Factor Analysis Research Methods and Statistics

Learning Outcomes At the end of this lecture and with additional reading you will be able to Describe the difference between Factor Analysis (FA) and Principal Component Analysis (PCA) Understand how to extract factors Describe the difference between orthogonal and oblique rotation

Research Question Difference Factor Analysis Association

Correlations between sets of variables are displayed in a correlation matrix often referred to as an R-matrix. The diagonal elements of an R-matrix are all 1 because each variable will correlate with itself perfectly. The off-diagonal elements are the correlation coefficients between pairs of variables

The existence of clusters of large correlation coefficients between subsets of variables suggest that those variables may be measuring aspects of the same underlying dimensions. These are called factors

Below is an example of two subsets of correlated variables TiSocIntT2SelLi T11 Soc.721 Int T Sel Li

A Factor plot Factor 1 Factor 2 1 1

Factor Analysis and Principal Component Analysis FA analyses only shared variances, unique variance is excluded and error variance is assumed PCA analyses all variance as it assumes there is no error PCA transforms data into smaller sets of uncorrelated components

FA and PCA FA is looked on to confirm a hypothesis where researchers believe a smaller set of factors cause/influence observed variables PCA is looked on as exploratory in order to reduce a large data set into smaller dimensions

Why do we use Factor Analysis By reducing a data set from a group of interrelated variables into a smaller set of uncorrelated factors factor analysis enables the researcher to explain the maximum amount of common variance in a correlation matrix using the smallest amount of explanatory concepts

So how does Factor Analysis work Factor analysis tries to find the common underlying dimensions within a set of data As such FA is interested in the amount of common variance (the amount of variance that is shared with other variables) The proportion of common variance present in a specific variable is known as communality, therefore FA assumes that all variance is communal and gives it a figure of 1

How to extract factors FA uses eigenvalues to establish how many factors to extract An eigenvalue represents the amount of variation explained by a factor, and an eigenvalue of 1 or more represents a substantial amount of variance (Kaiser 1960) Another method is to plot a scree plot. The point for selection should be at the point of inflection (Cattell 1966b)

Scree plot

Factor rotation I Once factors have been extracted it is possible to calculate to what degree variables load onto these factors Generally most variables have high loadings on the most important factor and low loadings on the others, this makes interpretation difficult To improve interpretation a technique called factor rotation is used

Factor rotation II Rotation allows variables to be loaded on to one factor There are two types of rotation Orthogonal: this means that as factors are rotated they are kept independent (they do not correlate at all) and the right angles are kept at 90° Oblique: this means that factors are allowed to correlate

Factor rotation III orthogonal oblique Factor 2 Factor 1

Factor loadings Once factors have been found the next stage is to decide what variables load onto which factors It is often thought that loadings greater than.3 are important, although this does depend on the sample size (Stephens 1992) i.e. – sample size 50 loading >.7 – Sample size of 100 loading >.5 – Sample size of 200 loading >.3

Now lets look at a real example

Naming Dimensions Once all variables have been loaded onto the four factors, have a look at the questions they represent and decide what you would call each dimension This is also where you may decide that the factor extraction is not correct and you may only want to extract 3 factors instead