Correlation coefficients of the ranks of Spearman Kendall Fechner. Kendall's rank correlation coefficient. See what "kendalla's rank correlation coefficient" is in other dictionaries

To calculate Kendall coefficient the values of the factor attribute are pre-ranked, that is, the ranks by X are recorded strictly in ascending order of quantitative values.

1) For each rank in Y, find the total number of subsequent ranks that are greater in value than the given rank. The total number of such cases is taken into account with the “+” sign and denoted by P.

2) For each rank in Y, the number of subsequent ranks that are less in value than the given rank are determined. The total number of such cases is counted with a “-” sign and denoted by Q.

3) Calculate S = P + Q = 9 + (- 1) = 8

4) The Kendall coefficient is calculated by the formula:

The Kendall coefficient can take values from -1 to +1 and the closer to, the stronger the connection between the features.

In some cases, to determine the direction of the relationship between two features, calculate Fechner coefficient... This coefficient is based on comparing the behavior of deviations of individual values of the factorial and effective characteristics from their average value. The Fechner coefficient is calculated by the formula:

; where the sum of C is the total number of coincidences of the signs of deviations, the sum of H is the total number of mismatches of the signs of deviations.

1) Calculate the average value of the factor attribute:

2) Determine the signs of deviations of the individual values of the factor attribute from the average.

3) Calculate the average value of the effective indicator: .

4) Find the signs of deviations of the individual values of the effective trait from the average value:

Output: the connection is direct, the coefficient does not indicate the tightness of the connection.

To determine the degree of tightness of the relationship between the three ranked features, the coefficient is calculated concordance. It is calculated using the formula:

, where m is the number of ranked features; n is the number of ranked observation units.

Industries	X1	X2	X3	R1	R2	R3
Power engineering			7,49
Fuel			12,70
Black M.			5,92
Tsvetnaya M.			9,48
Mechanical engineering			4,18
Outcome:

X1- the number of employees (thousand people); X2- volume of industrial sales (billion rubles); X3- average monthly salary.

1) We rank the values of all features and set the ranks strictly in ascending order of quantitative values.

2) The sum of the ranks is determined for each line. This column is used to calculate the final row.

3) Calculate .

4) For each row, find the squares of the deviations of the sums of ranks and values of T. For the same column, we calculate the final row, which we denote by S. The concordance coefficient can take values from 0 to 1, and the closer to 1, the stronger the relationship between the traits.

When ranking, the expert must arrange the evaluated elements in ascending (decreasing) order of their preference and assign each of them ranks in the form of natural numbers. In direct ranking, the most preferred item is rank 1 (sometimes 0) and the least preferred item is rank m.

If an expert cannot strictly rank due to the fact that, in his opinion, some elements are the same in preference, then it is allowed to assign the same ranks to such elements. To ensure that the sum of the ranks is equal to the sum of the places of the ranked elements, so-called standardized ranks are used. The standardized rank is the arithmetic mean of the numbers of elements in a ranked row that are the same in preference.

Example 2.6. The expert ordered the six elements by preference as follows:

Then the standardized ranks of these elements will be

Thus, the sum of the ranks assigned to the elements will be equal to the sum of the numbers in the natural series.

The accuracy of expressing preference by ranking the elements significantly depends on the cardinality of the set of presentations. The ranking procedure gives the most reliable results (in terms of the closeness of the revealed preference and the "true" one) when the number of evaluated elements is no more than 10. The limiting cardinality of the presentation set should not exceed 20.

The processing and analysis of rankings is carried out in order to build a group preference relationship based on individual preferences. In this case, the following tasks can be set: a) determination of the tightness of the relationship between the rankings of two experts on the elements of the set of presentations; b) determining the relationship between the two elements according to the individual opinions of the group members regarding the various characteristics of these elements; c) assessment of the consistency of opinions of experts in a group containing more than two experts.

In the first two cases, the coefficient is used as a measure of the tightness of the connection rank correlation... Depending on whether only strict or loose ranking is allowed, either Kendall's or Spearman's rank correlation coefficient is used.

Kendall's rank correlation coefficient for problem (a)

where m- the number of elements; r 1 i - rank assigned by the first expert i−th element; r 2 i - the same, by the second expert.

For problem (b), the components (2.5) have the following meaning: m is the number of characteristics of the two estimated elements; r 1 i(r 2 i) - rank i characteristics in the ranking of the first (second) element, set by a group of experts.

Strict ranking uses the rank correlation coefficient R Spearman:

whose components have the same meaning as in (2.5).

Correlation coefficients (2.5), (2.6) vary from -1 to +1. If the correlation coefficient is +1, it means that the rankings are the same; if it is -1, then they are opposite (the rankings are inverse to each other). Equality of the correlation coefficient to zero means that the rankings are linearly independent (uncorrelated).

Since with this approach (an expert is a “gauge” with a random error), individual rankings are considered as random, the problem arises of statistically testing the hypothesis about the significance of the obtained correlation coefficient. In this case, the Neumann-Pearson criterion is used: they are set by the significance level of the criterion α and, knowing the distribution laws of the correlation coefficient, determine the threshold value c α, with which the obtained value of the correlation coefficient is compared. The critical area is right-sided (in practice, the value of the criterion is usually first calculated and the level of significance is determined from it, which is compared with the threshold level α ).

Kendall's rank correlation coefficient τ has, for m> 10, a distribution close to normal with the following parameters:

where M [τ] - mathematical expectation; D [τ] - variance.

In this case, the tables of the standard normal distribution function are used:

and the boundary τ α of the critical region is defined as the root of the equation

If the calculated value of the coefficient τ ≥ τ α, then the rankings are considered to be in good agreement. Typically, the value of α is selected in the range 0.01-0.05. For t ≤ 10, the distribution of t is given in table. 2.1.

The check of the significance of the consistency of the two rankings using the Spearman coefficient ρ is carried out in the same order using the Student's distribution tables for m> 10.

In this case, the quantity

has a distribution well approximated by the Student's distribution with m- 2 degrees of freedom. At m> 30, the distribution of the quantity ρ is in good agreement with the normal distribution, which has M [ρ] = 0 and D [ρ] =.

For t ≤ 10, the significance of ρ is checked using table. 2.2.

If the rankings are not strict, then the Spearman coefficient

where ρ is calculated by (2.6);

where k 1, k 2 - the number of different groups of non-strict ranks in the first and second rankings, respectively; l i is the number of identical ranks in i group. In the practical use of the Spearman's ρ and Kendall's rank correlation coefficients, it should be borne in mind that the ρ coefficient provides a more accurate result in terms of the minimum variance.

Table 2.1.Distribution of Kendall's rank correlation coefficient

One of the factors limiting the application of criteria based on the assumption of normality is the sample size. As long as the sample is large enough (for example, 100 or more observations), you can assume that the sample distribution is normal, even if you are not sure that the distribution of the variable in the population is normal. However, if the sample is small, these criteria should only be used if there is confidence that the variable is indeed normally distributed. However, there is no way to test this assumption in a small sample.

The use of criteria based on the assumption of normality is also limited to a scale of measurements (see chapter Basic concepts of data analysis). Statistical methods such as t-test, regression, etc. assume that the original data is continuous. However, there are situations where the data are simply ranked (measured on an ordinal scale) rather than measured accurately.

A typical example is given by the ratings of sites on the Internet: the first position is taken by the site with the maximum number of visitors, the second position is taken by the site with the maximum number of visitors among the remaining sites (among sites from which the first site has been removed), etc. Knowing the ratings, we can say that the number of visitors to one site is greater than the number of visitors to another, but how much more is impossible to say. Imagine you have 5 sites: A, B, C, D, E, which are in the top 5 places. Suppose that in the current month we had the following arrangement: A, B, C, D, E, and in the previous month: D, E, A, B, C. The question is, there have been significant changes in site ratings or not? In this situation, obviously, we cannot use the t-test to compare these two groups of data, and move on to the area of specific probabilistic calculations (and any statistical criterion contains a probabilistic calculation!). We reason like this: how likely is it that the difference in the two site layouts is due to purely random reasons, or that the difference is too large and cannot be explained by pure chance. In this reasoning, we only use the ranks or permutations of sites and do not in any way use a specific form of distribution of the number of visitors to them.

For the analysis of small samples and for data measured on poor scales, nonparametric methods are used.

A quick tour of nonparametric procedures

Essentially, for each parametric criterion, there is at least, one nonparametric alternative.

In general, these procedures fall into one of the following categories:

distinction criteria for independent samples;
distinction criteria for dependent samples;
assessment of the degree of dependence between the variables.

In general, the approach to statistical criteria in data analysis should be pragmatic and not burdened with unnecessary theoretical reasoning. With a STATISTICA computer at your disposal, you can easily apply several criteria to your data. Knowing about some of the pitfalls of the methods, you will choose the right solution through experimentation. The development of the plot is quite natural: if you need to compare the values of two variables, then you use the t-test. However, it should be remembered that it is based on the assumption of normality and equality of variances in each group. Breaking free from these assumptions results in nonparametric tests that are especially useful for small samples.

The development of the t-test leads to analysis of variance, which is used when the number of compared groups is more than two. The corresponding development of nonparametric procedures leads to a nonparametric analysis of variance, although it is significantly poorer than the classical analysis of variance.

To assess the dependence, or, to put it somewhat pompously, the degree of tightness of the connection, the Pearson correlation coefficient is calculated. Strictly speaking, its application has limitations associated, for example, with the type of scale in which the data are measured and the nonlinearity of the dependence; therefore, alternatively, nonparametric, or so-called rank, correlation coefficients are also used, which are used, for example, for ranked data. If the data are measured on a nominal scale, then it is natural to present them in contingency tables that use Pearson's chi-square test with various variations and corrections for accuracy.

So, in essence, there are only a few types of criteria and procedures that you need to know and be able to use, depending on the specifics of the data. You need to determine which criterion should be applied in a particular situation.

Nonparametric methods are most appropriate when sample sizes are small. If there is a lot of data (for example, n> 100), it often doesn't make sense to use nonparametric statistics.

If the sample size is very small (for example, n = 10 or less), then the significance levels for those nonparametric tests that use the normal approximation can only be considered as rough estimates.

Differences between independent groups... If there are two samples (for example, men and women) that need to be compared with respect to some average value, for example, the mean pressure or the number of leukocytes in the blood, then the t-test can be used for independent samples.

Nonparametric alternatives to this test are the Val'd-Wolfowitz, Mann-Whitney series test) / n, where x i - i-th value, n is the number of observations. If the variable contains negative values or zero (0), the geometric mean cannot be calculated.

Harmonic mean

The harmonic average is sometimes used to average frequencies. The harmonic mean is calculated by the formula: ГС = n / S (1 / x i) where ГС is the harmonic mean, n is the number of observations, х i is the value of observation with the number i. If the variable contains zero (0), the harmonic mean cannot be calculated.

Dispersion and standard deviation

Sample variance and standard deviation are the most commonly used measures of variability (variation) in data. The variance is calculated as the sum of the squares of the deviations of the values of the variable from the sample mean, divided by n-1 (but not by n). The standard deviation is calculated as the square root of the variance estimate.

Swing

The range of a variable is an indicator of volatility, calculated as a maximum minus a minimum.

Quartile scope

The quarterly range, by definition, is: upper quartile minus lower quartile (75% percentile minus 25% percentile). Since the 75% percentile (upper quartile) is the value to the left of which 75% of cases are located, and the 25% percentile (lower quartile) is the value to the left of which 25% of cases are located, the quartile range is the interval around the median. which contains 50% of the cases (variable values).

Asymmetry

Asymmetry is a characteristic of the shape of the distribution. The distribution is skewed to the left if the skewness value is negative. The distribution is skewed to the right if the asymmetry is positive. The skewness of the standard normal distribution is 0. The skewness is associated with the third moment and is defined as: skewness = n × M 3 / [(n-1) × (n-2) × s 3], where M 3 is: (x i -x mean x) 3, s 3 is the standard deviation raised to the third power, n is the number of observations.

Excess

Kurtosis is a characteristic of the shape of a distribution, namely, a measure of the severity of its peak (relative to a normal distribution, the kurtosis of which is equal to 0). As a rule, distributions with a sharper peak than normal have a positive kurtosis; distributions whose peak is less acute than the peak of the normal distribution have negative kurtosis. The excess is associated with the fourth moment and is determined by the formula:

kurtosis = / [(n-1) × (n-2) × (n-3) × s 4], where M j is: (x-x mean x, s 4 is the standard deviation to the fourth power, n is the number of observations ...

Brief theory

Kendall's correlation coefficient is used when variables are represented by two ordinal scales, provided that there are no associated ranks. The calculation of Kendall's coefficient involves counting the number of matches and inversions.

This coefficient varies within and is calculated by the formula:

For calculation, all units are ranked by attribute; according to a number of other criteria, the number of subsequent ranks exceeding the given one (we denote them by) and the number of subsequent ranks below the given one (we denote them by) are calculated for each rank.

It can be shown that

and Kendall's rank correlation coefficient can be written as

In order to test the null hypothesis at the level of significance that the general Kendall's rank correlation coefficient is equal to zero under a competing hypothesis, it is necessary to calculate the critical point:

where is the sample size; Is the critical point of the two-sided critical region, which is found from the table of the Laplace function by the equality

If - there is no reason to reject the null hypothesis. The rank correlation between the features is insignificant.

If - the null hypothesis is rejected. There is a significant rank correlation between the features.

An example of solving the problem

The task

When recruiting seven candidates for vacant positions, two tests were offered. The test results (in points) are shown in the table:

Test

Candidate

Calculate Kendall's rank correlation coefficient between test results for two tests and assess its significance at the level.

The solution of the problem

Calculate Kendall's coefficient

The ranks of the factor attribute are arranged strictly in ascending order, and the corresponding ranks of the effective attribute are recorded in parallel. For each rank from among the ranks following it, the number of higher ranks is calculated (entered in the column) and the number of lower ranks (entered in the column).

Sum

To calculate Kendall's rank correlation coefficient r k it is necessary to rank the data for one of the attributes in ascending order and determine the corresponding ranks for the second attribute. Then, for each rank of the second feature, the number of subsequent ranks, greater in magnitude than the taken rank, is determined, and the sum of these numbers is found.

Kendall's rank correlation coefficient is determined by the formula

where R i- the number of ranks of the second variable, starting from i+1, the magnitude of which is greater than the magnitude i rank of this variable.

There are tables of percentage points of the distribution of the coefficient r k, allowing to test the hypothesis about the significance of the correlation coefficient.

For large sample sizes, critical values r k are not tabulated, and they have to be calculated using approximate formulas, which are based on the fact that under the null hypothesis H 0: r k= 0 and large n random value

distributed approximately according to the standard normal law.

40. Relationship between traits measured in nominal or ordinal scales

The problem often arises of checking the independence of two features measured on a nominal or ordinal scale.

Let some objects measure two features X and Y with the number of levels r and s respectively. The results of such observations are conveniently presented in the form of a table, called a contingency table.

In the table u i(i = 1, ..., r) and v j (j= 1, ..., s) - the values taken by the features, the value n ij- the number of objects from the total number of objects for which the attribute X took on the meaning u i, and the sign Y- meaning v j

We introduce the following random variables:

u i

- the number of objects that have a value v j

In addition, there are obvious equalities

Discrete random variables X and Y independent if and only if

for all couples i, j

Therefore, the conjecture about the independence of discrete random variables X and Y can be written like this:

As an alternative, as a rule, they use the hypothesis

The validity of the hypothesis H 0 should be judged on the basis of sample frequencies n ij contingency tables. In accordance with the law of large numbers at n→ ∞, the relative frequencies are close to the corresponding probabilities:

To test the hypothesis H 0, statistics are used

which, if the hypothesis is true, has the distribution χ 2 sec rs − (r + s- 1) degrees of freedom.

Independence criterion χ 2 rejects hypothesis H 0 with significance level α if:

41. Regression analysis. Basic concepts of regression analysis

For a mathematical description of the statistical relationships between the studied variables, the following problems should be solved:

ü choose a class of functions in which it is advisable to seek the best (in a certain sense) approximation of the dependence of interest;

ü find estimates of the unknown values of the parameters included in the equations of the required dependence;

ü to establish the adequacy of the obtained equation of the required dependence;

ü to identify the most informative input variables.

The totality of the listed tasks is the subject of research in regression analysis.

The regression function (or regression) is the dependence of the mathematical expectation of one random variable on the value taken by another random variable, which forms a two-dimensional system of random variables with the first.

Let there be a system of random variables ( X,Y), then the regression function Y on X

And the regression function X on Y

Regression functions f(x) and φ (y) are not mutually reversible if only the relationship between X and Y is not functional.

When n-dimensional vector with coordinates X 1 , X 2 ,…, X n you can consider the conditional mathematical expectation for any component. For example, for X 1

called regression X 1 on X 2 ,…, X n.

For a complete definition of the regression function, it is necessary to know the conditional distribution of the output variable for fixed values of the input variable.

Since in a real situation such information is not available, they are usually limited to the search for a suitable approximating function f a(x) for f(x), based on statistical data of the form ( x i, y i), i = 1,…, n... This data is the result n independent observations y 1 ,…, y n random variable Y for the values of the input variable x 1 ,…, x n, while the regression analysis assumes that the values of the input variable are specified accurately.

The problem of choosing the best approximating function f a(x), being the main one in regression analysis, and does not have formalized procedures for its solution. Sometimes the choice is determined based on the analysis of experimental data, more often from theoretical considerations.

If it is assumed that the regression function is sufficiently smooth, then the approximating function f a(x) can be represented as a linear combination of a set of linearly independent basis functions ψ k(x), k = 0, 1,…, m−1, i.e., in the form

where m- number of unknown parameters θ k(in the general case, the value is unknown, refined during the construction of the model).

Such a function is linear in parameters, therefore, in the case under consideration, we speak of a regression function model that is linear in parameters.

Then the problem of finding the best approximation for the regression line f(x) is reduced to finding such parameter values for which f a(x; θ) is the most adequate to the available data. One of the methods to solve this problem is the least squares method.

42. Least square method

Let the set of points ( x i, y i), i= 1,…, n located on a plane along some straight line

Then, as a function f a(x) approximating the regression function f(x) = M [Y|x] it is natural to take linear function argument x:

That is, the basis functions here are chosen ψ 0 (x) ≡1 and ψ 1 (x)≡x... This regression is called simple linear regression.

If the set of points ( x i, y i), i= 1,…, n is located along some curve, then as f a(x) it is natural to try to choose the family of parabolas

This function is non-linear in parameters θ 0 and θ 1, however, by functional transformation (in this case, taking the logarithm), it can be reduced to new function f ’a(x), linear in parameters:

43. Simple Linear Regression

The simplest regression model is simple (one-dimensional, one-way, paired) linear model, which has the following form:

where ε i- random variables (errors) uncorrelated with each other, having zero mathematical expectations and the same variances σ 2 , a and b- constant coefficients (parameters) that need to be estimated from the measured response values y i.

To find the parameter estimates a and b linear regression, determining the straight line most satisfying the experimental data:

the method of least squares is applied.

According to least squares parameter estimates a and b are found from the condition of minimizing the sum of squares of deviations of the values y i vertically from the “true” regression line:

Let there be ten observations of a random variable Y with fixed values of the variable X

To minimize D we equate to zero the partial derivatives with respect to a and b:

As a result, we obtain the following system of equations for finding estimates a and b:

Solving these two equations gives:

Expressions for Parameter Estimates a and b can also be represented as:

Then the empirical equation of the regression line Y on X can be written as:

Unbiased variance estimate σ 2 deviations of values y i from the fitted straight line of regression is given by the expression

Let's calculate the parameters of the regression equation

Thus, the regression line looks like:

And the estimation of variance of deviations of values y i from the fitted straight line of regression

44. Checking the Significance of the Regression Line

Found estimate b≠ 0 can be a realization of a random variable, the mathematical expectation of which is equal to zero, that is, it may turn out that there is actually no regression dependence.

To deal with this situation, you should test the hypothesis H 0: b= 0 with a competing hypothesis H 1: b ≠ 0.

The test of the significance of the regression line can be carried out using analysis of variance.

Consider the following identity:

The magnitude y i− ŷ i = ε i called the remainder and is the difference between two quantities:

ü deviation of the observed value (response) from the total average response;

ü deviation of the predicted response value ŷ i from the same average

The written identity can be written as

Having squared both parts of it and summed over i, we get:

Where the quantities are named:

the total (total) sum of squares of the SC n, which is equal to the sum of the squares of the deviations of observations relative to the mean value of observations

the sum of squares due to the regression of SK p, which is equal to the sum of squares of the deviations of the regression line values relative to the mean of observations.

residual sum of squares SK 0. which is equal to the sum of the squares of the deviations of the observations relative to the values of the regression line

So the spread Y-kov relative to their mean can be attributed to some extent to the fact that not all observations lie on the regression line. If this were the case, then the sum of squares relative to the regression would be zero. It follows that the regression will be significant if the sum of the squares of the SC p is greater than the sum of the squares of the SC 0.

Regression significance test calculations are performed in the following ANOVA table.

If errors ε i distributed according to the normal law, then if the hypothesis H 0 is valid: b= 0 statistics:

distributed according to Fisher's law with the number of degrees of freedom 1 and n−2.

The null hypothesis will be rejected at the significance level α if the calculated statistic value F will be greater than the α percentage point f 1;n−2; α of the Fisher distribution.

45. Checking the adequacy of the regression model. Residual method

The adequacy of the constructed regression model is understood as the fact that no other model gives a significant improvement in predicting the response.

If all values of the responses are obtained at different values x, i.e., there are no several response values obtained with the same x i, then only a limited test of the adequacy of the linear model can be carried out. The basis for such a check is the leftovers:

Deviations from the established pattern:

Insofar as X- one-dimensional variable, points ( x i, d i) can be plotted on a plane in the form of the so-called residual plot. Such a representation sometimes makes it possible to find some regularity in the behavior of the residuals. In addition, the analysis of the residuals allows you to analyze the assumption regarding the distribution of errors.

In the case when the errors are distributed according to the normal law and there is an a priori estimate of their variance σ 2 (an estimate obtained on the basis of previously performed measurements), then a more accurate assessment of the adequacy of the model is possible.

By using F-Fisher's criterion can be used to check whether the residual variance is significant s 0 2 differs from the a priori estimate. If it is significantly greater, then there is an inadequacy and the model should be revised.

If the prior estimate σ 2 no, but response measurements Y repeated two or more times with the same values X, then these repeated observations can be used to obtain another estimate σ 2 (the first is the residual variance). Such an estimate is said to represent a “pure” error, since if x are the same for two or more observations, then only random changes can affect the results and create a scatter between them.

The resulting estimate turns out to be a more reliable estimate of the variance than the estimate obtained by other methods. For this reason, when planning experiments, it makes sense to set up experiments with repetitions.

Suppose we have m different meanings X : x 1 , x 2 , ..., x m... Let for each of these values x i there is n i response observations Y... Total observations are obtained:

Then the simple linear regression model can be written as:

Let's find the variance of “pure” errors. This variance is the combined estimate of the variance σ 2, if we represent the values of the responses y ij at x = x i as sample volume n i... As a result, the variance of “pure” errors is:

This variance serves as an estimate σ 2 regardless of whether the fitted model is correct.

Let us show that the sum of squares of “pure errors” is a part of the residual sum of squares (the sum of squares included in the expression for the residual variance). Remaining for j th observation at x i can be written as:

If you square both sides of this equality and then sum them over j and by i, we get:

On the left of this equality is the residual sum of squares. The first term on the right is the sum of squares of “pure” errors, the second term can be called the sum of squares of inadequacy. The last amount has m−2 degrees of freedom, therefore, the variance of inadequacy

The statistics of the criterion for testing the hypothesis H 0: the simple linear model is adequate, against the hypothesis H 1: the simple linear model is inadequate, the random variable is

If the null hypothesis is true, the value F has a Fisher distribution with degrees of freedom m−2 and n−m... The hypothesis of linearity of the regression line should be rejected with a significance level α, if the obtained value of the statistic is greater than the α-percentage point of the Fisher distribution with the number of degrees of freedom m−2 and n−m.

46. Checking the adequacy of the regression model (see 45). ANOVA

47. Checking the adequacy of the regression model (see 45). Determination coefficient

Sometimes, to characterize the quality of the regression line, a sample coefficient of determination is used R 2, showing what part (fraction) of the sum of squares, due to the regression, SK p is in the total sum of squares SK n:

The closer R 2 to one, the better the regression approximates the experimental data, the closer the observations are adjacent to the regression line. If R 2 = 0, then the changes in the response are completely due to the influence of unaccounted factors, and the regression line is parallel to the axis x-ov. In the case of simple linear regression, the coefficient of determination R 2 is equal to the square of the correlation coefficient r 2 .

The maximum value R 2 = 1 can be achieved only in the case when the observations were carried out at different values of x-ov. If there are repeated experiments in the data, then the value of R 2 cannot reach unity, no matter how good the model is.

48. Confidence Intervals for Simple Linear Regression Parameters

Just as the sample mean is an estimate of the true mean (the population mean), so are the sample parameters of the regression equation a and b- nothing more than an estimate of the true regression coefficients. Different samples give different estimates of the mean - just as different samples will give different estimates of the regression coefficients.

Assuming that the error distribution law ε i are described by the normal law, the parameter estimate b will have a normal distribution with parameters:

Since the parameter estimate a is a linear combination of independent normally distributed quantities, it will also have a normal distribution with mean and variance:

In this case, the (1 - α) confidence interval for estimating the variance σ 2 taking into account that the ratio ( n−2)s 0 2 /σ 2 distributed by law χ 2 with the number of degrees of freedom n−2 will be determined by the expression

49. Confidence intervals for the regression line. Confidence interval for dependent variable values

We usually do not know the true values of the regression coefficients. a and b... We only know their estimates. In other words, the true regression line can go higher or lower, be steeper or shallower than the one constructed from the sample data. We calculated the confidence intervals for the regression coefficients. You can also calculate the confidence region for the regression line itself.

Let for simple linear regression it is necessary to construct (1− α ) confidence interval for the mathematical expectation of the response Y at value NS = NS 0. This mathematical expectation is a+bx 0, and its estimate

Since, then.

The obtained estimate of the mathematical expectation is a linear combination of uncorrelated normally distributed values and therefore also has a normal distribution centered at the point of the true value of the conditional mathematical expectation and variance

Therefore, the confidence interval for the regression line at each value x 0 can be represented as

As you can see, the minimum confidence interval is obtained at x 0 equal to the mean and increases as x 0 “moves away” from the middle in any direction.

To obtain a set of joint confidence intervals suitable for the entire regression function, along its entire length, in the above expression instead of t n −2,α / 2 must be substituted