- Motivating example: The SAQ
- Pearson correlation formula
- Partitioning the variance in factor analysis
- Extracting factors
- principal components analysis
- common factor analysis
- principal axis factoring
- maximum likelihood
- Rotation methods
- Simple Structure
- Orthogonal rotation (Varimax)
- Oblique (Direct Oblimin)
- Generating factor scores
Suppose you are conducting a survey and you want to know whether the items in the survey have similar patterns of responses, do these items “hang together” to create a construct? The basic assumption of factor analysis is that for a collection of observed variables there are a set of underlying variables called factors (smaller than the observed variables), that can explain the interrelationships among those variables. Let’s say you conduct a survey and collect responses about people’s anxiety about using SPSS. Do all these items actually measure what we call “SPSS Anxiety”?
Motivating Example: The SAQ (SPSS Anxiety Questionnaire)
Let’s proceed with our hypothetical example of the survey which Andy Field terms the SPSS Anxiety Questionnaire. For simplicity, we will use the so-called “SAQ-8” which consists of the first eight items in the SAQ. Click on the preceding hyperlinks to download the SPSS version of both files. The SAQ-8 consists of the following questions:
- Statistics makes me cry
- My friends will think I’m stupid for not being able to cope with SPSS
- Standard deviations excite me
- I dream that Pearson is attacking me with correlation coefficients
- I don’t understand statistics
- I have little experience of computers
- All computers hate me
- I have never been good at mathematics
Pearson Correlation of the SAQ-8
Let’s get the table of correlations in SPSS Analyze – Correlate – Bivariate:
|**. Correlation is significant at the 0.01 level (2-tailed).|
|*. Correlation is significant at the 0.05 level (2-tailed).|
From this table we can see that most items have some correlation with each other ranging from r=−0.382r=−0.382 for Items 3 and 7 to r=.514r=.514 for Items 6 and 7. Due to relatively high correlations among items, this would be a good candidate for factor analysis. Recall that the goal of factor analysis is to model the interrelationships between items with fewer (latent) variables. These interrelationships can be broken up into multiple components
Partitioning the variance in factor analysis
Since the goal of factor analysis is to model the interrelationships among items, we focus primarily on the variance and covariance rather than the mean. Factor analysis assumes that variance can be partitioned into two types of variance, common and unique
- Common variance is the amount of variance that is shared among a set of items. Items that are highly correlated will share a lot of variance.
- Communality(also called h2h2) is a definition of common variance that ranges between 00 and 11. Values closer to 1 suggest that extracted factors explain more of the variance of an individual item.
- Unique varianceis any portion of variance that’s not common. There are two types:
- Specific variance: is variance that is specific to a particular item (e.g., Item 4 “All computers hate me” may have variance that is attributable to anxiety about computers in addition to anxiety about SPSS).
- Error variance:comes from errors of measurement and basically anything unexplained by common or specific variance (e.g., the person got a call from her babysitter that her two-year old son ate her favorite lipstick).
The figure below shows how these concepts are related:
The total variance is made up to common variance and unique variance, and unique variance is composed of specific and error variance. If the total variance is 1, then the communality is h2h2 and the unique variance is 1−h21−h2. Let’s take a look at how the partition of variance applies to the SAQ-8 factor model.
Here you see that SPSS Anxiety makes up the common variance for all eight items, but within each item there is specific variance and error variance. Take the example of Item 7 “Computers are useful only for playing games”. Although SPSS Anxiety explain some of this variance, there may be systematic factors such as technophobia and non-systemic factors that can’t be explained by either SPSS anxiety or technophbia, such as getting a speeding ticket right before coming to the survey center (error of meaurement). Now that we understand partitioning of variance we can move on to performing our first factor analysis. In fact, the assumptions we make about variance partitioning affects which analysis we run.