Editing Factor analysis (section)

===Rotation methods===
The output of PCA maximizes the variance accounted for by the first factor first, then the second factor, etc. A disadvantage of this procedure is that most items load on the early factors, while very few items load on later variables. This makes interpreting the factors by reading through a list of questions and loadings difficult, as every question is strongly correlated with the first few components, while very few questions are strongly correlated with the last few components.

Rotation serves to make the output easier to interpret. By [[Change of basis|choosing a different basis]] for the same principal components{{snd}}that is, choosing different factors to express the same correlation structure{{snd}}it is possible to create variables that are more easily interpretable.

Rotations can be orthogonal or oblique; oblique rotations allow the factors to correlate.<ref name="StackExchangeRotation">{{cite web |title=Factor rotation methods |url=https://stats.stackexchange.com/q/185216 |website=Stack Exchange |access-date=7 November 2022}}</ref> This increased flexibility means that more rotations are possible, some of which may be better at achieving a specified goal. However, this can also make the factors more difficult to interpret, as some information is "double-counted" and included multiple times in different components; some factors may even appear to be near-duplicates of each other. 

==== Orthogonal methods ====
Two broad classes of orthogonal rotations exist: those that look for sparse rows (where each row is a case, i.e. subject), and those that look for sparse columns (where each column is a variable).

* Simple factors: these rotations try to explain all factors by using only a few important variables. This effect can be achieved by using ''Varimax'' (the most common rotation).
* Simple variables: these rotations try to explain all variables using only a few important factors. This effect can be achieved using either ''Quartimax'' or the unrotated components of PCA.
* Both: these rotations try to compromise between both of the above goals, but in the process, may achieve a fit that is poor at both tasks; as such, they are unpopular compared to the above methods. ''Equamax'' is one such rotation.

====Problems with factor rotation====
It can be difficult to interpret a factor structure when each variable is loading on multiple factors. 
Small changes in the data can sometimes tip a balance in the factor rotation criterion so that a completely different factor rotation is produced. This can make it difficult to compare the results of different experiments. This problem is illustrated by a comparison of different studies of world-wide cultural differences. Each study has used different measures of cultural variables and produced a differently rotated factor analysis result. The authors of each study believed that they had discovered something new, and invented new names for the factors they found. A later comparison of the studies found that the results were rather similar when the unrotated results were compared. The common practice of factor rotation has obscured the similarity between the results of the different studies.<ref name="Fog2022">{{cite journal |last1=Fog |first1=A |title=Two-Dimensional Models of Cultural Differences: Statistical and Theoretical Analysis |journal=Cross-Cultural Research |date=2022 |volume=57 |issue=2–3 |pages=115–165 |doi=10.1177/10693971221135703|s2cid=253153619 |url=https://backend.orbit.dtu.dk/ws/files/292673942/Two_dimensional_models_of_culture.pdf }}</ref>