# What are the assumptions of factor analysis?

I want to check if I really understood [classic, linear] factor analysis (FA), especially assumptions that are made before (and possibly after) FA.

Some of the data should be initially correlated and there is a possible linear relation between them. After doing factor analysis, the data are normally distributed (bivariate distribution for each pairs) and there is no correlation between factors (common and specifics), and no correlation between variables from one factor and variables from other factors.

Is it correct?

Input data assumptions of linear FA (I'm not speaking here about internal assumptions/properties of the FA model or about checking the fitting quality of results).

1. Scale (interval or ratio) input variables. That means the items are either continuous measures or are conceptualized as continuous while measured on discrete quantitative scale. No ordinal data in linear FA (read). Binary data should also be avoided (see this, this). Linear FA assumes that latent common and unique factors are continuous. Therefore observed variables which they load should be continuous too.
2. Correlations are linear. Linear FA may be performed based on any SSCP-type association matrix: Pearson correlation, covariance, cosine, etc (though some methods/implementations might restrict to Pearson correlations only). Note that these are all linear-algebra products. Despite that the magnitude of a covariance coefficient reflects more than just linearity in relation, the modeling in linear FA is linear in nature even when covariances are used: variables are linear combinations of factors and thus linearity is implied in the resulting associations. If you see/think nonlinear associations prevail - don't do linear FA or try to linearize them first by some transformations of the data. And don't base linear FA on Spearman or Kendall correlations (Pt. 4 there).
3. No outliers - that's as with any nonrobust method. Pearson correlation and similar SSCP-type associations are sensitive of outliers, so watch out.
4. Reasonably high correlations are present. FA is the analysis of correlatedness, - what's its use when all or almost all correlations are weak? - no use. However, what is "reasonably high correlation" depend on the field of study. There is also an interesting and varied question whether very high correlations should be accepted (the effect of them on PCA, for example, is discussed here). To test statistically if the data are not uncorrelated Bartlett's test of sphericity can be used.
5. Partial correlations are weak, and factor can be enough defined. FA assumes that factors are more general than just loading pairs of correlated items. In fact, there even an advice not to extract factors loading decently less than 3 items in explotatory FA; and in confirmatory FA only 3+ is guaranteed-identified structure. A technical problem of extraction called Heywood case has, as one of the reasons behind, the too-few-items-on-factor situation. Kaiser-Meyer-Olkin (KMO) "sampling adequacy measure" estimates for you how weak are partial correlations in the data relative the full correlations; it can be computed for every item and for the whole correlation matrix. Common Factor analysis model assumes that pairwise partial correlations are enough small not be bothered about and modelled, and they all fall into that population noise for individual correlation coefficients which we don't regard any differently from the sample noise for them (see). And read also.
6. No multicollinearity. FA model assumes that all items each posesses unique factor and those factors are orthogonal. Therefore 2 items must define a plane, 3 items - a 3d space, etc: p correlated vectors must span p-dim space to accomodate their p mutually perpendicular unique components. So, no singularity for theoretical reasons$$^1$$ (and hence automatically n observations > p variables, without saying; and better n>>p). Not that complete multicollinearity is allowed though; yet it may cause computational problems in most of FA algorithms (see also).
7. Distribution. In general, linear FA does not require normality of the input data. Moderately skewed distributions are acceptable. Bimodality is not a contra-indication. Normality is indeed assumed for unique factors in the model (they serve as regressional errors) - but not for the common factors and the input data (see also). Still, multivariate normality of the data can be required as additional assumption by some methods of extraction (namely, maximum likelihood) and by performing some asymptotic testing.

$$^1$$ ULS/minres methods of FA can work with singular and even non p.s.d. correlation matrix, but strictly theoretically such an analysis is dubious, for me.

• ,could you read this post,it seemed little different. Jul 28, 2017 at 14:27
• If Binary data should also be avoided,what else factor analysis method can we do for binary data? Apr 22, 2019 at 8:23
• dear ttnphns; I notice that you don't mention that the data are assumed normal and other online indicate that normality is not required. My query is if the latent variables are assumed normal, and the observations are modelled as a weighted sum of the factors does this then not imply a normal distribution on the observations? (I'm sorry im sure this is a dumb question) Feb 16, 2020 at 12:27
• @user2957945, Paragraph 7 says about normality. Normality assumption is necessary for some methods of factor extraction and for performin some statistical tests facultatively accompanying factor analysis. To your question: Yes, if factors are distributed normally and errors normally too, that will mean manifest variables are also normal. Feb 16, 2020 at 12:39
• ah, thanks @ttnphns; sorry to bother you -- I dont quite know how I managed to miss that. Appreciate your help. Feb 16, 2020 at 12:49

Much of the time, factor analysis is conducted without any statistical tests per se. It is much more subjective and interpretive than methods such as regression, structural equation modelling, and so on. And generally it is inferential tests that come with assumptions: in order for p values and confidence intervals to be correct, those assumptions must be met.

Now, if the method for choosing the number of factors is set to be the maximum likelihood method, then there is an assumption that goes with this: that the variables input into the factor analysis will have normal distributions.

That the input variables will have nonzero correlations is a sort of assumption in that without it being true, factor analysis results will be (probably) useless: no factor will emerge as the latent variable behind some set of input variables.

As far as there being "no correlation between factors (common and specifics), and no correlation between variables from one factor and variables from other factors," these are not universally assumptions that factor analysts make, although at times either condition (or an approximation of it) might be desirable. The latter, when it holds, it known as "simple structure."

There is another condition that is sometimes treated as an "assumption": that the zero-order (vanilla) correlations among input variables not be swamped by large partial correlations. What this means in a nutshell is that relationships should be strong for some pairings and weak for others; otherwise, results will be "muddy." This is related to the desirability of simple structure and it actually can be evaluated (though not formally "tested") using the Kaiser-Meyer-Olkin statistic, or the KMO. KMO values near .8 or .9 are usually considered very promising for informative factor analysis results, while KMOs near .5 or .6 are much less promising, and those below .5 might prompt an analyst to rethink his/her strategy.

• As i read, that factor analysis starts with some correlation with variables and we try to make this correlation more and more clear Nov 11, 2012 at 19:29
• After application of Factor analysis, if we have used orthogonal rotation, we will be sure that there is no correlation between factors Nov 11, 2012 at 19:32

Assumptions underlying exploratory factor analysis are:
• Interval or ratio level of measurement
• Random sampling
• Relationship between observed variables is linear
• A normal distribution (each observed variable)
• A bivariate normal distribution (each pair of observed variables)
• Multivariate normality
Above from the SAS file