The general strategy for evaluating statistical hypotheses discussed above primarily determines the use of the so-called parametric methods of mathematical statistics.

Parametric methods are based on some, as a rule, quite probable assumptions about the nature of the distribution of a random variable. Typically, parametric methods used in the analysis of experimental data are based on the assumption that the distribution of these data is normal. A consequence of this assumption is the need to estimate the distribution parameters under study. Thus, in the case of the following t -Student's test such estimated parameters are the mathematical expectation and variance. In some cases, additional assumptions are made about how the parameters characterizing the distribution of a random variable in different samples correlate with each other. So, in the Student's test, which is often used to compare average values ​​( mathematical expectation) two data series for their homogeneity or heterogeneity, an additional assumption is made about the homogeneity of the variances of the distribution of random variables in the two general populations from which these data were extracted.

The advantage of parametric data analysis methods is the fact that they have a fairly high power. Under test power have in mind its ability to avoid errors of the second kind, or β-errors. The smaller the β-error is, the higher the power of the test. In other words, test power = 1 - β.

The high power of parametric tests, or criteria, is due to the fact that these methods require that the available data be described in metric scale. As you know, the metric scales include the interval scale and the ratio scale, which is sometimes also called the absolute scale. Interval scale allows the researcher to find out not only the relations of equality or inequality of the elements of the sample (as it allows to do name scale ) and not only order relations (as it allows to do order scale ), but also evaluate the equivalence of intervals. Absolute scale in addition to this, it allows you to evaluate the equivalence of relations between the elements of the set obtained during the measurement. That is why metric scales are referred to as strong measuring scales. Due to this power, parametric methods allow more accurate expression of differences in the distribution of a random variable under the condition that the bullet or alternative hypotheses are true.

It should also be noted that, in general, parametric methods of statistics are more developed in the theory of mathematical statistics and therefore are used much more widely. Almost any experimental result can be evaluated using any of these methods. It is these methods that are considered mainly in textbooks and manuals on statistical analysis data.

At the same time, the difficulties associated with the use of parametric analysis methods in statistics are that in some cases a priori assumptions about the nature of the distribution of the random variables under study may turn out to be incorrect. And these cases are very typical for psychological research in certain situations.

So, if we compare two samples using t -Student's test, you can find that the distribution of our data differs from normal, and the variances in the two samples differ significantly. In this case, the use of a parametric Student's test may, to some extent, distort the conclusions that the researcher wants to draw. This danger increases if the values ​​of the calculated statistics turn out to be close to the boundary values ​​of the quantiles that are used to accept or reject the hypotheses. In most cases, however, as, for example, in the case of using t -test, some deviations from theoretically given assumptions are not critical for reliable statistical inference. In other cases, such deviations may pose a serious threat to such a conclusion. Then the researchers can develop special procedures that can adjust the decision-making procedure about the truth of statistical hypotheses. The purpose of these procedures is to circumvent or relax the overly stringent requirements of the parametric models of the statistics used.

One of the options for such actions of the researcher, when he discovers that the data he received differs in its parameters from what is specified in the structural model of the parametric test used, may be to try to convert these data to the right kind. For example, as noted in Chap. 1, by measuring the reaction time, can be avoided high value the asymmetry of its distribution, if we use the logarithms of the obtained values ​​for analysis, and not the values ​​of the reaction time themselves.

Another option is to refuse to use any a priori assumptions about the nature of the distribution of a random variable in the general population. And this means the rejection of parametric methods of mathematical statistics in favor of non-parametric ones.

Nonparametric are called methods of mathematical statistics, in which no a priori assumptions are made about the nature of the distribution of the data under study and no assumptions are made about the ratio of the distribution parameters of the analyzed values. This is the main advantage of these methods.

The advantage of nonparametric statistics is fully revealed when the results obtained in the experiment are presented in a weaker form. non-metric scale, representing the ranking results. Such a scale is called order scale. Of course, in some cases, the researcher can convert these data to a stronger interval scale using data normalization procedures, but, as a rule, the best option in this situation is to use nonparametric tests specially designed for statistical analysis.

As a rule, tests of non-parametric statistics involve estimating the available ratios of rank sums in two or more samples, and on the basis of this, a conclusion is formulated about the ratio of these samples. Examples of such tests are sign test, Wilcoxon signed rank test, and Mann U-test – whitney, which are used as an analogue of the parametric t -Student's test.

At the same time, if the measurement results are presented in more than strong scale, the use of non-parametric statistics means the rejection of some of the information contained in the data. The consequence of this is the danger of an increase in the error of the second kind inherent in these methods.

Thus, the methods of nonparametric statistics are more conservative than the methods of parametric statistics. Their use threatens to a greater extent with an error of the second kind, i.e. a situation where the researcher, for example, cannot detect differences between two samples, when such differences actually take place. In other words, such methods turn out to be less powerful than parametric methods. Therefore, the use of parametric statistics in the analysis of experimental data other than simple ranking is generally preferred.

Statistical scales

Statistical processing of research data

Statistical data are used in the processing of psychological research materials in order to extract as much useful information as possible from the quantitative data obtained in the experiment.

The use of certain statistical methods is determined by which statistical scale the received material belongs to.

Name scale. This scale includes materials in which the studied objects differ from each other in their quality, and the order is not important. For example, the distribution of conference participants. In the statistical processing of such materials, one must take into account the number of units each object is represented.

Order scale. The order of objects is the focus. This scale in statistics includes such research materials in which objects belonging to one or several classes are subject to consideration, but differ when comparing one with another: more - less, higher - lower, etc.

The easiest way to show the typical features of the scale of order, if we turn to the results of any sports competitions. They sequentially list the participants who took the first, second, third and other positions, respectively.

in order of place, and information about the actual achievements of athletes fade into the background, or is absent.

Interval scale. It includes such materials in which a quantitative assessment of the object under study is given in fixed units. Materials corresponding to the scale of intervals must have a unit of measurement that was identical to itself for all repeated measurements.

Relationship scale. This scale includes materials that take into account not only the number of fixed units , as in the scale of intervals, but also the ratios of the total results obtained among themselves. To work with such relationships, you need to have some absolute point, from which the countdown is conducted.

If the data available to the researcher, upon closer examination, only slightly diverge from the Gaussian normal distribution curve, then this gives the researcher the right to use parametric methods in statistical processing, the initial provisions of which are based on the Gaussian normal distribution curve. The normal distribution is called parametric because to construct and analyze the Gaussian curve, it is enough to have only two parameters: the arithmetic mean, the value of which should correspond to the height of the perpendicular restored at the center of the curve, and the so-called root mean square, or standard deviation, a value that characterizes the range of fluctuations this curve.

If it is impossible to apply parametric methods, it is necessary to turn to non-parametric ones.

Parametric estimation methods

The use of parametric methods presupposes a priori knowledge of the theoretical law of distribution of the quantity under study or its determination from empirical data, which necessitates checking the consistency of the ED and the chosen theoretical law. Parametric estimation from censored samples is based on traditional methods mathematical statistics (maximum likelihood, moments, quantiles), linear estimation methods and a number of others.

Processing of Heavily Censored Samples maximum likelihood method allowed under the following conditions:

6 < N<10, 10 < = N<20, 20 < = N<50, 50 < = N<100,

r /N> = 0,5; r/ N> = 0,3; r/ N> = 0,2; r/ N>= 0,1.

When these constraints are not met, only a lower confidence bound on the distribution parameters can be computed.

Maximum likelihood estimates are asymptotically efficient, not biased, and asymptotically normally distributed under relatively loose constraints. If a continuous variable with a density function f(x, t) is censored at the points A And b(a<b), then the distribution density function under censoring is defined as

Likelihood function at N observations

.

If a variable is double-censored at fixed points a And b, so that are not observed k 1 smallest and k 2 largest elements of the sample, then the likelihood function

Where k 1 and k 2 are random variables.

When censoring with constant values k =r 1 and k 2=r 2 the likelihood function is equal to

where v1= xr 1+1, v2 = xN - r 2

Solving the likelihood equation for various censoring schemes is a rather difficult task. Such solutions can be obtained explicitly only for one-parameter distribution laws. Equations are known for finding the parameters of typical laws for the distribution of reliability indicators for left-censored samples.

Exponential Distribution. Point estimates of the distribution parameter l for different observation plans:

where F( X) is the normal distribution function, f(x) is the density function of the normal distribution.

The system of equations (8.7) admits only a numerical solution. When solving equations in this way, estimates of the mathematical expectation and standard deviation calculated from the combined sample are usually taken as initial approximations of unknown parameters.

log-normal distribution. Parameter estimates are calculated according to the formulas for the normal distribution law with the replacement of the operating time values ​​with their natural logarithms.

RWeibull distribution. Estimates of the parameters d and b for the design [ NUz] are calculated based on the system of equations

Where tm = tr for plan [ NUR], tm = T for plan [ NUT].

Systems of equations (8.8) - (8.9) do not have an analytical solution and require the use of numerical methods: first, the root of the first equation is found (estimate of the parameter b), then by direct substitution, the value of the estimate of the parameter d. For a two-parameter Weibull distribution, large (b>4) or small (b<0,5) значения параметра свидетельствуют о том, что ЭД не подчиняются этому закону или отношение r/N few. In such cases, one should apply nonparametric estimation methods or switch to the three-parameter Weibull distribution law.

Difficulties in applying the maximum likelihood method lead to the development of other methods. The method of moments usually leads to simple computational procedures, allows one to obtain asymptotically efficient, unbiased, and normally distributed estimates, but requires consideration of the type of censoring and is applicable for a relatively large sample size (at least 30). The use of the quantile method for estimating the parameters of distribution laws is less critical to the type of censoring. The high accuracy of the estimates is achieved by the optimal selection of quantiles, although such a selection is not always possible.

The method of linear estimates is used with a small sample size, it provides high efficiency, consistency and unbiased estimates of distribution parameters. This method is based on finding a linear function of order statistics (ordered elements of the sample), which would be an unbiased estimate of the desired parameter. The application is associated with the need to use special types of distributions, which causes certain inconveniences and makes it difficult to automate calculations.

Student's t-test for independent and
dependent samples.
F-Fisher criterion.
U-Mann-Whitney test.
T-Wilcoxon test, etc.

Statistical criteria are
RULE that ensures acceptance
true and rejection of a false hypothesis with
high probability.
Statistical criteria is a METHOD
calculating a certain number.
Statistical criteria is NUMBER.

Parametric criteria are
criteria that include in the calculation formula
distribution parameters (mean and
dispersion).
Non-parametric criteria are
criteria not included in the formula
calculation of distribution parameters and
frequency based
or ranks.

Allows direct assessment of differences in mean,
obtained in two samples (t-test
Student)
Allows direct assessment of differences in variances
(F-Fisher criterion)
Allows you to identify trends in the change of a trait
when passing from condition to condition (dispersive
univariate analysis)
Allows you to evaluate the interaction of two or more
factors and their influence on the change in the trait
(two-way analysis of variance)

Possibilities and Limitations of Parametric Criteria

Experimental data should correspond to two, and
sometimes three conditions:
a) the values ​​of the attribute are measured according to the interval
scale;
b) the distribution of the feature is normal;
c) in the analysis of variance must be observed
the requirement of equality of variances in the cell of the complex.
If the above conditions are met, then
parametric criteria are more
powerful than non-parametric ones.

Allows you to evaluate only average trends, for example,
answer the question whether more often in sample A there are
higher, and in sample B - lower values
sign (Rosenbaum, Mann-Whitney criteria,
Fisher angular transformation, etc.).
Allows you to evaluate only the differences in the ranges
trait variability (criterion angular
Fisher transform).
Allows you to identify trends in the change of a trait when
transition from condition to condition for any
feature distribution (trend criteria
Page, Jonkyra).

Possibilities and limitations of nonparametric tests

There is no way to evaluate the interaction
two or more factors.
Experimental data may not answer
none of the parametric statistics conditions:
a) feature values ​​can be represented in
any scale, starting from the scale of names;
b) the distribution of the feature can be any and
its coincidence with any theoretical law
distribution is optional and does not need
verification;
c) there is no requirement for equality of variances.

The statistical test has an empirical and
critical value.
The empirical value of the criterion is the number obtained
according to the criterion calculation rule.
The critical value of the criterion is the number that
defined for a given criterion with given variables
(for example, the number of people in the sample), highlighting
zone of significance and insignificance for a feature. Cm.
Tables of critical values ​​of the criterion.
According to the ratio of empirical and critical values
criterion, the level of statistical significance is revealed and
concludes whether it is confirmed or refuted
null hypothesis.

Statistical inference rule

1) on the basis of the obtained experimental
data calculate empirical value
Criteria Camp
2) according to the tables corresponding to the criteria
find the critical values ​​of K1kr and K2kr, which
correspond to significance levels of 5% and 1%
3) write down the critical value in the form:
K1cr for p ≤ 0 05 and K2cr for p ≤ 0 01

10. 4) place the empirical value of the Camp criterion and the critical values ​​of K1kr and K2kr on the significance axis (abscissa axis Ox

Cartesian coordinate system, on
which three zones are allocated: left (insignificance),
medium (uncertainties, p ≤ 0.05), right
(significance, p ≤ 0.01)

11. The rule of acceptance of statistical inference

5) formulate decision making:
if Camp is in the zone of insignificance, then
the hypothesis H0 about the absence of differences is accepted;
if Camp is in the zone of uncertainty, then
there is a possibility of making a wrong decision
(it is necessary to increase the sample or use
another criterion)
if Camp is in the zone of significance, then the hypothesis
about the absence of differences H0 is rejected and
hypothesis H1 about the presence of differences is accepted

12. The rule of recognition of the significance of differences

In most cases, to recognize the differences
significant EMPIRICAL (obtained)
THE CRITERION VALUE MUST EXCEED
CRITICAL (table) according to
the number of degrees of freedom for two independent
samples df = (n1 + n2) – 2, for two dependent
samples df = (n1 + n2) – 1 or sample size
(n).
Exception: U-Mann-Whitney test, test
G-signs, Wilcoxon T-test, in which you need
follow the opposite rule.

13. Dependent and independent samples

Dependent samples are those samples that
which each respondent of one sample
aligned to a specific
the characteristic of the respondent of another sample.
Independent samples are those samples that
which the probability of selecting any
respondent of one sample does not depend on
selection of any of the respondents by another
samples.

14. Choosing a criterion for comparing two samples

Correspondence
distributions
normal law
(parametric)
Mismatch
distribution(s)
normal law
(non-parametric)
Independent
samples
t - criterion
Student
For
independent
samples
U-criterion
Manna-Whitney;
Dependent
samples
t - criterion
Student for
dependent
samples
Criterion
series
Criterion of signs
T-test
Wilcoxon;

15. Student's t-test for independent samples

general populations from which are extracted
independent samples differ from each other.
Initial assumptions:
1.
One sample is drawn from one population
aggregate, the other - from another (values
measured traits should not hypothetically
correlate with each other).
2.
In both samples, the distribution is approximately
corresponds to the normal law.
3.
The variances of features in two samples are approximately
are the same.

16. Student's t-test for independent samples

Initial data structure: under study
trait(s) measured in respondents, each
of which belongs to one of
compared samples.
Restrictions:
1. The distributions do not differ significantly
from the normal law in both samples.
2. With a different number of samples, the variance
are not statistically significantly different
(checked by the F-Fisher criterion or by
Lieven's criterion).

17. Formula for calculations

Where,
is the mean value of the first sample
is the mean value of the second sample
– standard deviation on the first sample
- standard deviation for the second sample

18. Student's t-test for dependent samples

Tests the hypothesis that the means of two
general populations, from which are extracted
compared dependent samples differ from each other
friend.
Initial assumptions:
1.
Each representative of one sample was given a
matching representative of another sample.
2.
The data of the two samples are positively correlated.
3.
The distribution in both samples corresponds to
normal law.
Input data structure: there are two values
the trait(s) under study.

19. F-Fisher test

Used to test the hypothesis of equality
variances of two samples. It is included in the criteria
scattering.
*Make sense before using Student's t-test
preliminarily test the hypothesis of equality of variances.
If it is correct, then to compare the means, you can
use the t-Student criterion (hypotheses about equality
average values ​​in two samples).
Fisher's criterion is based on additional
independence and normality assumptions
data samples. Before using it
it is recommended to perform a normality check
trait distribution.

20. F-Fisher test

In regression analysis, Fisher's test
allows you to evaluate the significance of linear
regression models.
In particular, it is used in stepping
regression to test the feasibility
inclusion or exclusion of independent
variables (features) into the regression model.
In analysis of variance, Fisher's test
allows assessing the significance of factors and their
interactions.

21. Mann-Whitney U-test for independent samples

Shows how much two rows coincide (intersect)
values ​​of the measured trait(s).
Conditions for application:
1.
The distribution in at least one sample differs from
normal look.
2.
Small sample size (more than 100 people)
use parametric criteria, less than 10
people are non-parametric, but the results
considered preliminary).
3.
No homogeneity of variances when comparing means
values.

22. Wilcoxon T-test for dependent samples

It is based on the ordering of values
differences (shifts) of attribute values ​​in
each pair of its dimensions.
The idea of ​​the criterion is to count
the probability of obtaining the minimum of
positive and negative
differences provided that the distribution
positive or negative
differences equiprobably and equally

23. Kruskal-Wallis H-test for 3 or more independent samples

Used to assess differences in degree
the severity of the analyzed trait
simultaneously between three, four and
more samples.
Allows you to identify the degree of change
feature in the samples, without indicating
the direction of these changes.

24. Kruskal-Wallis H-test

Conditions for application:
1. The measurement must be taken in the scale
order, intervals or ratios.
2. Samples must be independent.
3. A different number of respondents is allowed in
compared samples.
4. When comparing three samples, it is allowed,
so that one of them has n=3, and the other two
n=2. But in this case, the differences can be
recorded only at the level of the average
significance.

25. Fisher's criterion φ* (phi) (Fisher's Angular Transform)

The criterion φ (phi) is intended for
comparison of two series of sample
values ​​according to the frequency of occurrence of any feature.
This criterion can be applied to any
samples - dependent and independent. A
it is also possible to estimate the frequency
occurrence of the trait and quantitative,
and quality variable.

26. Fisher's criterion φ*

Conditions for application:
1. The measurement can be taken at any
scale.
2. Characteristics of samples can be any.
3. Lower bound - in one of the samples,
be only 2 observations, while in the second
there must be at least 30 observations. Upper
the border is not defined.
4. For small sample sizes, the lower bounds
samples must contain at least 5
observations each.

27. Classification of problems and methods for their solution

Tasks
Conditions
Methods
1. Identification
a) 2 samples
Q - Rosenbaum criterion;
differences in the level of subjects
U - Mann-Whitney test;
researched
φ* - criterion (angular
sign
Fisher transform)
b) 3 or more selections - Jonkyr's trend criterion;
rock test subjects
H - Kruskal-Wallis test.
2. Evaluation of the shift a) 2 measurements on one
T - Wilcoxon test;
values
and the same sample
G - sign criterion;
researched
test subjects
φ* - criterion (angular
sign
Fisher transform).
b) 3 or more measurements
χl2 - Friedman's criterion;
on the same
L - Page tendencies criterion.
sample of subjects

28. Classification of problems and methods for their solution

Tasks
3. Identification
differences in
distribution
4.Identification
degrees
consistency
changes
Conditions
Methods
a) when compared
empirical
sign distribution
theoretical
χ2 - Pearson's criterion;

m - binomial criterion
b) when comparing
two empirical
distributions
χ2 - Pearson's criterion;
λ - Kolmogorov-Smirnov criterion;
φ* - criterion (angular
Fisher transform).
rs - rank coefficient
Spearman's correlations.
rs - rank coefficient
spearman correlations
a) two features
b) two hierarchies or
profiles

29. Classification of problems and methods for their solution

Tasks
Conditions
5. Analysis
a) influenced
changes
one factor
sign under
influence
controlled
conditions
b) under the influence
two factors
simultaneously
Methods
S - trend criterion
Jonkyra;
L - Page tendencies criterion;
single-factor dispersive
Fisher's analysis.
Two-factor dispersive
Fisher's analysis.