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Factors Analysis – Statistical Method

Factor analysis is that statistical method which is used for describing validity among various observed variables in terms of the unobserved variables, which are also called factors. The factor analysis is in search of the joint variations which are in response to the unobserved latent variables. The observed variables are then modeled as the linear combinations of all the potential factors, along with the error terms. The information relating to the interdependencies of the observed variables is used for reducing sets of variables in a data set. When we have a set of p random observable variables, where x₁, ....x_pand the means are u₁,....,u_p ,

We get



The e_i is independently distributed items of error having zero mean and a finite variance, which might not be same for all the i’s.



For n observations, the dimensions will be x_p*n, L_p*k, F_k*n Here the value of a particular observation is denoted by x and F. The matrix L does not vary across observations.

We can assume the assumptions on F

1. f and e are independent

2. E(F)=0

3. Cov(F)=1

A solution of above set of equations having the constraints of F is denied as the factors and L is called the loading matrix.

Assuming Cov (x − μ) = Σ, then form the conditions just imposed on F we get,



For any orthogonal matrix Q , if we set L=LQ and F=Q^TF, the criteria for being factors F, the criteria for being factors and the factor holdings hold, therefore the set of factor and the factor holding is identical only up to the orthogonal transformations.

Type of factor analysis

1. Exploratory data analysis or EFA- is used for covering underlying structure of a relatively larger set of variables. The a priori assumption is that any indicator might be indicated with any factor. This can be termed as the most common form of factor analysis found. Factor loadings are used for intuiting the factor structure of the data and there is no prior theory. ctor analysis or2. Confirmatory factor analysis or CFA is used for determining whether the number of factors as well as the loadings of indicator variables lying on them confirm to what is expected in real on the axis of a pre established theory. The selection of independent variables is done on the basis of prior theory and we use the factor analysis to see whether they as per the predicted manner on expected number of all factors.

Types of factoring

1. The principal component analysis- it can be termed as the most common form of factor analysis which seeks linear combination of variables such that we can extract the maximum variances from the variables. r analysis- it is also known as the Rao’s canonical factor analysis, which is entirely a different method of computing and seeks factors that have the highest canonical correlation with the observed variables.

3. The common factor analysis-. The other names of common factor analysis are the principal factor analysis and the principal axis factoring. It seeks the least number of factors which account for some common variances of the set of variable.

4. Image factoring-this is based on the correlation matrix of the predicted variables rather than the actual variables, and each sample is predicted from other variables by using multiple regression.

5. The alpha factoring- this is based on the reliability of factors by assuming that the variables are sampled randomly from the universe of the variables. This makes it different from the other methods which assumed the case samples and variables to be fixed.

6. Factor regression model- this is a ki6. Factor regression model- this is a kind of combinational mole obtained form combining regression model and factor model. It can also be termed as a hybrid model whose factors are only partially known.

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