1. Statistical Reference Guide
  2. Factor analysis (FA)

Factor Analysis (FA) in ExcelFactor analysis (FA) explores the underlying factors (latent variables) among a large number of interrelated variables.

Factor analysis is related to principal component analysis (PCA), but the two are not identical and have different purposes. FA makes the assumption that an underlying model exists, whereas PCA is a descriptive variable reduction technique.

  • EFA
    Exploratory factor analysis (EFA) identifies the underlying relationships between a large number of interrelated variables when there are no prior hypotheses about factors or patterns amongst the variables.
  • Factor rotation
    Rotations minimize the complexity of the factor loadings to make the structure simpler to interpret.

EFA

Exploratory factor analysis (EFA) identifies the underlying relationships between a large number of interrelated variables when there are no prior hypotheses about factors or patterns amongst the variables.

EFA is a technique based on the common factor model which describes the measured variables by a function of the common factors, unique factors, and error of measurements. Common factors are those that influence two or more measured variables, while unique factors influence only one measured variable.

Pattern matrix

The factor pattern matrix loadings are the linear combinations of the factors that make up the original standardized variables.

Structure matrix

The factor structure matrix loadings are the correlation coefficients between the factors and the variables.

Correlation matrix

The factor correlation matrix coefficients are the correlation coefficients between the factors.

Note: When the factors are not rotated, or the rotation is orthogonal, there is no correlation between the factors and the correlation matrix is equal to the identity matrix. Also, the loadings in the pattern matrix and structure matrix are identical, although it can be useful to remember the different interpretations - as linear coefficients or correlation coefficients.

Factor rotation

Rotations minimize the complexity of the factor loadings to make the structure simpler to interpret.

Factor loading matrices are not unique, for any solution involving two or more factors there are an infinite number of orientations of the factors that explain the original data equally well. Rotation of the factor loading matrices attempts to give a solution with the best simple structure.

There are two types of rotation:
  • Orthogonal rotations constrain the factors to be uncorrelated. Although often favored, in many cases it is unrealistic to expect the factors to be uncorrelated, and forcing them to be uncorrelated makes it less likely that the rotation produces a solution with a simple structure.
  • Oblique rotations permit the factors to be correlated with one another. Often produces solutions with a simpler structure.

Matrix rotations

Orthogonal and oblique matrix rotations.

p = number of variables, m = number of factors.

Method Parameters Purpose
Varimax   Orthogonal only. A computational faster equivalent to CF-Varimax.
Promax power 0 ... 4 Oblique only.
Crawford-Ferguson kappa 0 ... 1 Smaller kappa minimizes variables complexity and larger kappa minimizes factor complexity.
CF-Varimax Crawford-Ferguson kappa = 1/p. Spread variances across factors. Each factor tends to have either large or small loadings on a particular variable making it easy to identify each variable with a single factor.
CF-Quartimax Crawford-Ferguson kappa = 0. Minimizes variable complexity. Works well with distinct clusters without cross-loadings.
CF-Equamax Crawford-Ferguson kappa = m/(2p).  
CF-Parsimax Crawford-Ferguson kappa = (m-1) / (p+m-2).  
CF-Factor Parsimony Crawford-Ferguson kappa = 1. Minimizes factor complexity. Primarily of theoretical interest.
Oblimin gamma 0 ... 1  
O-Quartimin Oblimin gamma = 0. Equivalent to quartimax.
O-Biquartimin Oblimin gamma = 0.5.  
O-Covarimin Oblimin gamma = 1. Equivalent to varimax.
Geomin delta > 0 Minimizes variable complexity.

Extracting factors

Extract the underlying factors (latent variables) among a large number of interrelated variables in a multidimensional dataset.

  1. Select a cell in the dataset.
  2. On the Analyse-it ribbon tab, in the Statistical Analyses group, click Multivariate, and then click Common Factors.
    The analysis task pane opens.
  3. In the Variables list, select the variables.
  4. In the Factors to extract edit box, type the number of underlying factors to attempt to extract.
  5. Optional: To highlight the coefficients depending on their sign and magnitude, select the Color maps check box.
  6. Optional: To rotate the extracted factors to allow easier interpretation, in the Rotation drop-down list, select the type of rotation, and then in the Method drop-down list, select the rotation and type any additional required parameters.
  7. Click Calculate.

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