Method of main components

The method of the main components is based on attempts to explain the maximum level of variance in a certain set of variables, and is oriented to the elements located in the correlation matrix along the diagonal. There is another method based on factor analysis aimed at approximating the correlation matrix using a certain number of factors (less than a given number of variables), but the methods of approximation differ substantially from the first proposed method.

So, the method of factor analysis allows us to explain the correlation between the variables themselves, and is oriented on the elements of a matrix of correlation type that are outside its diagonal.

Based on practical application, let's try to understand the necessity of applying this or that method. Factor analysis is used when there is an interest of the researcher in studying the interrelation between variables, the method of the main components is used if there is a need to reduce the dimensionality of the data and to a lesser extent requires an interpretation of them.

Based on practice, we can see that the methods of factor analysis use a fairly large number of observations. At the same time, this quantity should be higher by an order of magnitude than the number of factors detected.

The method of main components is very popular in marketing research, since it can be used in the presence of multicollinear initial data. In the process of such marketing research, the questionnaires contain similar questions, and the answers received will correspond to the principles of multicollinearity.

It is advisable to consider the method of the main components in a set of indicators, which should be a guide for the researcher in the preliminary choice of the number of components or factors. The most important of these are eigenvalues, expressing the level of variance of variables, explained by this factor. There is also one important empirical rule, which is very useful for estimating the number of factors (there must be as many factors as there are eigenvalues over one). It is possible to explain this rule in a somewhat simpler way - the eigenvalues express the fraction of normalized variances of variables that are explained by the factor, and in the case of exceeding one they must express these variances contained in more than one variable.

It is necessary to clarify once again that the rule of "individual eigenvalues" is empirical, and the question of the necessity of its application can be solved only by the researcher himself. For example, an eigenvalue has a value that is less than one, but it explains the spread that is distributed between the variables. For a specialist in the marketing field, it is very important that, when segmenting, the identified factors have a meaningful meaning. And those factors containing eigenvalues greater than one, but not having a meaningful interpretation, will not be taken into account. And the situation can arise quite the opposite.

Another important issue concerning the practical application of methods of factor analysis is the question of rotations. Such variants of rotation can be considered. The most popular of these is the varimax method. It is based on achieving the maximum level of variance of variables for each individual factor. This method helps to find a rotation in which some variables take high values, while others are low enough for each individual factor.

Another method of rotation is a kilometax, it helps to find a certain turn in which the factors for each individual variable have both low and high loads.

The method of rotation equimax is some compromise between the two methods discussed above.

All these methods refer to orthogonal with mutually perpendicular axes; when used, there is a lack of correlation between individual factors.