Personality and Individual Differences
Positive Manifold Visualiser
Six fictional tasks, a correlation matrix you control, and a first factor that appears whatever structure you put underneath it.
Simulated — fictional tasks, no real intelligence-test data
Learning objective
By the end you should be able to say why a first factor with positive loadings follows arithmetically from the positive manifold, and settles nothing about what produces it.
About 25 minutes. Nothing you do here is saved or sent anywhere.
What a factor is here
A factor in this tool is a property of a correlation matrix — a summary of numbers you generated with sliders. It is not a biological entity, a brain structure or a measured quantity.
First, a judgement
Across a century of research, scores on almost any pair of cognitive tasks correlate positively. This is called the positive manifold, and factor analysing such a matrix always produces a first factor on which everything loads positively.
The visualiser
Need a hand?
Nothing here changes the controls for you, and none of it says what you are going to find.
Key terms
- Positive manifold
- The finding that scores on diverse cognitive tasks tend to correlate positively with one another.
- General factor
- A factor on which every task loads positively.
- Group factor
- Variance shared by some tasks and not others — the three verbal-ish ones here, for instance.
- Specific variance and measurement error
- What belongs to one task alone, and what is noise. Neither of them contributes to any correlation.
Six fictional cognitive tasks
Three verbal-ish, three spatial-ish. You decide how much of each task's performance is shared with every other task, shared only within its group, or specific and error.
The correlation matrix
Where that variance came from, and other ways to read it
Where each task's variance comes from
The illustrative first factor
| Task | Group | First-factor loading |
|---|
| Pair | Relationship | Model | Observed in sample |
|---|
What this configuration shows
Challenge — what would tell you apart?
What this demonstrates
Every cognitive task correlates positively with every other, and that pattern is one of the most reliable findings in psychology. What it means is not settled by the pattern. A single shared source produces it; so does a web of overlapping group factors with no general one at all. The first factor is arithmetic performed on the matrix, not evidence about what caused it.
For teaching elsewhere: take this activity as one self-contained block of HTML, on the clipboard or as a file. Either way it is styled so that it will not disturb the page you put it into.