groups. History says that the experiment was a success.

Thanks to the Fisher test, the probability that Bristol was acting intuitively was reduced to 0.01428. That is, it was possible to correctly identify the cup in one case out of 70. But still, there is no way to reduce to zero the chances that Madame determines by chance. Even if you increase the number of cups.

This story gave impetus to the development of the “null hypothesis”. At the same time, Fisher's exact criterion was proposed, the essence of which is to enumerate all possible combinations of dependent and independent variables.

2. What is Fisher's exact test used for?

Fisher's exact test is mainly used to compare small samples. There are two good reasons for this. Firstly, the calculation of the criterion is quite cumbersome and can take a long time or require powerful computing resources. Secondly, the criterion is quite accurate (which is reflected even in its name), which allows it to be used in studies with a small number of observations.

A special place is given to Fisher's exact test in medicine. This is an important method for processing medical data and has found its application in many scientific studies. Thanks to it, it is possible to study the relationship between certain factors and outcomes, compare the frequency of pathological conditions between two groups of subjects, etc.

3. In what cases can Fisher's exact test be used?

The variables being compared should be measured on a nominal scale and have only two values, for example, blood pressure is normal or elevated, the outcome is favorable or unfavorable, there are postoperative complications or not.
Fisher's exact test is designed to compare two independent groups divided by factor. Accordingly, the factor should also have only two possible values.
The criterion is suitable for comparing very small samples: Fisher's exact test can be used to analyze four-complete tables in the case of values of the expected phenomenon less than 5, which is a limitation for application Pearson chi-square test, even taking into account the Yates amendment.
Fisher's exact test can be one-sided or two-sided. With a one-sided option, it is known exactly where one of the indicators will deviate. For example, a study compares how many patients recovered compared to a control group. It is assumed that therapy cannot worsen the condition of patients, but only either cure it or not.
A two-tailed test evaluates frequency differences in two directions. That is, the likelihood of both a higher and lower frequency of the phenomenon in the experimental group compared to the control group is assessed.

An analogue of Fisher's exact test is Pearson chi-square test, while Fisher's exact test has higher power, especially when comparing small samples, and therefore has an advantage in this case.

4. How to calculate Fisher's exact test?

Let's say we are studying the dependence of the frequency of births of children with congenital malformations (CDD) on maternal smoking during pregnancy. For this, two groups of pregnant women were selected, one of which was an experimental group, consisting of 80 women who smoked in the first trimester of pregnancy, and the second was a comparison group, including 90 women leading a healthy lifestyle throughout pregnancy. The number of cases of fetal congenital malformation in the experimental group was 10, in the comparison group - 2.

First, we create a four-field contingency table:

Fisher's exact test is calculated using the following formula:

where N is the total number of subjects in two groups; ! - factorial, which is the product of a number and a sequence of numbers, each of which is less than the previous one by 1 (for example, 4! = 4 3 2 1)

As a result of calculations, we find that P = 0.0137.

5. How to interpret the value of Fisher's exact test?

The advantage of the method is that the resulting criterion corresponds to the exact value of the significance level p. That is, the value of 0.0137 obtained in our example is the level of significance of the differences between the compared groups in the frequency of development of congenital malformations of the fetus. It is only necessary to compare this number with the critical level of significance, usually taken in medical research as 0.05.

If the value of Fisher's exact test is greater than the critical value, the null hypothesis is accepted and a conclusion is made that there are no statistically significant differences in the frequency of the outcome depending on the presence of the risk factor.
If the value of Fisher's exact test is less than the critical value, the alternative hypothesis is accepted and a conclusion is made that there are statistically significant differences in the frequency of the outcome depending on the exposure to the risk factor.

In our example P< 0,05, в связи с чем делаем вывод о наличии прямой взаимосвязи курения и вероятности развития ВПР плода. Частота возникновения врожденной патологии у детей курящих женщин статистически значимо выше, чем у некурящих.

ODDS RATIO

Odds ratio is a statistical indicator (in Russian its name is usually abbreviated as OR, and in English - OR from "odds ratio"), one of the main ways to describe in numerical terms how much the absence or presence of a certain outcome is related to the presence or absence of a certain factor in a specific statistical group.

1. History of the development of the odds ratio indicator

The term “chance” comes from the theory of gambling, where this concept was used to denote the ratio of winning positions to losing ones. In the scientific medical literature, the odds ratio indicator was first mentioned in 1951 in the work of J. Kornfield. Subsequently, this researcher published papers that noted the need to calculate a 95% confidence interval for the odds ratio. (Cornfield, J. A Method for Estimating Comparative Rates from Clinical Data. Applications to Cancer of the Lung, Breast, and Cervix // Journal of the National Cancer Institute, 1951. - N.11. - P.1269–1275.)

2. What is the odds ratio used for?

The odds ratio estimates the association between a particular outcome and a risk factor.

The odds ratio allows you to compare study groups according to the frequency of detection of a certain risk factor. It is important that the result of applying the odds ratio is not only the determination of the statistical significance of the relationship between the factor and the outcome, but also its quantitative assessment.

3. Conditions and limitations for the use of odds ratios

Outcome and factor indicators must be measured on a nominal scale. For example, the effective sign is the presence or absence of a congenital malformation in the fetus, the studied factor is the mother’s smoking (smokes or does not smoke).
This method allows for the analysis of only four-field tables, when both the factor and the outcome are binary variables, that is, they have only two possible values (for example, gender - male or female, arterial hypertension - presence or absence, disease outcome - with or without improvement ...).
The groups being compared must be independent, that is, the odds ratio is not suitable for before-after comparisons.
The odds ratio indicator is used in case-control studies (for example, the first group is patients with hypertension, the second is relatively healthy people). For prospective studies, when groups are formed based on the presence or absence of a risk factor (for example, the first group is smokers, the second group is non-smokers), it can also be calculated relative risk.

4. How to calculate odds ratio?

The odds ratio is the value of a fraction in which the numerator contains the odds of a certain event for the first group, and the denominator contains the odds of the same event for the second group.

Chance is the ratio of the number of subjects who have a certain characteristic (outcome or factor) to the number of subjects who do not have this characteristic.

For example, a group of patients operated on for pancreatic necrosis was selected, the number of which was 100 people. After 5 years, 80 of them were still alive. Accordingly, the chance of survival was 80 to 20, or 4.

A convenient way is to calculate the odds ratio by summarizing the data in a 2x2 table:

	There is an outcome (1)	No outcome (0)	Total
There is a risk factor (1)	A	B	A+B
No risk factor (0)	C	D	C+D
Total	A+C	B+D	A+B+C+D

For this table, the odds ratio is calculated using the following formula:

It is very important to assess the statistical significance of the identified association between the outcome and the risk factor. This is due to the fact that even with low values of the odds ratio, close to unity, the relationship, nevertheless, may turn out to be significant and should be taken into account in statistical conclusions. Conversely, with large OR values, the indicator turns out to be statistically insignificant, and, therefore, the identified relationship can be neglected.

To assess the significance of the odds ratio, the boundaries of the 95% confidence interval are calculated (the abbreviation 95% CI or 95% CI from the English "confidence interval" is used). Formula for finding the upper limit value of 95% CI:

Formula for finding the value of the lower limit of 95% CI:

5. How to interpret the odds ratio value?

If the odds ratio is greater than 1, this means that the chances of finding a risk factor are greater in the group with the outcome present. Those. the factor has a direct connection with the probability of the outcome occurring.
An odds ratio less than 1 indicates that the chances of detecting a risk factor are greater in the second group. Those. the factor has an inverse relationship with the probability of the outcome occurring.
With an odds ratio equal to one, the chances of detecting a risk factor in the compared groups are the same. Accordingly, the factor does not have any impact on the probability of the outcome.

Additionally, in each case, the statistical significance of the odds ratio is necessarily assessed based on the values of the 95% confidence interval.

If the confidence interval does not include 1, i.e. both values of the boundaries are either higher or lower than 1, a conclusion is drawn about the statistical significance of the identified relationship between the factor and the outcome at the significance level p<0,05.
If the confidence interval includes 1, i.e. its upper limit is greater than 1, and its lower limit is less than 1, it is concluded that there is no statistical significance of the relationship between the factor and the outcome at a significance level of p>0.05.
The size of the confidence interval is inversely proportional to the level of significance of the relationship between the factor and the outcome, i.e. the smaller the 95% CI, the more significant the identified relationship is.

6. Example of calculating the odds ratio indicator

Let's imagine two groups: the first consisted of 200 women who were diagnosed with a congenital malformation of the fetus (Exodus+). Of these, 50 people smoked during pregnancy (Factor+) (A), were non-smokers (Factor-) - 150 people (WITH).

The second group consisted of 100 women without signs of congenital malformation of the fetus (Outcome -) among whom 10 people smoked during pregnancy (Factor+) (B), did not smoke (Factor-) - 90 people (D).

1. Let’s create a four-field contingency table:

2. Calculate the value of the odds ratio:

OR = (A * D) / (B * C) = (50 * 90) / (150 * 10) = 3.

3. Find the boundaries of 95% CI. The value of the lower limit calculated using the above formula was 1.45, and the upper limit was 6.21.

Thus, the study showed that the chances of meeting a woman who smokes among patients with diagnosed congenital malformation of the fetus are 3 times higher than among women without signs of congenital malformation of the fetus. The observed dependence is statistically significant, since the 95% CI does not include 1, the values of its lower and upper limits are greater than 1.

RELATIVE RISK

Risk is the likelihood of a particular outcome, such as illness or injury, occurring. Risk can take values from 0 (there is no probability of the outcome occurring) to 1 (an unfavorable outcome is expected in all cases). In medical statistics, as a rule, changes in the risk of an outcome are studied depending on some factor. Patients are conditionally divided into 2 groups, one of which is affected by the factor, the other is not.

Relative risk is the ratio of the frequency of outcomes among subjects who were influenced by the factor being studied to the frequency of outcomes among subjects who were not influenced by this factor. In the scientific literature, the abbreviated name of the indicator is often used - RR or RR (from the English "relative risk").

1. History of the development of the relative risk indicator

The calculation of relative risk is borrowed by medical statistics from economics. Correct assessment of the influence of political, economic and social factors on the demand for a product or service can lead to success, and underestimation of these factors can lead to financial failure and bankruptcy of the enterprise.

2. What is relative risk used for?

Relative risk is used to compare the likelihood of an outcome depending on the presence of a risk factor. For example, when assessing the effect of smoking on the incidence of hypertension, when studying the dependence of the incidence of breast cancer on the use of oral contraceptives, etc. Relative risk is the most important indicator in prescribing certain treatment methods or conducting studies with possible side effects.

3. Conditions and limitations for the application of relative risk

Factor and outcome indicators should be measured on a nominal scale (for example, patient gender - male or female, arterial hypertension - present or not).
This method allows for the analysis of only four-field tables, when both the factor and the outcome are inary variables, that is, they have only two possible values (for example, age younger or older than 50 years, the presence or absence of a certain disease in the anamnesis).
Relative risk is used in prospective studies, when study groups are formed based on the presence or absence of a risk factor. In case-control studies, the relative risk should be used instead of odds ratios.

4. How to calculate relative risk?

To calculate relative risk you need:

5. How to interpret the relative risk value?

The relative risk indicator is compared with 1 in order to determine the nature of the relationship between the factor and the outcome:

If the RR is equal to 1, we can conclude that the factor under study does not affect the probability of the outcome (no relationship between the factor and the outcome).
For values greater than 1, it is concluded that the factor increases the frequency of outcomes (direct relationship).
For values less than 1, it indicates a decrease in the probability of the outcome when exposed to the factor ( Feedback).

The values of the boundaries of the 95% confidence interval are also necessarily estimated. If both values - both the lower and the upper limit - are on the same side of 1, or, in other words, the confidence interval does not include 1, then a conclusion is drawn about the statistical significance of the identified relationship between the factor and the outcome with an error probability of p<0,05.

If the lower limit of the 95% CI is less than 1, and the upper limit is greater, then it is concluded that there is no statistical significance of the influence of the factor on the frequency of the outcome, regardless of the value of the RR (p>0.05).

6. Example of calculating the relative risk indicator

In 1999, a study was conducted in Oklahoma on the incidence of stomach ulcers in men. Regular consumption of fast food was chosen as an influencing factor. In the first group there were 500 men who constantly ate fast food, among whom stomach ulcers were diagnosed in 96 people. The second group included 500 supporters of a healthy diet, among whom stomach ulcers were diagnosed in 31 cases. Based on the data obtained, the following contingency table was constructed:

PEARSON CORRELATION CRITERION

The Pearson correlation test is a method of parametric statistics that allows you to determine the presence or absence of a linear relationship between two quantitative indicators, as well as evaluate its closeness and statistical significance. In other words, the Pearson correlation test allows you to determine whether one indicator changes (increases or decreases) in response to changes in another? In statistical calculations and inferences, the correlation coefficient is usually denoted as r xy or R xy.

1. History of the development of the correlation criterion

The Pearson correlation test was developed by a team of British scientists led by Karl Pearson(1857-1936) in the 90s of the 19th century, to simplify the analysis of the covariance of two random variables. In addition to Karl Pearson, people also worked on the Pearson correlation criterion Francis Edgeworth And Raphael Weldon.

2. What is the Pearson correlation test used for?

The Pearson correlation test allows you to determine the closeness (or strength) of the correlation between two indicators measured on a quantitative scale. Using additional calculations, you can also determine how statistically significant the identified relationship is.

For example, using the Pearson correlation criterion, you can answer the question of whether there is a connection between body temperature and the content of leukocytes in the blood during acute respiratory infections, between the height and weight of the patient, between the fluoride content in drinking water and the incidence of dental caries in the population.

3. Conditions and limitations for applying the Pearson chi-square test

Comparable indicators should be measured on a quantitative scale (for example, heart rate, body temperature, white blood cell count per 1 ml of blood, systolic blood pressure).
Using the Pearson correlation criterion, you can only determine the presence and strength of a linear relationship between quantities. Other characteristics of the relationship, including the direction (direct or reverse), the nature of the changes (rectilinear or curvilinear), as well as the presence of dependence of one variable on another, are determined using regression analysis.
The number of compared quantities must be equal to two. In the case of analyzing the relationship of three or more parameters, you should use the method factor analysis.
The Pearson correlation criterion is parametric, and therefore the condition for its application is the normal distribution of each of the compared variables. If it is necessary to perform a correlation analysis of indicators whose distribution differs from normal, including those measured on an ordinal scale, you should use Spearman's rank correlation coefficient.
The concepts of dependence and correlation should be clearly distinguished. The dependence of quantities determines the presence of a correlation between them, but not vice versa.

For example, the height of a child depends on his age, that is, the older the child, the taller he is. If we take two children of different ages, then with a high degree of probability the growth of the older child will be greater than that of the younger one. This phenomenon is called dependence, implying a cause-and-effect relationship between indicators. Of course, there is also a correlation between them, meaning that changes in one indicator are accompanied by changes in another indicator.

In another situation, consider the relationship between a child’s height and heart rate (HR). As is known, both of these values directly depend on age, so in most cases, children of greater height (and therefore older age) will have lower heart rate values. That is, a correlation will be observed and may be quite close. However, if we take children of the same age, but different heights, then, most likely, their heart rate will differ insignificantly, and therefore we can conclude that heart rate is independent of height.

The above example shows how important it is to distinguish between the concepts of connection and dependence of indicators, fundamental in statistics, in order to draw correct conclusions.

4. How to calculate the Pearson correlation coefficient?

The Pearson correlation coefficient is calculated using the following formula:

5. How to interpret the value of the Pearson correlation coefficient?

Pearson correlation coefficient values are interpreted based on their absolute values. Possible values of the correlation coefficient vary from 0 to ±1. The greater the absolute value of r xy, the higher the closeness of the relationship between the two quantities. r xy = 0 indicates a complete lack of communication. r xy = 1 – indicates the presence of an absolute (functional) connection. If the value of the Pearson correlation criterion turns out to be more than 1 or less than -1, an error was made in the calculations.

To assess the tightness, or strength, of a correlation, generally accepted criteria are usually used, according to which the absolute values of r xy< 0.3 свидетельствуют о weak connection, r xy values from 0.3 to 0.7 - about connection average tightness, values of r xy > 0.7 - o strong communications.

A more accurate assessment of the strength of the correlation can be obtained by using the Chaddock table:

The statistical significance of the correlation coefficient r xy is assessed using the t-test, calculated using the following formula:

The obtained t r value is compared with the critical value at a certain significance level and the number of degrees of freedom n-2. If t r exceeds t crit, then a conclusion is drawn about the statistical significance of the identified correlation.

6. Example of calculating the Pearson correlation coefficient

The purpose of the study was to identify, determine the closeness and statistical significance of the correlation between two quantitative indicators: the level of testosterone in the blood (X) and the percentage of muscle mass in the body (Y). The initial data for a sample consisting of 5 subjects (n = 5) are summarized in the table:

SPEARMAN'S CRITERION

Spearman's rank correlation coefficient is a non-parametric method that is used for the purpose of statistically studying the relationship between phenomena. In this case, the actual degree of parallelism between the two quantitative series of the studied characteristics is determined and an assessment of the closeness of the established connection is given using a quantitatively expressed coefficient.

1. History of the development of the rank correlation coefficient

This criterion was developed and proposed for correlation analysis in 1904 Charles Edward Spearman, English psychologist, professor at the Universities of London and Chesterfield.

2. What is the Spearman coefficient used for?

Spearman's rank correlation coefficient is used to identify and evaluate the closeness of the relationship between two series of compared quantitative indicators. If the ranks of indicators, ordered by degree of increase or decrease, in most cases coincide (a greater value of one indicator corresponds to a greater value of another indicator - for example, when comparing a patient’s height and his body weight), a conclusion is made about the presence straight correlation connection. If the ranks of indicators have the opposite direction (a higher value of one indicator corresponds to a lower value of another - for example, when comparing age and heart rate), then they speak of reverse connections between indicators.

The Spearman correlation coefficient has the following properties:

The correlation coefficient can take values from minus one to one, and with rs=1 there is a strictly direct relationship, and with rs= -1 there is a strictly feedback relationship.
If the correlation coefficient is negative, then there is a feedback relationship; if it is positive, then there is a direct relationship.
If the correlation coefficient is zero, then there is practically no connection between the quantities.
The closer the module of the correlation coefficient is to unity, the stronger the relationship between the measured quantities.

3. In what cases can the Spearman coefficient be used?

Due to the fact that the coefficient is a non-parametric analysis method, testing for normality of distribution is not required.

Comparable indicators can be measured both on a continuous scale (for example, the number of red blood cells in 1 μl of blood) and on an ordinal scale (for example, points expert assessment from 1 to 5).

The effectiveness and quality of the Spearman assessment decreases if the difference between the different values of any of the measured quantities is large enough. It is not recommended to use the Spearman coefficient if there is an uneven distribution of the values of the measured quantity.

4. How to calculate the Spearman coefficient?

Calculation of the Spearman rank correlation coefficient includes the following steps:

5. How to interpret the Spearman coefficient value?

When using the rank correlation coefficient, the closeness of the connection between characteristics is conditionally assessed, considering coefficient values less than 0.3 to be a sign of weak connection; values greater than 0.3 but less than 0.7 are a sign of moderate closeness of connection, and values of 0.7 or more are a sign of high closeness of connection.

It can also be used to assess the tightness of the connection. Chaddock scale.

The statistical significance of the obtained coefficient is assessed using Student's t-test. If the calculated t-test value is less than the tabulated value for a given number of degrees of freedom, the observed relationship is not statistically significant. If it is greater, then the correlation is considered statistically significant.

KOLMOGOROV-SMIRNOV METHOD

The Kolmogorov-Smirnov test is a nonparametric goodness-of-fit test, in the classical sense it is intended to test simple hypotheses about whether the analyzed sample belongs to some known distribution law. The best known application of this criterion is to check the populations under study for normality of distribution.

1. History of the development of the Kolmogorov-Smirnov criterion

The Kolmogorov-Smirnov criterion was developed by Soviet mathematicians Andrey Nikolaevich Kolmogorov And Nikolai Vasilievich Smirnov.
Kolmogorov A.N. (1903-1987) - Hero of Socialist Labor, professor of Moscow state university, academician of the USSR Academy of Sciences - the greatest mathematician of the 20th century, is one of the founders modern theory probabilities.
Smirnov N.V. (1900-1966) - Corresponding Member of the USSR Academy of Sciences, one of the creators of nonparametric methods of mathematical statistics and the theory of limit distributions of order statistics.

Subsequently, the Kolmogorov-Smirnov goodness-of-fit test was modified to be used to test populations for normality of distribution by an American statistician, professor at George Washington University Hubert Lilliefors(Hubert Whitman Lilliefors, 1928-2008). Professor Lilliefors was one of the pioneers in the use of computer equipment in statistical calculations.

Hubert Lilliefors

2. Why is the Kolmogorov-Smirnov criterion used?

This criterion allows us to assess the significance of the differences between the distributions of two samples, including the possibility of using it to assess the compliance of the distribution of the sample under study with the law of normal distribution.

3. In what cases can the Kolmogorov-Smirnov criterion be used?

The Kolmogorov-Smirnov test is designed to test for normal distribution of sets of quantitative data.

For greater reliability of the data obtained, the volumes of the samples under consideration should be large enough: n ≥ 50. When the size of the estimated population is from 25 to 50 elements, it is advisable to use the Bolshev correction.

4. How to calculate the Kolmogorov-Smirnov criterion?

The Kolmogorov-Smirnov criterion is calculated using special statistical programs. It is based on statistics of the form:

Where sup S- the supremum of the set S, Fn- distribution function of the population under study, F(x)- normal distribution function

The inferred probability values are based on the assumption that the mean and standard deviation of a normal distribution are known a priori and are not estimated from the data.

However, in practice, parameters are usually calculated directly from the data. In this case, the test of normality involves a composite hypothesis (“how likely is it to obtain a D statistic of this or greater significance depending on the mean and standard deviation calculated from the data”) and Lilliefors probabilities are given (Lilliefors, 1967).

5. How to interpret the value of the Kolmogorov-Smirnov test?

If D Kolmogorov-Smirnov statistics is significant (p<0,05), то гипотеза о том, что соответствующее распределение нормально, должна быть отвергнута.

RUSSIAN ACADEMY OF NATIONAL ECONOMY AND PUBLIC SERVICE under the PRESIDENT OF THE RUSSIAN FEDERATION

ORYOL BRANCH

Department of Mathematics and Mathematical Methods in Management

Independent work

Mathematics

on the topic “Variation series and its characteristics”

for full-time students of the Faculty of Economics and Management

areas of training "Human Resources Management"

Goal of the work: Mastering the concepts of mathematical statistics and methods of primary data processing.

An example of solving typical problems.

Task 1.

The following data was obtained through the survey ():

1 2 3 2 2 4 3 3 5 1 0 2 4 3 2 2 3 3 1 3 2 4 2 4 3 3 3 2 0 6

3 3 1 1 2 3 1 4 3 1 7 4 3 4 2 3 2 3 3 1 4 3 1 4 5 3 4 2 4 5

3 6 4 1 3 2 4 1 3 1 0 0 4 6 4 7 4 1 3 5

Necessary:

1) Compile a variation series (statistical distribution of the sample), having previously written down a ranked discrete series of options.

2) Construct a frequency polygon and cumulate.

3) Compile a series of distributions of relative frequencies (frequencies).

4) Find the main numerical characteristics of the variation series (use simplified formulas to find them): a) arithmetic mean, b) median Meh and fashion Mo, c) dispersion s 2, d) standard deviation s, e) coefficient of variation V.

5) Explain the meaning of the results obtained.

Solution.

1) To compile ranked discrete series of options Let's sort the survey data by size and arrange them in ascending order

0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

5 5 5 5 6 6 6 7 7.

Let’s compose a variation series by writing the observed values (variants) in the first row of the table, and the corresponding frequencies in the second (Table 1)

Table 1.

2) A frequency polygon is a broken line connecting points ( x i; n i), i=1, 2,…, m, Where m X.

Let us depict the polygon of frequencies of the variation series (Fig. 1).

Fig.1. Frequency polygon

The cumulative curve (cumulate) for a discrete variation series represents a broken line connecting the points ( x i; n i nak), i=1, 2,…, m.

Let's find the accumulated frequencies n i nak(the accumulated frequency shows how many variants were observed with a characteristic value less X). We enter the found values in the third row of Table 1.

Let's build a cumulate (Fig. 2).

Fig.2. Cumulates

3) Let's find the relative frequencies (frequencies), where , where m– number of different characteristic values X, which we will calculate with equal accuracy.

Let us write down the distribution series of relative frequencies (frequencies) in the form of Table 2

table 2

4) Let's find the main numerical characteristics of the variation series:

a) Find the arithmetic mean using a simplified formula:

where are conditional options

Let's put With= 3 (one of the average observed values), k= 1 (the difference between two neighboring options) and draw up a calculation table (Table 3).

Table 3.

x i	n i	u i	u i n i	u i 2 n i
		-3	-12
		-2	-26
		-1	-14





Sum			-11

Then the arithmetic mean

b) Median Meh variation series is the value of the characteristic that falls in the middle of the ranked series of observations. This discrete variation series contains an even number of terms ( n=80), which means that the median is equal to half the sum of the two middle options.

Fashion Mo variation series is called the option that corresponds to the highest frequency. For a given variation series, the highest frequency n max = 24 corresponds to option X= 3, means fashion Mo=3.

c) Variance s 2, which is a measure of the dispersion of possible values of the indicator X around its average value, we find it using a simplified formula:

, Where u i– conditional options

We will also include intermediate calculations in Table 3.

Then the variance

d) Standard deviation s we find it using the formula:

e) Coefficient of variation V: (),

The coefficient of variation is an immeasurable quantity, so it is suitable for comparing scattering variation series, variants of which have different dimensions.

The coefficient of variation

5) The meaning of the results obtained is that the value characterizes the average value of the characteristic X within the sample under consideration, that is, the average value was 2.86. Standard deviation s describes the absolute spread of indicator values X and in this case amounts to s≈ 1.55. The coefficient of variation V characterizes the relative variability of the indicator X, that is, the relative spread around its average value, and in this case is .

Answer: ; ; ; .

Task 2.

The following data is available on the equity capital of the 40 largest banks in Central Russia:

12,0	49,4	22,4	39,3	90,5	15,2	75,0	73,0	62,3	25,2
70,4	50,3	72,0	71,6	43,7	68,3	28,3	44,9	86,6	61,0
41,0	70,9	27,3	22,9	88,6	42,5	41,9	55,0	56,9	68,1
120,8	52,4	42,0	119,3	49,6	110,6	54,5	99,3	111,5	26,1

Necessary:

1) Construct an interval variation series.

2) Calculate the sample mean and sample variance

3) Find the standard deviation and coefficient of variation.

4) Construct a histogram of frequency distributions.

Solution.

1) Let's choose an arbitrary number of intervals, for example, 8. Then the width of the interval is:

Let's create a calculation table:

Interval option, x k –x k +1	Frequency, n i	Middle of the interval x i	Conditional option, and i	and i n i	and i 2 n i	(and i+ 1) 2 n i
10 – 25		17,5	– 3	– 12
25 – 40		32,5	– 2	– 10
40 – 55		47,5	– 1	– 11
55 – 70		62,5
70 – 85		77,5
85 – 100		92,5
100 – 115		107,5
115 – 130		122,5
Sum				– 5

The value selected as false zero is c= 62.5 (this option is located approximately in the middle of the variation series) .

Conditional options are determined by the formula

When processing large amounts of information, which is especially important when carrying out modern scientific developments, the researcher faces the serious task of correctly grouping the source data. If the data is discrete in nature, then, as we have seen, no problems arise - you just need to calculate the frequency of each feature. If the characteristic under study has continuous nature (which is more common in practice), then choosing the optimal number of feature grouping intervals is by no means a trivial task.

To group continuous random variables, the entire variational range of the characteristic is divided into a certain number of intervals To.

Grouped interval (continuous) variation series are called intervals ranked by the value of the attribute (), where the numbers of observations falling into the r"th interval, or relative frequencies (), are indicated together with the corresponding frequencies ():

Characteristic value intervals
mi frequency

bar chart And cumulate (ogiva), already discussed in detail by us, are an excellent means of data visualization, allowing you to get a primary idea of the data structure. Such graphs (Fig. 1.15) are constructed for continuous data in the same way as for discrete data, only taking into account the fact that continuous data completely fills the region of their possible values, taking on any values.

Rice. 1.15.

That's why the columns on the histogram and the cumulate must touch each other and have no areas where the attribute values do not fall within all possible(i.e., the histogram and cumulates should not have “holes” along the abscissa axis, which do not contain the values of the variable being studied, as in Fig. 1.16). The height of the bar corresponds to frequency – the number of observations falling within a given interval, or relative frequency – the proportion of observations. Intervals must not intersect and are usually the same width.

Rice. 1.16.

The histogram and polygon are approximations of the probability density curve (differential function) f(x) theoretical distribution, considered in the course of probability theory. Therefore, their construction is so important in the primary statistical processing of quantitative continuous data - by their appearance one can judge the hypothetical distribution law.

Cumulate – a curve of accumulated frequencies (frequencies) of an interval variation series. The graph of the cumulative distribution function is compared with the cumulate F(x), also discussed in the probability theory course.

Basically, the concepts of histogram and cumulate are associated specifically with continuous data and their interval variation series, since their graphs are empirical estimates of the probability density function and distribution function, respectively.

The construction of an interval variation series begins with determining the number of intervals k. And this task is perhaps the most difficult, important and controversial in the issue under study.

The number of intervals should not be too small, as this will make the histogram too smooth ( oversmoothed), loses all the features of variability of the original data - in Fig. 1.17 you can see how the same data on which the graphs in Fig. 1.15, used to construct a histogram with a smaller number of intervals (left graph).

At the same time, the number of intervals should not be too large - otherwise we will not be able to estimate the distribution density of the studied data along the numerical axis: the histogram will be under-smoothed (undersmoothed), with empty intervals, uneven (see Fig. 1.17, right graph).

Rice. 1.17.

How to determine the most preferable number of intervals?

Back in 1926, Herbert Sturges proposed a formula for calculating the number of intervals into which it is necessary to divide the original set of values of the characteristic being studied. This formula has truly become extremely popular - most statistical textbooks offer it, and many statistical packages use it by default. How justified this is and in all cases is a very serious question.

So, what is the Sturges formula based on?

Consider the binomial distribution)

Grouping data and constructing a distribution series. Statistical study of variation series and calculation of average values ​​How to find variation series in statistics

Glossary of statistical terms

General statistics questions

WHAT ARE MEDICAL STATISTICS?

CREATION OF A DATABASE IN MS EXCEL

WHICH METHOD TO CHOOSE FOR STATISTICAL ANALYSIS?

Measurement scale

Number of compared populations

Great Statisticians

Karl Pearson (March 27, 1857 – April 27, 1936)

Florence Nightingale (12 May 1820 – 13 August 1910)

Johann Carl Friedrich Gauss (April 30, 1777 – February 23, 1855)

Lev Semyonovich Kaminsky (1889 – 1962)

Criteria and methods

STUDENT'S t-CRITERION FOR INDEPENDENT POPULATIONS

2. What is the Student's t-test used for?

3. In what cases can the Student’s t-test be used?

4. How to calculate Student's t-test?

PAIRED STUDENT'S t-TEST

1. History of the development of the t-test

2. What is the paired Student's t-test used for?

3. In what cases can you use the paired Student's t-test?

4. How to calculate paired Student's t-test?

5. How to interpret the Student's t-test value?

6. Example of calculating Student's t-test

MANN-WHITNEY U-CRITERION

1. History of the development of the U-criterion

2. What is the Mann-Whitney U test used for?

3. In what cases can the Mann-Whitney U test be used?

4. How to calculate the Mann-Whitney U test?

5. How to interpret the value of the Mann-Whitney U test?

WILCOxon CRITERION

1. History of the development of the Wilcoxon test for related samples

2. What is the Wilcoxon test used for?

3. Conditions and limitations of using the Wilcoxon T-test

4. How to calculate the Wilcoxon T-test for related samples?

5. How to interpret the value of the Wilcoxon test?

Example of calculating the Wilcoxon test for related samples

PEARSON CHI-SQUARE CRITERION

1. History of the development of the χ 2 criterion

2. Why is Pearson's χ 2 test used?

4. How to calculate the Pearson chi-square test?

5. How to interpret the value of the Pearson chi-square test?

6. Example of calculating the Pearson chi-square test

FISCHER'S EXACT CRITERION

1. History of the development of the criterion

2. What is Fisher's exact test used for?

3. In what cases can Fisher's exact test be used?

4. How to calculate Fisher's exact test?

5. How to interpret the value of Fisher's exact test?

ODDS RATIO

1. History of the development of the odds ratio indicator

2. What is the odds ratio used for?

3. Conditions and limitations for the use of odds ratios

4. How to calculate odds ratio?

5. How to interpret the odds ratio value?

6. Example of calculating the odds ratio indicator

RELATIVE RISK

1. History of the development of the relative risk indicator

2. What is relative risk used for?

3. Conditions and limitations for the application of relative risk

4. How to calculate relative risk?

5. How to interpret the relative risk value?

6. Example of calculating the relative risk indicator

PEARSON CORRELATION CRITERION

1. History of the development of the correlation criterion

2. What is the Pearson correlation test used for?

3. Conditions and limitations for applying the Pearson chi-square test

4. How to calculate the Pearson correlation coefficient?

5. How to interpret the value of the Pearson correlation coefficient?

6. Example of calculating the Pearson correlation coefficient

SPEARMAN'S CRITERION

1. History of the development of the rank correlation coefficient

2. What is the Spearman coefficient used for?

3. In what cases can the Spearman coefficient be used?

4. How to calculate the Spearman coefficient?

5. How to interpret the Spearman coefficient value?

KOLMOGOROV-SMIRNOV METHOD

1. History of the development of the Kolmogorov-Smirnov criterion

2. Why is the Kolmogorov-Smirnov criterion used?

Grouping data and constructing a distribution series. Statistical study of variation series and calculation of average values How to find variation series in statistics