Factor is confusing much in the same way as hierarchical and beta, because it too has different meanings in different contexts. Factor might be a little worse, though, because its meanings are related.
In both meanings, a factor is a variable. But a factor has a completely different meaning and implications for use in two different contexts.
Factor in Factor Analysis
In factor analysis, a factor is an latent (unmeasured) variable that expresses itself through its relationship with other measured variables.
Take for example a variable like leadership. We may want to measure a person’s or an organization’s leadership style, but this is the kind of construct that would be impossible to measure using a single variable. It’s just too abstract and multifaceted, although it does represent a single concept.
So instead, you may have to develop the scale with many items, each of which measures some more measurable part of leadership. The idea would be that there is an underlying unmeasurable factor, leadership, that causes people to respond in certain patterns on the many items on the scale.
The purpose of factor analysis is to analyze these patterns of response as a way of getting at this underlying factor. Factor analysis also allows you to use the weighted item responses to create what are called factor scores. These represent a single score for each person on the factor.
Factor scores are nice because they allow you to use a single variable as a measure of the factor in the other analyses, rather than a set of items.
Factor as a Categorical Predictor Variable
Contrast that to the use of a factor in a linear model or a linear mixed model. In this context, a factor is still a variable, but it refers to a categorical independent variable. So you may have heard of fixed factors and random factors. In both cases, those are referring to a categorical independent variable.
Like covariates, factors in a linear model can be either control variables or important independent variables. The model uses them the same way in either case. The only difference is how you are going to interpret the results.
This all gets especially tricky when the continuous factor scores from a factor analysis are used as predictors in a linear model. Technically, since they are continuous, they wouldn’t be factors in the model, in the second definition. They would be covariates.
Read other posts in the series on Confusing Statistical Terms.