The Problem of Sample Representativeness for Conducting Experimental and Broad Psychological Research

Error of sample survey (accuracy) - difference (deviation) between the values of the parameters in general population and its sample value (Yadov & Semenova, 1998);

Share of feature (expected frequency of the result) - expected share of feature, for which the error is estimated. If no information on the share of feature is available, it is necessary to use a value that equals 50% when the maximum error occurs.

Standard deviation (dispersion) - variability of observations;

Minimal clinically important effect - minimal changes and differences that we do not want to ignore.

Therefore, we intend to give examples of how a sample size can be calculated taking into account the main and additional criteria. It should be noted separately that, in addition to universal approaches, the emphasis has been put on techniques that take into account the form of representation of data. In our case, this is the frequency (percentage) of qualitative data.

An approach similar to the one discussed above, when the researcher can rely on predecessors instead of going deep into calculations, is present in more precise methods of determining the quantitative composition of a sample.

1. For example, Altman's Nomogram is often used to determine the quantitative composition of the sample, which gives an approximate sample size depending on the power of the criterion, the level of significance and the value of the effect (Altman, 1991). In our case, according to its data, we get to the number of 100 dates (values, persons), as it was suggested in some studies, only in case of a significant reduction in the power of the criterion (80% and lower).

2. Rather approximate, though mathematically grounded data presented by sociologist V. Paniotto, who proposed the formula for calculating the sample from the volume of the general population and the acceptable error (level of reliability) (Paniotto, 1984).



Paniotto's formula has the following form:

where Д - sample error; n - sample size;

N - the volume of the general population.

Using this formula, V. Paniotto proposed the following approximate ratio of the volume of the general population and that of the sample (Table 1) with an acceptable statistical error Д at 5%.

Table 1. The Ratio of General Population and Sample

The volume of general population	500	1000	2000	3000
The volume of sample	222	286	333	350
The volume of general population	4000	5000	10000	100000
The volume of sample	360	370	385	400

As we can see from the table, Paniotto's calculations confirm the insufficiency of 100 people for large general populations, and this data is most closely related to the number of respondents we have chosen (according to Paniotto, a large population is the one that has more than 5,000 people). This pattern persists and if we pay attention to Sample 2, where individual sample groups (strata) are formed from the main general population and are smaller than the main general population, but they are still related to large general populations due to their size - in our study, they include 400 people.

The calculation by the formula based on our general population in these groups of subjects also gives the required 400 people, so this number of respondents is minimal and sufficient, and a further increase in the quantitative parameters of the sample is not essential. And the level of reliability of 5% is a minimum threshold for humanitarian areas. First of all, it indicates statistically significant reliability, and secondly, increasing the level of reliability even up to 1% increases the sample exponentially, complicating the research.

The essential decrease of the quantitative indicators of the sample can lead to the following consequences:

- it can affect the objectivity of the results due to narrowing dispersion;

- it can violate the representativeness of the sample in terms of general population and thus distort the result;

- it reduces the possibility to transfer the results, which changes the population represented by the sample;

- it calls for maximum justification for the selection of the subjects with maximum consideration of the most important and additional characteristics of the general population, which, in turn, requires a thorough analysis of the latter and complicates the selection even further.

It is also necessary to note that Paniotto's formula is not the only formula available for such calculations. There are other formulas as well, but they are usually adapted for quantitative data and take into account the scope of data, standard deviations, etc. In our case, we deal with the analysis of associations, which means that we deal mainly with qualitative analysis, so quantitative data are used only indirectly, as the frequency (quantity of observations) of qualitative entities.

Let us cite several examples of methods of calculation of the quantitative composition of the sample that meet our requirements to the form of data presentation and the hypothesis of our research.

3. The formula for estimating sample size at one single frequency (Bland, 2000):

n = 15.4 * p * (1-p) / W2; (5)

where n - the required sample size,

p - the expected frequency of the result,

W - the width of the confidence interval.

We select our data at the minimum levels: the controlled variable is the frequency of deviations in associative responses, p = 50% (0.5); W - +/- 5%, i.e. 10% (or 0.1). We calculate the sample size according to our data:

n = 15.4 * 0,5 * (1-0.5) / 0.12 = 385

Thus, to get the result within the statistical error 0,05 (5%), with the estimated frequency of observing productive associations being 50%, the size of the sample is 385 people. And given the possible 10% withdrawal from the group, we obtain the quantitative composition of the sample is 385 x 1.1 = 424 persons.

4. In case if it is necessary to compare frequencies of two groups (nests) within a single sample, the following method of calculation can be applied. But before that, it is necessary to calculate the number of respondents for each group represented in the research.

The following data is selected for the calculation: controlled variable - the frequency of compliance of the associations; the value of significant differences - 20% (0.2); the level of significance - 5%; power - 80%; for a two-way test (accepting or denying an alternative hypothesis).

A formula for calculating the sample size when comparing two frequencies (Bland M., 2000):

n = [Zd+Zp]2 х [(p х (l-p1) + (p2 х (1-p2))] / [p^]2; (6)

where n - the sample size for each group (the total sample size is twice as large),

p1 - the first frequency, we select its index as 60% (0.60);

p2 - the second frequency, we select it at the level of 40% (0.40);

p1-p2 = clinically important differences, chosen as 20% (0.2);

Zd - depends on the level of significance, determined by special tables (the table of critical values of Student's coefficient, t) - is 1.96 at the level of significance is 0.05;

Zp - depends on the selected power (determined by the tables of critical values) - in our case, it is 0.84 with power being 80%.

As we process our data with the help of this formula, we obtain the following:

n = [1,96 + 0,84] 2 x [(0,6 x 0,4) + (0,4 x 0.6)] / [0,2] 2 = 95

Thus, we receive the number of observations needed to be included in each group. Consider possible withdrawals of 10% - 95 x 1,1 = 105 people in the group. The total sample size will be twice as large, i.e., 210 people. It means that in this case it can be argued that a sample consisting of 210-230 persons will be sufficient to detect differences in the frequency of associations, with 80% power, 5% confidence, and 20% level of minimally important differences.

5. Let us show another formula for the case of a partial (share) representation of the results (percentage). For that, let us determine the sample size on the basis of the estimated confidence interval (Koichubekov, Sorokina, Mkhitaryan, 2014). The initial information necessary for the implementation of this approach is the magnitude of variation, which is believed to be inherent to the population; desired accuracy; the level of reliability, which should correspond to the results of the conducted research.

The size of the sample is determined by the following formula:

where n - sample size;

z - normalized deviation, which is determined according to the chosen level of significance (the table of critical values of Student's t-coefficient),

p - estimated frequency of variation for the sample, q = (100-p), e - permissible error.

For the assumed minimal variation of 50%, the permissible error is 5%, z - for the power of 80% - 1,29, and for the power of 95% - 1.96, we get the sample size of 166 respondents at the power of 80%, and 384 respondents if the power is 95%.

6. For the breadth of our analysis of the approaches to determining the quantitative composition of the sample, let us give some examples of such calculations using modern Network Calculators that use similar approaches and formulas, but also have a possibility to automate certain processes:

6.1. «Calculator» website (http://allcalc.ru/node/100) with minimum possible criteria (trustworthiness, confidence, reliability (power) - 85%; confidence interval, error - 5% and higher) for the chosen general population (young people aged 18-35, approximately 10,448,900 people), yields the result of 384 respondents for the necessary sample.

6.2. «Sociopolis» website (http://sociopolis.ua/ru/servisy/ kalkulator-vybirky/) with minimum possible criteria (trustworthiness, confidence, reliability (power) - 95%, error - 0,05 and higher) for the chosen general population (young people aged 18-35, approximately

10,448,900 people) yields the result of 384 respondents for the necessary sample.

6.3. «Medical Statistics» website (http://medstatistic.ru/ calculators/calcsize.ht) provides wider opportunities for considering various criteria and methods of their calculation (with due references to sources). Here we once again applied our selection of minimum possible limitations and rigidity of the criteria:

- level of precision - medium;

- significance rate - 0.05;

- variable А - 1.96;

- research power - 80%;

- variable В - 0.84;

- confidence interval - 2;

- permissible error - 5;

- the width of confidence interval - 10%;

Pi -- pa

- standardized variability ^{100 - p) = 0.4;

- minimum important difference between values (shares or means) - 20 %;

- type of sample formation - random;

- units for data presentation - percentage;

- expected frequency of the phenomenon in the experimental group - 60%;

- withdrawal rate in the experimental group - 40%;

- expected frequency of the phenomenon in the control group - 40%;

- withdrawal rate in the control group - 60%;

- general population - young people (aged 18-35), approximately

10,448,900 people.

Table 2 demonstrates the results of sample size calculation.

As we can see from the table 2, four out of seven available variants suggest using samples consisting of 400 and more respondents. Two variants deal with the size of a separate group within the sample as frequencies or strata are compared (see formula (6)). This leaves only one remaining variant that suggests the size of a sample consisting of 100 respondents, according to the values given the author in the table. So, as far as the problem of defining the size of the sample of the general population is considered, the analysis described above proves our assumptions about the minimal size of the experimental sample consisting of 400 respondents.

Statistical analysis of the obtained results is another empirical and objective proof that testifies not only to the above-mentioned assumptions but also to the feasibility of using this approach in the empirical psycholinguistic association experiment.

Table 2. Results of Sample Size Calculation Using Different Methods

Method	Formula used for calculation	Result
K. Otdelnova's method	According to the table	100
Formula for repeated selection (one sample)	i2 x p x q 71 ~ Д2	384
Formula without repeated selection (one sample)	JVx^xpXj П JVxi^^xpxj	385
Plokhinskiy's formula for comparing two groups	Ј n = ^2 X (pi^l + P29a)	768
Lera formula for relative values (determines the size of each group which is compared)	16 (Рі-Р2)/л/т(10°-т)	101
Sample calculation formula where one single frequency is estimated	15.4 x pi x {100 -- pi} 71 “ wa	371
Sample calculation formula where two frequencies are estimated	(A + B)2 x (pi x (100-pi) 4-pa x (100 - pa))	95
	" Ы-Р2?

In the first place, it is necessary to mention that the statistical analysis of empirical data obtained via association experiments is aggravated by the fact that this data is usually presented in a quantitative form, which means that they have to be converted into numbers in order to apply statistical coefficients. In this case, we can only use those coefficients that are compatible with statistical analysis based on frequency parameters. The number of these coefficients is rather limited, the most common out of which are: Pearson coefficient x2, binomial criterion m, Kolmogorov-Smirnov criterion X, and ф* criterion (Fisher z-transformation). As long as we evaluate the feasibility of applying these coefficients in our empirical research, let us analyze the peculiarities of representation and distribution of this data and the goal of our study considering the conditions of applying this or that coefficient (this analysis can be useful for other experiments as well). It should not be forgotten that the core idea of our empirical research was to show that «playfulness» is a relevant lexeme in the linguistic consciousness of respondents, as well as to determine the influence of such factors as gender, age, and profession on the stimulus «playfulness». All this being considered, the general population and the sample appear to have a multi-level structure (randomized experimental sample (Sample 1) - stratified sample (Sample 2) - cluster sample (Sample 3)). The goal of this scientific paper was to provide evidence for the importance of determining the sufficient minimum of the sample size in experimental research and in this way to confirm the relevance of the chosen quantity of respondents in the sample groups used in our psycholinguistic association experiment.

The analysis of the peculiarities and structure of research, as well as the peculiarities of presenting the data, make it possible to choose a relevant statistical procedure (i.e., the coefficient). Thus, the binominal criterion (m) is used if research uses only one sample and the number of respondents does not exceed 300 people, which does not meet our requirements. Kolmogorov-Smirnov criterion X assumes that the categories according to which the analysis is supposed to be made should be sorted out in ascending or descending order, and this codification cannot be random. This condition does not satisfy the peculiarities of presenting data in our research. In the first place, it does not have a precise distinction between different categories (levels), if one considers frequencies of reactions as categories. Secondly, the distribution of categories is a random value, so it would be wrong to speak about the accumulation of frequencies. As far as Pearson coefficient x2 is concerned, it applies to our research, but it has certain limitations. Notably, in our study, it is necessary to consider the number of reactions rather than the number of respondents by way of the number of surveys. It is explained by the fact that a respondent gives several associative responses, and this very fact does not meet the main goal of the present paper, which is to provide rationalization for the selecting a particular number of respondents rather than reactions.

That leaves us only one statistical criterion for this case, which is Ф* criterion (Fisher z-transformation), which has minimal restrictions and is aimed at comparing samples according to the frequency of the observed effect, which makes this approach useful for our conditions. The only significant requirement here is to make sure that all reactions are reduced to the alternative scale «effect present - effect absent». In our case, this problem is solved using dividing all reactions into high frequency and low-frequency reactions so that groups can be further compared according to these very criteria.

As long as we deal with social groups in our research, we decided to turn to the fundamentals of social psychology for the definition of the notion of a «group» to explain why certain reactions are considered to be high-frequency reactions. The minimal threshold of his notion is claimed to be 3-5 people. In this way, reactions, whose frequencies allowed us to define them as a group within certain parameters (starting from 3 people), can be considered as high-frequency reactions.

It would be worthwhile to mention that groups that are being compared, as far as the aim of the research is concerned (general population - a sample of representation - representative sample - stratified sample - cluster sample), can be perceived as nested one inside the other, as long as they were selected consecutively. It should also be noted that to prevent cluster homogeneity from influencing the results, randomization was used for all statistical calculations when creating strata and clusters (nests). This explains identical associative reactions and possible frequency distributions (high and low-frequency reactions) and allows us to compare these groups according to the ways this parameter is distributed. It means that the ratio of high frequency and low-frequency reactions were calculated within each of these key groups, and later these groups were statistically compared with each other with the help of ф* criterion. This helped to obtain data about statistical relevance (equivalence) of the groups with the possibility of transferring results from one group into another to confirm the relevance of reactions and their distribution, which also proves the representativeness of these groups. The results of statistical analysis are displayed in Table 3 and Figure 1.

When analyzing and comparing the data of table 3 about the declared issue, we see the sample of 100 respondents showing sufficient differences at the level of p<0,01 with all other experimental groups. Therefore, this quantity of the respondents is absolutely insufficient for ensuring transfer of the results obtained to the general population. In other words, a nested sample, even on condition of monitoring the homogeneity factor does not reflect the characteristics of the general population, which entirely confirms the calculations above.

Table 3. Comparison of the Samples by the ratio of the Reactions Frequency (ф* criterion)

Sample, Quantity	100 Nested sample	400 Stratified sample	1600 Randomized sample	3000 Sample of randomization
400	4 441**		1.754*	2.758**
3000	7.198**	2.758**	1,004

Note: * - ambiguity area (p<0,05), ** - relevance area (p<0,01)

Fig. 1. Relevance areas upon criterion ф*

Comparing the group of 400 respondents (stratified sample), we observe some distinctions from the randomized sample (1600 people) within ambiguity area (p<0,05) at the edge of the non-relevance area. That is to say, showing specific differences from general randomized sample, that limits representativeness even on this level of correlation. It is also confirmed by above substantiations in differences of the strata formation and their influence on the result. As for correlation with the standard generalized sample, we have got insufficient distinctions, albeit at the edge of the non-relevance area. In this way, it is possible to state firmly, that even the sample level substantiated above of 400500 people might be considered as a minimum, but it is not always sufficient to ensure representativeness of the sample, as well as a result in comparison with the general population.

Statistical analysis of the randomized experimental sample compliance (1,600 people) with the randomization sample (3,000 people) and the general population has not revealed substantial distinctions in the distribution of the results. We can safely say that the given quantity of people is sufficient to confirm representativeness of the respondents, and the data as well; therefore the sample keeps the characteristics of the general sample.

It is possible to observe certain regularity: while reducing the sample level, distinctions increase, and groups of less quantity of respondents do not reflect all characteristics of the general population, even if the influence of homogeneity factor experience targeted prevention. A substantial level of quantity combination of the experimental sample in case of large general populations lies within an interval from 400 to 1500 people and depends on quantity and quality combination of general population, the goal, and features of the research arrangement.

Therefore, we have confirmed our assumption, that to obtain a representative result, the sample should be sufficient to meet the requirements on reflection of main characteristics of the general population. The quantity of 100 people or close to it, cannot meet the given demand, minimal quantity composition of the sample (but not always sufficient!), should not be less than 400-500 people. These data are confirmed by above theoretical substantiation, approaches to the calculation of quantity combination of the experimental sample, and first of all, by statistical estimates based on the results of practical research of the associative reactions.

Conclusions

There is a rich variability of methods in the calculation of the sample quantitative combination, and one should choose the method depending on the list of the following research indicators: general population, the quantity of required final results, the hypothesis, the features of data representation, preciseness, level of representativeness, level of relevance, etc. But, in any case, despite of the calculation methods, the basic issue is that the sample should reflect the characteristics of the general population, and this requires the substantiation of the latter, and the presence of minimal limit, which imply that if the sample is less, then it is impossible to create a substantiated representative sample.

The procedure is complicated because of the absence of stable and agreed methodological criteria and requirements to the determination of the quantitative combination of the sample. Therefore, even attempts to get as much as close to the existing methodological approaches and purity of research, we couldn't entirely avoid specific deviations in the formation of the representative samples. Since approaching to maximal representativeness violates the peculiarities of strata, formation of clusters and keeps us from the goal of our psycholinguistic study (definition of distinctions). Therefore, to form maximal representative sample one needs an accurate definition of the general population with all its characteristics, and this is exceedingly difficult to execute. Analysis of these possibilities is the perspective of our further experimental and methodological work in this direction. Also, there is a certain terminological confusion brought by a variety of terms and names applied to the same indicators, which is also should be agreed.

So, the issue about the volume of the general population is uncertain, unsolved and challenging to be solved. Some principles of the sample volume definition have been considered above, but a researcher, while choosing the sample volume, is influenced by some other factors, including the resources - time, finances, and from another side, one's wish to engage as many people in the survey, as it will be necessary for obtaining the maximally reliable information.

As it is mentioned above, the sample volume directly depends on width and homogeneity of the aggregate (population) under study. The less homogeneous the population in research is, the more peculiarities it has. In most cases, the members who constitute it, either group of people, or particular individuals always differ (by gender, age, education, profession or other distinguishing marks). Of course, there is no need to reflect within the sample all the qualities of the researched object, but it is necessary for the most relevant ones. The more informative and detailed the analysis of general population will be, the more qualities of the given object we take into consideration, the more massive should be the sample volume. It is because the respondents divide into subgroups, which can be compared on statistical basis only under the condition of their adequate quantity. Thus, the number of subgroups within the sample influences their volumes directly. But, at the same time, excessive detail of the marks can prevent the study of the large groups; transfer the vector of research to the category of branch-oriented, or even individual. This problem is solved directly by the author of the particular study about its goal, hypothesis, and tasks, which allows to ignore part of the characteristics of general population under the condition of substantiation and to observe the critical methodological requirements. And one of the essential stages, which mostly define the qualities of the results obtained, is the formation of a representative sample of the respondents.

In many cases, it is reasonable to draw on the experience, by the researchers while carrying out sample surveys, and focus on it, depending on the scope and character of research. Thus, «typical» samples for national surveys vary within 1000 - 2500 respondents (depending on quantity of subgroups (strata), which is under analysis), the strata in frame of these researches and regional surveys - from 200 to 500 (while analyzing of numerous subgroups the volume of regional sample can increase to 1000 person and more), and nests within the strata defined in quantity from 100 to 200 person (Osipov, 2009).

Empirical research confirmed the given calculations with the application of statistical procedures. There is a regularity revealed that under the condition of decreasing volume of the sample the distinctions increase and the groups with fewer numbers of the respondent don't reflect all the characteristics of the general population. The assumption that quantity of 100 people or close to it, cannot meet this requirement, the minimal quantitative composition of the sample (not always sufficient!), should be not less than 400-500 people. The adequate level of the quantitative composition of the experimental sample is an interval from 400 to 1500 people and depends on the quantitative and qualitative composition of the general sample and the features of research. Therefore it should be sufficient to meet requirements in the reflection of the general population features, as well as the goal of research.

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