This article is linked to the research carried out within a Doctoral Thesis Project related to the quality of the Audit service. The selection of territories is necessary to carry out the investigations, the selection of which should not be a random phenomenon, but the result of a process where statistical tools that justify it are used, for them the Cluster Analysis technique is used, relating this to the defined variables based on the Audit service.
Introduction to cluster analysis.
Cluster analysis, also known as Cluster Analysis, Numerical Taxonomy or Pattern Recognition, is a multivariate statistical technique whose purpose is to divide a set of objectives into groups, so that the profiles of the objectives in the same group are very similar between each other (internal group cohesion) and those of different cluster objectives are different (external group isolation).
Academics and market researchers often find the best solution to solve their studies by defining homogeneous groups of objects, be they individuals, firms, products, or even behaviors.
Strategic options based on the identification of groups within the population such as targeting or marketing would not be possible without a methodological goal. The same need is found in other areas, ranging from the physical sciences (for example, classification of various groups of animals, such as insects or mammals) to the social sciences (for example, analysis of various psychiatric profiles).
In all of these examples, the analyst tries to find a natural structure through observations based on a multivariate profile. The most commonly used technique for this purpose is Conglomerate Analysis (Cluster Analysis, AC, as of now).
It should be clear from the beginning:
- That the technique has no vocation / inferential properties That, therefore, the results achieved for a sample serve only for that design (its value concerns only the analyst's objectives): choice of individuals, relevant variables used, criterion similarity used, level of grouping chosen ending…. They define different solutions. That Cluster and discriminant do not have too much in common: the discriminant tries to explain a structure and the Cluster tries to determine it.
Two Basic Objectives:
- "Taxonomic" analysis for exploratory or confirmatory purposes. Change (simplification) of the data dimension (that described at the beginning of this document: grouping of individual objects in new (group) study structures)
What is Cluster Analysis?
AC is nothing more than a set of techniques used to classify objects or cases into relatively homogeneous groups called conglomerates (clusters).
The objects in each group (conglomerates) tend to be similar to each other (high internal homogeneity, within the Cluster) and different from the objects of the other groups (high external heterogeneity, between Clusters) with respect to some predetermined selection criteria.
In this way, if the classification is successful, the objects within the cluster will be very close to each other in the geometric representation, and the different clusters will be far apart. This analysis is also known as a classification analysis or numerical taxonomy.
AC has the essential purpose of grouping together objects that have identical characteristics, that is, it thus becomes an exploratory analysis technique designed to reveal natural groupings within a data collection. This analysis does not make any distinction between dependent variables (DV) and independent variables (VI), but calculates the interdependent relationships of the entire set of variables.
AC is used in marketing for various purposes, including:
- Market segmentation
For example, consumers can be grouped using the benefits derived from purchasing a product as a basis. Each group will consist of relatively homogeneous consumers in terms of the benefits they seek. This approach is known as profit segmentation. Understanding Buyer Behavior
AC can be used to identify homogenous buyer groups. Thus, the behavior of each group can be studied separately. This analysis is also used to identify the kind of strategies buyers use to obtain external information. Identification of opportunities for new products
By grouping brands and products, competitive sets within the market can be determined. Brands in the same group compete more with each other than with other groups. A company can analyze its current offerings compared to those of its competitors to identify potential opportunities for new products.Test Market
Selection By dividing cities into homogeneous groups, it is possible to select comparable cities in order to test various strategies. marketing data reduction
AC can be used as a general data reduction tool to develop subgroups of data that are easier to manage than individual observations. Subsequent multivariate analysis is performed based on subgroups, rather than individual observations. For example, to describe differences in product usage behavior, consumers can first be grouped. Differences between clusters can be studied with the use of multiple discriminant analysis.
AC is a useful tool for analyzing data in many different situations. For example, a researcher who is collecting data using a questionnaire might encounter a large number of observations that are meaningless unless they are within manageable groups.
CA can carry out this data grouping objectively by reducing the information of an entire population or the sample of information on specific small groups. For example, if we can understand the attitudes of a population by identifying the largest groups within the population, then we can reduce the data of an entire population within a large number of groups. In this way, the researcher achieves greater conciseness and a more understandable description of the observations, with minimal loss of information.
AC can also be used when an investigator wishes to develop hypotheses concerning the nature of the data. For example, a researcher might believe that attitudes about consuming low-sugar soft drinks vis-à-vis normal soft drink consumers could be used to separate consumers into logical groups or segments. AC can classify soft drink consumers by their attitudes toward normal or low-calorie soft drinks, and the resulting clusters can be outlined based on similarities and demographic differences.
However, a single and definitive solution to the problem of cluster creation should never be expected.
In practice, various solutions derived from the application of the many techniques offered by this analysis will be considered and, after their corresponding comparison, we will lean towards the most useful for the type of research proposed. The researcher's knowledge of her clients, products or services will play an important role when deciding between the different alternative solutions.
Basic concepts of Cluster Analysis
Most of the procedures used in this multivariate technique are relatively simple, as they are not supported by statistical reasoning. Most of the grouping methods are heuristic, algorithm-based. Thus, AC presents a strong contrast with the analysis of variance, regression, discriminant analysis, and factor analysis, which are based on statistical reasoning.
The fundamental principles involved in any CA are:
- Agglomeration Report
Provides information on the objects or cases that are combined at each stage of a hierarchical grouping process. Grouping Centers
The mean (mean) values of the variables for all cases or objects in a particular group. Grouping Centers
They are the initial starting points in the non-hierarchical grouping. Groups are built around these centers or seeds. Group
Participation Indicates the group to which each object or case belongs. Dendrogram
Also called a tree graph, it is a graphical device to present the results of the cluster.
The vertical lines represent the groups that are joined. The position of the line on the scale indicates the distances at which the groups joined. It is read from left to right. Distances between Group Centers
Indicate the separation between the individual pairs of the groups. The widely separated groups are distinct and therefore desirable. Icicles Diagram
It is a graphic representation of the results of the cluster, it is so named because it resembles a row of icicles hanging from the eaves of a house. The columns correspond to the objects that are grouped and the rows correspond to the number of clusters. Read from bottom to top Distance / Similarity Coefficients Matrix
It is a lower triangle matrix that contains the distances in the paired direction between the objects or cases.
Cluster Analysis Steps
The decision of the number of clusters requires the criteria of the researcher. The clusters obtained should be interpreted in terms of the additional salient variables. Finally, the researcher needs to assess the validity of the cluster process.
1. Formulation of the Problem
Perhaps the most important part of formulating the QA problem is the selection of the variables on which the grouping is based. Including one or more irrelevant variables can distort a grouping solution that might otherwise be useful. Basically, the selected set of variables should describe the similarity between the objects in relevant terms for the market research problem. Variables should be selected based on previous research, theory, or a consideration of the hypotheses being tested. In exploratory research, the researcher must apply judgment and intuition.
2. Selection of a Similarity Measure
Since the object of the cluster is to group similar objects together, some measure is needed to assess the differences and similarities between objects. The concept of similarity is fundamental in Cluster Analysis. Similarity (similarity) is a measure of correspondence or similarity between the objects to be grouped. The most common strategy is to measure equivalence in terms of the distance between pairs of objects. Objects with reduced distances between them are more similar to each other than those with greater distances and will therefore be grouped within the same cluster.
In this way, any object can be compared to any other object through the similarity measure.
In the measurement of the similarity between the objects of an AC there are three methods:
- Correlation Measures Distance Measures Association Measures
Each of these methods represents a particular perspective on similarity, depending on both the objectives and the type of data. Correlation and distance measures require metric data, while association measures require non-metric data.
Many computer programs have limited support for association measures, and the researcher is often forced to calculate the similarity measures first, and then to introduce the similarity matrix into a cluster program.
3. Data Standardization
Once the measure has been selected to quantify the similarity between pairs of objects, the researcher must ask one last question… should the data be standardized before calculating the similarities? In order to answer this question adequately, the researcher must take into account that most distance measurements are quite sensitive to the differences in scales or magnitudes made between the variables.
In general, variables with a large dispersion (large values of their standard deviations) have more impact on the final value of similarity.
Consider as an example that you want to group the individualities of a series of people into three variables, which are: attitude towards a product, age and income. We are supposed to measure attitude on a seven-point like-dislike scale, while age was measured in years and earnings in dollars.
If we plot the results of the relevant survey on a three-dimensional graph, the distance between the points (and their similarities) would be based almost entirely on income differences.
The explanation is very simple, while the possible differences in attitude towards the product are found in a range of attitudes ranging from one to seven, those produced in income can have a range one hundred times greater.
In this way, we would not be able (graphically) to observe any difference in the dimension associated with the attitude towards the product. For this reason, the researcher must be aware of the implicit weight of the variables that participate in the research study.
The most common form of standardization is the conversion of each variable into typical scores (also known as Z scores). The way of calculation is that each observation of each variable is subtracted from its corresponding mean and the result of this operation is divided by the standard deviation (standard) of the variable in question.
This process converts the score of each original data into a standardized value with a mean of zero and a standard deviation of one. Ultimately, what is achieved with this is to eliminate, one by one, the prejudices introduced by the differences in the scales of the different attributes (variables) used in the analysis.
4. Assumptions of the Analysis
AC is a methodological objective to quantify the characteristics of a set of observations. Therefore, it has strong mathematical properties, but no statistical foundations. The requirements of normality, linearity and homoscedasticity (so relevant in other techniques), have little consistency in AC.
However, the researcher must focus his attention on two other essential questions for this type of analysis, such as: the representativeness of the sample and multicollinearity.
In many cases, a population census is available to make use of cluster analysis. A sample of cases is then obtained and the clusters obtained from it are expected to be representative of the structure of the original population. The analyst must always bear in mind that the cluster analysis will be as good as the representativeness of the sample.
Thus, all efforts should focus on ensuring that representativeness, so that the results can be generalized to the population of interest.
Multicollinearity was a result of other multivariate techniques, since it was difficult to differentiate the true impact of multicollinear variables. In the cluster analysis, on the other hand, the effect is different, since the multicollinear variables are implicitly weighted in a more severe way.
Suppose, for example, that respondents are grouped into ten variables related to a certain service. When examining multicollinearity, we appreciate that there really are two clearly differentiated groups of variables.
The first is made up of eight elements (variables) and the second of the remaining two.
If what we want is to really group the respondents in the dimensions of the service analyzed (in this case represented by the two groups of variables), we will not be able to consider the ten variables as a whole, since that would mean weighting each variable equally.
In other words, when weighting the cluster analysis uniformly for each variable, the first dimension would have four times more opportunities (eight items versus two) to affect the similarity measure than the second dimension would have.
Thus, the act of multicollinearity is a process of weighting hidden from the observer, but which nevertheless affects the analysis. For this reason, the analyst should encourage an exhaustive study of the variables used in the cluster analysis in order to find possible multicollinearity.
If multicollinearity is found in the variables used for the study, it will be necessary to obtain the same number of them in each set or use one of the distance measures, such as the Mahalanobis Distance, to compensate for the existing correlation discovered.
Once the variables have been selected and the similarities matrix has been calculated, the partitioning process begins. First, the researcher must select the grouping algorithm to be used to form the clusters (groups) and then make the decision on the number of groups to be formed.
Both decisions have substantial implications not only in the results obtained, but also in the interpretation that could be derived from them.
5. Selection of the Grouping Procedure
There are two types of procedures: hierarchical and non-hierarchical. The hierarchical conglomerate is characterized by the development of a hierarchy or tree-like structure.
An important feature of hierarchical procedures is that the results of the first stage can be nested with the results of the last stage, leading to a tree-like similarity. For example, the solution of cluster six is obtained thanks to the union of two of the clusters found in phase seven of the cluster.
In this way, clusters are formed only by the union of existing groups, so any member of a cluster can trace their relationship on an unbreakable path that would start with a simple relationship.
The hierarchical methods can be by Agglomeration or by Division. Agglomeration clustering begins with each object in a separate group.
Clusters are formed by grouping objects into increasingly larger sets. This process continues until all objects are part of a single group.
Division clustering begins with all objects grouped into a single set. Clusters are divided until each object is an independent group.
Within clusters by clustering are clustering methods, which are frequently used in market research.
They consist of Link methods, Variance methods or sums of error squares and the Centroid method. Link Methods include single link, full link and average link.
The simple link method is based on the minimum distance or closest neighbor rule. The first two conglomerate objects are those that have the shortest distance from each other. The next shortest distance is identified, whether the third object is grouped with the first two or a new cluster of two objects is formed.
At each stage, the distance between two clusters is the distance between their two closest points.
At any stage, two clusters emerge from the shortest single link between them. This process continues until all objects are in a cluster.
The simple link method does not work properly when clusters are not well defined.
The full link method is similar to the single link except that it is based on the maximum distance or the farthest neighbor strategy. In this case, the distance between two clusters is calculated as the distance between their furthest points.
The average link method works similarly, but in this method, the distance between two clusters is defined as the average of the distances between all pairs of objects, where a member of the pair of each of the clusters is found (See Figure Methods of Link for the Cluster). As can be seen, the average link method uses information on all pairs of distances, not just the minimum or maximum. For this reason, it is generally preferred over simple and complete linking methods.
The Variance Methods try to generate clusters in order to reduce the variance within the groups. A frequently used method of variance is the Ward Procedure.
For each cluster, the means for all variables are calculated. Then, for each object, the square Euclidean distance is calculated for the group means (Figure Other Agglomeration Clustering methods); these distances are added to all objects. At each stage, the two clusters are combined with the smallest increase in the total sum of the squares of the distances within the clusters.
In the Centroid Method, the distance between two groups is the distance between their centroids (means for all variables), as shown in Figure Other Agglomeration Clustering methods.
Every time objects are grouped, a new centroid is calculated. Of the hierarchical methods, the Average Link method and the Ward Procedure have shown better performance than the others.
The second type of cluster procedure, the non-hierarchical cluster method, is often referred to as the K-Means Clustering.
These methods include the Sequential Threshold, Parallel Threshold, and the Division for Optimization.
In the Sequential Threshold method, a group center is selected and all objects are grouped within a threshold value that is specified in advance from the center.
Then a new group center or seed is selected and the process is repeated for the ungrouped points. Once an object is grouped with a seed, it is no longer considered for its cluster with subsequent seeds. The Parallel Threshold method works similarly, except that multiple group centers are selected simultaneously and threshold level objects are grouped within the closest center.
The Division for Optimization method differs from the other two threshold procedures in that objects can later be reassigned to other groups, in order to optimize a general criterion, such as the average distance within groups for a given number of clusters.
There are two basic ways of knowing the grouping mode of the objects in question:
- Icicles Chart
Its columns correspond to the objects that are grouped (interviewees,…) and the rows to the number of groups. This figure is read from the bottom up. Initially all cases are considered as individual groups. In the first case, the two closest objects are combined.
Each subsequent step leads to the formation of a new group in one of the following three ways: (1) two individual cases are grouped, (2) one case joins an existing group, (3) two groups join.
It is read from left to right. The vertical lines represent the joined groups. The position of the line on the scale indicates the distances at which the groups join.
Because many distances are similar in magnitude in the early stages, it is difficult to determine the sequence in which some of the earliest clusters form. However, it is evident that in the last two stages, the distances in which the conglomerates are combined are great. This information is useful in deciding the number of clusters.
It is also possible to obtain information on the participation of case clusters by specifying the number of groups. Although this information can be deduced from the icicle trace, a tabular representation is useful.
Hierarchical and Non-Hierarchical Methods
6. Decision of the Number of Clusters
A big problem in all agglomeration techniques is how to select the number of groups (clusters). Unfortunately, there is no objective selection process.
In the case of hierarchical cluster analysis, the distances between the clusters reflected in the different stages of the agglomeration process can serve as a useful guide, the analyst could thus establish a cap to stop the process at his convenience (this information can be obtained from the program agglomeration or dendrogram).
For example, you could do this when the distance between the groups exceeds a specific value or when the successive distances between the steps mark a sudden jump.
However, the most used option is to calculate different agglomeration solutions (two, three, four groups, for example) and then decide between alternative solutions with the help of a pre-established criterion, common sense, or theoretical foundations.
These distances are often called measures of error variability.
In the case of non-hierarchical cluster analysis, a graph can be drawn that compares the number of groups with the relationship between the total variance of the groups and the variance between the groups.
The point on the graph where a marked bend or bend occurs will indicate the appropriate number of groups. In general, it will not be worth increasing the number of groups beyond this point. Another possibility to decide the optimal number of groups is to define some kind of intuitive conceptualization of the theoretical relationship of the data.
Researchers should examine the variation produced between group sizes from a conceptual perspective, comparing the results obtained with the expectations created in the study objectives.
Another problem that can arise in this type of analysis is the presence of one-person groups, that is, clusters formed by a single individual. They are a problem because they could be outliers (outliers) not detected in the debugging process of our data source.
If a single-member group appears, the analyst should study whether it represents a valid structural component in the sample or, on the contrary, should be deleted because it is not representative. If any observation is removed from the analysis, the researcher should rerun the cluster analysis for the new valid observations and thus be able to define new groups.
7. Interpretation and Preparation of the Cluster Profile
The interpretation and the profile of the groups includes the analysis of the group centroids. The centroids represent the mean values of the objects that the group contains in each of the variables. Centroids allow us to describe each group by assigning a name or label.
If the cluster program does not provide this information, it can be obtained through discriminant analysis.
The objective of this stage is essentially to examine the variation of the clusters to assign labels that truthfully describe their nature.
It is useful to prepare the profile of the groups in terms of the variables used for the cluster, such as demographic, psychographic, product use, media use, or other variables.
Let's consider an example to better understand how the process works. Suppose we are interested in studying the effective diet against regular intake of light drinks.
For this, a scale of evaluation of the respondent's attitude was prepared, which was made up of seven different statements. Thus, the interviewed individuals gave values of 1 to 7 points. The statements that were part of the seven-point scale were of the type: light diet drinks taste stronger, diet drinks are healthier, etc.
It was agreed to collect demographic data and consumption data for soft drinks due to their relevance to the study.
As previously stated, in this phase the averages of the profile scores are examined. For our specific case, based on the attitude scale designed for each group and thus being able to assign a descriptive label to each of them.
Suppose two of the groups resulting from the cluster analysis had favorable attitudes towards light diet drinks and a third group negative attitudes. One could manage the possibility that, of the two groups favorable in attitude, one of them was favorable only to light diet drinks and the other favorable to both light and normal soft drinks.
We would then assess the attitudes of each cluster and develop substantive interpretations to facilitate labeling for each group. For example, one of the clusters could be labeled as health and calorie conscious individuals and the other as individuals indifferent to a rise in sugar.
Regarding the profiling of the clusters or groups, it should be said that it is only the description of the characteristics of each cluster to explain how they could infer in relevant dimensions.
To achieve this, the use of Discriminant Analysis or some other appropriate statistic is normally used. The analyst uses the data not previously included in the agglomeration procedure to outline the characteristics of each cluster.
These data are usually demographic characteristics, psychographic profiles, consumption patterns, etc.
Applying this process and extrapolating it to the example of beverages, we would conclude that the cluster of health and calorie-conscious individuals lies in better education or higher professional incomes as they are moderate consumers of soft drinks.
In summary, the profile analysis focuses on describing not what the clusters directly determine but (once the different groups have been determined) their own characteristics.
For this reason, special emphasis is placed on the characteristics that define the groups and on the capacity of the members of each conglomerate to predict a particular attitude of the cluster in question.
8. Validation of Clusters Obtained
Given the general criteria included in the AC, no grouping solution should be accepted without an assessment of its confidence and validity. Validation is the analyst's attempt to ensure that the clusters obtained are representative of the original population and that they are generalizable to other objects and stable over time.
The following procedures provide adequate reviews of the quality of grouping results:
- Carry out the AC with the same data and use different distance measurements. Compare the results with all measurements to determine the stability of the solutions. Use various clustering methods and compare the results. Divide the data in half at random. Perform the AC separately in each half (subsample). Compare the solutions of the two analyzes and evaluate the correspondence of the results or compare the group centroids of the two subsamples. Delete the variables randomly. Perform grouping based on the reduced set of variables. Compare the results based on the complete set with those obtained when performing the cluster. In the non-hierarchical cluster the solution may depend on the order of the cases in the data set. To study this,it is recommended to carry out multiple runs and use different orders of the cases until the solution is stabilized.
Definition of Variables
Variables are defined as the property that can vary and whose variation can be measured. Examples: sex, motivation to work, personality, exposure to a campaign, quality of service.
In working with hypothetical formulations, the definition of the type of relationships established between the variables is of vital importance, since the verification depends on the degree to which these relationships can be demonstrated. This requires great precision in the use of the logical terms that link to the variables, since a misused expression can completely distort the meaning of the formulation.
The relationships between variables can be classified as follows:
- Reversible: if X, then Y, and if Y then X, or irreversible: if X, then Y, but yes Y, there is no conclusion regarding X. Deterministic: if X, then always Y, or stochastic: if X, then probably Y. Sequential: if X, then later Y, or coexistent: if X, then also Y. Enough: if X, then independently of anything else, Y, or contingent: if X, then Y, but only if Z. Necessary: if X, and only X, then Y, or substitutable: if X, then Y, but if Z, also Y. Interdependent: when the attributes of reversibility, contingency and sequentiality are combined, for example: if X, it varies to Xi, Xii, Xiii…., then Y also varies to Yi, Yii, Yiii…, etc.
Once the relationships between the variables have been established with precision, it is necessary to carry out their operationalization, that is, to identify the dimensions, indicators, scales and categories with which the presence of the variable to be measured can be verified.
Three types of variables have been defined in research theory, which commonly appear in hypothetical designs: dependent, independent and foreign variables. We do not believe it necessary to go into more detail about the characteristics of each one, even though there are many examples where the relationship between the first two is confused or those that are parallel to the main relationship are not taken into account.
Quantitative and qualitative dimensions can be distinguished in any variable. Variable dimensions can also be spatial and temporal. Other types of dimensions are contextual and situational, in addition the dimensions of the variable can be individual and group.
This matter requires a deep prior analysis of the object of study, so that all the dimensions and indicators that identify the selected variable can be established with precision. From here a solid and verifiable hypothesis can be elaborated.
The following diagram presents the analysis of the possible dimensions for two variables selected in an example from a group of students, where the Performance and School Discipline are selected:
| Dimensions | performance | Discipline |
| Quantitative | Passed percentage | Amount of facts |
| Qualitative | Quality of ratings | Transcendence, magnitude |
| Objective | Ratings | Sanctions and incentives |
| Subjective | Performance criteria (of teachers, students, parents, officials, and others). | Discipline criteria (from teachers, students, parents, officials, and others). |
| Space | In exams, class work, extra class assignments and others. | In the classroom, workshop, sports area, library and others. |
| Temporary | Weekly, monthly, semester, school year, career, etc. | Weekly, monthly, semester, school year, career, etc. |
| Contextual | In scheduled or surprise exercises. | Under the control of the teacher, or other people. |
| Situational | In the partial, final or selectivity controls. | In teaching, extra-teaching, recreational activities, etc. |
| Individual | Individual performance, by subject, period and course. | Individual compliance with disciplinary rules. |
| Group | Group performance, by subject, period and course. | Individual compliance with disciplinary rules. |
From these studied elements we proceed to the selection or definition of variables, for which various bibliographies were consulted.
Based on the existing relationship with the research being carried out and the final elements that we are pursuing, a study by Guimaraes, Sandy and McKeen (2003) was selected for the Quality Managetmen Journal Vol.10. Issue4. October 2003, entitled "Empirical verification of some factors related to the development of quality systems", in which an in-depth study is carried out with data from more than 228 quality systems to select variables that affect the quality of the service selected by them Taking into account the aspects treated there and taken as the fundamental basis for determining the variables of this research, the following have been defined as variables to carry out the Cluster Analysis:
- Number of audits carried out: This variable indicates the number of audits carried out by each territory, taking into account their size and territorial characteristics. Number of auditors that the system has: It allows us to know the Human Resources that the system has in the different territories, it also indicates the deficit or satisfaction of the Human Resources needs involved in the provision of the service. Auditor Qualification: It is the experience acquired from the service provider, not all providers have the same training. The provider's experience is obtained through training in the preparation of the tasks to be performed and in the execution of the same.Characteristics of the business system: The business system of the territories can be homogeneous or heterogeneous, depending on the characteristics and development of each territory, this implies that the Audit service provided is different in terms of the number of audits to be carried out, the number of auditors involved, the resources that are committed, the execution time of Audits, requalification and training of Human Resources.
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