## Interpreting SPSS output on loadings

SPSS has a range of not very consistent terminology in its output when performing principal component and factor analyses. Principal components assumes that each item is completely predicted by the other items in the model; factor analysis assumes that there is a non-negligible amount of item variance which is not explained by other items in the model.

The explanation of the terminology is given below for three different scenarios.

**Using Principal components (no oblique rotation)**Component coefficient is the item-factor loading. This represents the correlation between the item and each factor (since the components are assumed uncorrelated). This matrix is outputted even when an oblique rotation is specified.

Component scores are the standardised item-factor score regression weights. This represents the unique contribution of each item in determining the factor adjusted for all other items. It is equivalent to the beta estimate in a linear regression of items on each factor score.

These correlations and regression coefficients are the same for an unrotated or orthogonal rotation.

**Using Factor analysis (maximum likelihood, no oblique rotation)**Factor matrix is the item-factor loading. This represents the correlation between the item and each factor when the factors are specified as uncorrelated. This matrix is produced assuming the factors are not correlated and is outputted even if an oblique rotation, specifying non-zero factor intercorrelations, is asked for.. Factor score coefficient matrix is the standardised item-factor score regression weight. This represents the unique contribution of each item in determining the factor adjusted for all other items. It is equivalent to the beta estimate in a linear regression of items on each factor score.**An oblique rotation (assumes non-zero correlations between factors)**

A factor inter-correlation matrix (**F**) is produced since correlations are now assumed to be non-zero..

Factor pattern matrix (**P**) is the standardised item-factor loading adjusted for all other items in the model. Hair et al (1998) and others suggest reporting the factor pattern matrix when it is given.

Factor structure matrix (**S**) is the correlation between the item and the factor not adjusting for any other items.

The factor pattern and factor structure matrices are identical if the factors are not assumed to be correlated (Loehlin, 1992). This is because when factors are correlated an item can influence a factor both directly and indirectly, through its associations with other factors.

**S = PF**

Factor score coefficient matrix (*Beta*) is the standardised item-factor score regression weight. This represents the unique contribution of each standardised item in determining the factor adjusted for all other items. It is equivalent to the beta estimate in a linear regression of items on each factor score.

**Beta =** R^{-1 } S = R^{-1 } **PF**

Where **R** is the item correlation matrix.

There are cut-offs which have been suggested to aid interpretation of factor loadings.