Personality & Individual Differences · Simplified
Turn the Axes, Change Every Number
Rotation is the step where a factor analysis stops being arithmetic and starts being a decision.
Step 1 of 2
First, a prediction
A researcher extracts two factors from twelve fictional adjective ratings, then rotates them. Before you see it happen: what does rotating the factors do to how well the model fits the data?
Step 2 of 2
Twelve markers, two factors, one dial
The twelve points came out of the extraction and they never move. The only thing the dial moves is the pair of axes you are reading them against.
Key terms
- Loading
- How strongly one marker relates to one factor. It is the marker's position measured along that axis.
- Communality
- The share of a marker's variance accounted for by the two factors together. It is the marker's distance from the centre, which no rotation can change.
- Simple structure
- A solution in which each marker loads clearly on one factor and near zero on the other. It is a criterion researchers choose, not a property the data has.
- Cross-loading
- A marker that relates substantially to both factors at once.
Each numbered point is one marker, at the position the extraction gave it. The two lines are the factor axes. A marker's loading on a factor is how far along that axis it sits.
| # | Marker | Factor 1 | Factor 2 | Communality | Reads as |
|---|
What this shows
Rotation changes the description, not the data
Key idea: Every loading in the table changed as you turned the dial, and three things did not: the position of each point, each marker's communality, and the total variance the two factors account for. The rotated solution reproduces the correlations exactly as well as the unrotated one, so no rotation is more correct than another in any sense the data can settle. What rotation buys is readability, and simple structure is a criterion researchers impose because a solution where each marker belongs to one factor is easier to talk about. That is a good reason. It is not evidence. The same point applies twice over to the names: the labels under the chart are the highest-loading markers and nothing more, and a factor is a summary of how a particular set of items covaried in a particular sample, not a thing found inside people.
Twelve invented markers with illustrative coordinates, chosen to make rotation legible rather than drawn from a real dataset. Three limits worth stating. The axes here stay at right angles, which forces the two factors to be uncorrelated; letting them tilt is a further choice that allows correlated factors, and it too leaves the fit unchanged. How many factors to extract is a separate decision made before any of this, and it changes the answer far more than rotation does. And nothing here speaks to whether two factors is the right number for these markers, or whether the same two would appear in another sample, in another language, or with a different set of adjectives.
The longer version adds a second marker set, an oblique control that lets the axes come off 90 degrees, and a closing challenge. It is at Factor Rotation Playground in the main collection.