Sten Score Method And Cluster Analysis: Identifying Respondents Vulnerable To Drug Abuse

Sten scores

First, the authors interpreted the results by using sten scores, sten being an abbreviation for 'Standard Ten'. The procedure is described as follows.

• Answers for direct markers of DAB (drag-addict behaviour) are translated into ‘response scores’: answers ‘yes’ is interpreted as ‘4’, answers ‘rather yes’ – as ‘3’, answers ‘rather no’ – as ‘2’, and answers ‘no’ – as ‘1’.

• Answers for inverse markers of DAB (drag-addict behaviour) are translated into ‘response scores’: answers ‘yes’ are interpreted as ‘4’, answers ‘rather yes‘ – as ‘3’, answers ‘rather no’ – as ‘2’, and answers ‘no’ – as ‘1’.

• The ‘raw score’ for every i-th respondent is calculated as the sum $x_i$ of all 35 response scores.

• The mean M and the standard deviation s is calculated by the raw scores of all respondents.

• By using the formula $z_i=\left(x_i-M\right)/s$, the raw scores are translated into ‘standard scores’.

• By using the sten formula ${sten}_i={2z}_i+\mathrm{5,5}$, the obtained ‘standard scores’ are translated into ‘standard tens’ or ‘stens’.

• Calculating the percentage of respondents whose results in stens are above 7.5 (high level of psychological propensity for DAB). Calculating this value for different universities and other subgroups of interest (age, gender, year, program of study, etc.).

• Calculating the percentage of respondents whose results in stens are below 3.5 (low level of psychological propensity for DAB). Calculating this value for different universities and other subgroups of interest (age, gender, year, program of study, etc.).

• Calculating the percentage of respondents whose results in stens are within the range from 3.5 to 7.5 (middle level of psychological propensity for DAB). Calculating this value for different universities and other subgroups of interest (age, gender, year, program of study, etc.).

• Figure 01 presents the distribution of these three levels of psychological propensity for DAB in different A and B universities. Labels A, ..., H of the horizontal axis stand for the universities, the percentage of the respondents with low, middle and high scores are shown in blue, red and green accordingly. One can see that the percentage of the high score respondents varies from 12.13% to 24.79%; the percentage of the middle score respondents varies from 64.23% to 69.7%; the percentage of the low score respondents varies from 9.6% to 21.42%, depending on the university. Therefore, the percentage of middle score respondents is much less changeable in comparison with high and low score respondents.

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The main objective of the study was to estimate the total number of respondents with a high level of psychological propensity for drag-addict behaviour (Shelehov, Kornetov & Grebennikova, 2016), or high-risk population, as a percentage of the sample. Figure 02 demonstrates these results. Here the percentage of the respondents with low, middle and high scores are shown in blue, red and green accordingly. One can see that the percentage of the high score respondents is 15.95%, the percentage of the middle score respondents is 66.76%, the percentage of the low score respondents is 17.29%.

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These results are very close to the results of similar studies conducted in Kazan, Birsk and Izhevsk (Cheverikina, 2012.). However, the authors can identify a number of problems arising when the tens scores for obtaining this objective are used.

First, the sten scores method needs information in numerical form, whereas in the test, respondents should answer ‘yes’, ‘rather yes’, ‘rather no’ and ‘no’, which means that the information is given in an order scale. Hence, at steps 1 and 2, answers are translated into numbers; consequently, the order scale is transformed into the numerical scale. From the statistical point of view, such transformations are unfounded and they result in the misrepresentation of information.

Then, the sten scores are in fact linearly transformed into normalised standard scores (see steps 46); after normalising, the authors investigate not absolute but relative numbers. Hence, the sten score indicates an individual's approximate position with respect to the population of values and, therefore, to other people in that population. So, the high-risk population can be estimated only in reference to the sample. In other words, the percentage does not depend significantly on the sample, if the authors investigate two groups of people, one with a high level of DAB, and another with a low level; in every group the authors will most probably obtain the same percentage of the high-risk population. This argument is supported by the investigation, the authors conducted using the same data. At step 5, the authors calculated the means $M_k$ and standard deviations $s_k$ for all universities $K\in \left\{A,\dots ,H\right\}$ and then the authors recalculated standard scores by using the formula $z_i^k=\left(x_i^k-M_k\right)/s_k$, where $x_i^k$ is the raw score of the i-th respondent from the k-th university. After that, the authors performed steps 69. Figure 03 presents the results. One can see that the percentage of the high score respondents varies now from 14.4% to 17.42%; the percentage of the middle score respondents varies from 66.43% to 70.72%; the percentage of the low score respondents varies from 14.79% to 17.60%, depending on the university. So, the percentages of high and low score respondents are much less changeable in comparison with Figure 01 . They are close to 16% in all cases, which can be explained as follows. The sten scores method is based on the hypothesis than the initial distribution of the raw scores is close to normal distribution (Kline, 2000); hence, the sten scores have normal distribution, too. The bounds of 3.5 stens and 7.5 stens were calculated by using this hypothesis; in fact, these are 16% quantiles of normal distribution. Taking into account the large number of respondents and questions, the authors can expect that the distribution of the raw scores, according to the central limit theorem, tends to be normal, and the percentages of high-level score and low-level score respondents should be close to 16%.

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Consequently, the main objective of the stens method is to determine not the percentage of high-level score and low-level score respondents, but to recognize the respondents belonging to these 16% groups. It is the reason of the wide application of the method in the interpretation of IQ and other similar tests (Anastasi, 1988). In other studies, it can be applied in certain conditions (Kline, 2000): (1) where the original distribution resembles normal distribution; (2) where the authors have some theoretical basis to expect normal distribution, and (3) where the authors are confident that their normalization group is sufficiently large and representative to be a true reflection of the population.

Besides, sometimes the results are essentially different with respect to the group where the mean and the variance are calculated. For example, for university H, Figure 01 shows a relatively big percentage of high-level scores and relatively small percentage of low-level scores, but the results shown in Figure 03 are divisive, as the authors note a relatively small percentage of high-level scores and relatively big percentage of low-level scores. The authors think that the main reason for this is that the number of students at the university is rather small (only 136 respondents were questioned). So, the condition (3) set by Kline does not hold true. It stresses the fact that this method does not fit well for small-size samples.

Cluster analysis

Cluster analysis divides data into groups (clusters) based only on information that describes the objects and their relationships. The requirement is that the objects within a cluster must be similar to one another and significantly different from the objects in other groups. The greater the similarity within a cluster, and the greater the difference between clusters, the better is the clustering. In cluster analysis, the characteristics of object (variables) can be of any type: nominal, order, quantitative. The objective is to define a distance between objects and between clusters (Everitt, Landau, Leese & Stahl, 2011).

In this case, the authors have 35 questions, which is a rather big number of variables. Besides, all direct markers are supposed to be equal, and all inverse markers too. Consequently, the authors chose as variables the numbers of every kind of responses for the direct and inverse markers; as a result, the authors obtained eight variables. After that, the authors divided all respondents within universities into three clusters, as in the experiment described above, by using software package “Statistica 10” (StatSoft, 2013). Here Figure 04 presents the results for university H (with a small number of respondents), and Figure 05 presents the results for university E (with a large number of respondents). Here the following variables are used:

• Var1 – number of answers “yes” for the inverse markers;

• Var2 – number of answers “rather yes” for the inverse markers;

• Var3 – number of answers “rather no” for the inverse markers;

• Var4 – number of answers “no” for the inverse markers;

• Var5 – number of answers “yes” for the direct markers;

• Var6 – number of answers “rather yes” for the direct markers;

• Var7 – number of answers “rather no” for the direct markers;

• Var8 – number of answers “no” for the direct markers.

Points mark mean values of the variables for each cluster.

One can see that in both cases, Cluster 3 (green line) corresponds to the low-risk groups. The number of answers “yes” for the inverse markers (Var 1) and the number of answers “no” for the direct markers (Var 1) exceed such values in other clusters, and exceed the numbers of other answers within the clusters. For university H, Cluster 1 (blue line) corresponds to the high-risk group: it has the least (between the clusters) number of answers “yes” for the inverse markers (Var 1) and number of answers “no” for the direct markers (Var 8), and the greatest numbers of answers “yes” and “rather yes” for the direct markers (Var 5 and Var 6). For university E, Cluster 2 (red line) corresponds to the high-risk group: the number of answers “yes” for the direct markers (Var 6) essentially exceeds similar values for other clusters. Cluster 2 (red line) for university H and cluster 1 (blue line) for university E correspond to the middle-risk groups. The analysis of the variable variances shows that the probability of incorrect clustering is not more than 0.001.

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Table 01 presents means of variables for clusters. One can see that the direct markers describe the situation better than the inverse markers. For example, the number of answers “no” for the direct markers is sufficiently greater for the low-risk groups than for other groups in both universities. The mean number is above 17; so, the low-risk respondents deny the direct markers in 17 cases from 27 in average. For both high-risk groups, the number of answers “yes” exceeds this numbers in other groups, and for university E, the difference is essential (7.22 against 1.88 and 1.55). The number of answers “yes” and “rather yes” is about 12-13; so, the high-risk respondents admit the direct markers in 12-13 cases from 27 in average, whereas in the middle-risk groups, this number is about 7; in the low-risk groups, it is about 5-6. So, the numbers of different answers for direct markers are suitable variables for clustering, both from mathematical and from psychological points of view.

As for the inverse markers, the number of answers “yes” is the biggest for the most cases (about 34), except for the high-risk group at university H. But in this case, the number of answers “rather yes” is the biggest. Therefore, the variables do not characterize well the groups with different DAB. Most probably, it is connected with a small number of markers.

Figure 06 presents the distribution of respondents within universities H and E. One can see that the percentages of the low-risk and high-risk respondents differ from 16% usual for sten scores methodology. The reason is that in cluster analysis the number of cluster members is not prescribed; the goal is to divide the respondents into a given number of clusters. Consequently, cluster analysis can be used to estimate the percentage of groups. The obtained percentages of the high-risk groups and the low-risk groups are higher than ones obtained by sten scores method (see Figure 01 for universities H and E). They are 30.48 against 24 for university H, and 18.58 against 14.39 for university E. One should note that in cluster analysis, no extra information was used, whereas in the sten score method, the mean and the variance of the raw score was calculated by the total sample.

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Remembering that the high-risk respondents of both universities admit the direct markers in 12-13 cases from 27 in average, it is rather a large number, and it is similar in both universities, in spite of the fact, that the number of respondents in these two cases differs a lot. Hence, cluster analysis can be used to estimate the number of high-risk respondents both for gross and for small samples.