This is my way of clarifying some terms that are often thrown around in psychometrics. It extends commonly used terms “factor” and “component”. These are all terms for the “dimensionality” or “structure” of a questionnaire’s scores across a good number of respondents in a psychometric exploration but they have different specificities.

#### Details #

**Domains**: any collection of items sharing some content characteristics such that we group them together and may add up those item scores separately from items on other “domains”. The classic example for me is the domains of the CORE-OM whose 34 items we partitioned into well-being, problems, functioning and risk when we were designing the measure. (This was to a large extent required by the funding but we were happy with that!) We never thought the items would show very clean factor structure in factor analyses though we did show that in large enough help-seeking and non-help-seeking UK samples that the risk items clearly correlated with each other more strongly than they did with the other 28 items and that there was some weak factor separation and some weak specific (i.e. divergent validity) correlations against other scales you might expect them too though that specificity was definitely weak.**Factors**: best restricted to any items clearly grouping together in a formal factor analysis, whether that is an exploratory or a “confirmatory” factor analysis. Factor analysis is a set of statistical ways of analysing data that partititions the variance across the variables into variance that is shared on a set of variables: a common factor (common as there is covariance in common across those items and not shared with other items not on that factor), as well as one or more common factors it also separates that variance from “unique” or “error” variance for each variable which is variance that is not shared with the common factor(s) nor with any other variable. This is a very clean and interesting model that has dominated one of the two main schools of psychometrics from probably the 1950s getting more sophisticated (and easier to execute) pretty much every year.**Components**: in psychometric use this always means something a bit similar to a factor analysis (FA): principal component analysis (PCA). However, unlike FA it’s not a statistical model, it’s a “simple” rearrangement of the score data into (initially) q orthogonal components. The value q here is less than or equal to the number of variables or to n – 1 where n is the number of observations. The rearrangement is “simple” to electronic computers but in the mid-20th C it was often done by hand and it was easier to do than all but rather simplified FA method so dominated FA for a period. In the late 20th Century PCA done by psychologists in SPSS was often reported as “factor analysis”, unsurprisingly because the default method in SPSS’s FACTOR ANALYIS menu was a PCA! Sometimes the initially orthogonal (perpendicular/independent of one another) components can be rotated. Rotation can be “orthogonal” which, as the name suggests, keep s the components perpendicular to one another. Rotation can also be “oblique” which loses theat neat mapping to Cartesian (typical graph) mapping but often makes the way that items map to the components clearer. This same “rotation”, orthogonal or oblique can be applied after an initial exploratory factor analysis but in confirmatory factor analysis oblique relationship (correlation) of the factors with one another is either allowed or disallowed in setting up the CFA.**Dimensions**: I see this an overarching idea, rather similar to orthogonal principal component analysis that simply assumes that it can be useful, either as an cartoon/sketch/approximation to assume that are areas of interest can be seen as mapping into multidimensional spaces such as the famous “extraversion”, “neuroticism” and “psychoticism” of the Eysenck’s personality questionnaires.

A final related issue is that of “multidimensional scaling” (MDS) which is probably best summarised here as a mathematical model that rearranges data somewhat in the way that PCA does but which doesn’t assume that the numbers can be treated as having linear (interval and/or ratio scaling in Steven’s categories of scaling). MDS only assumes that the raw numbers do reflect an ordinal/rank relationship between values (and may or may not allow ties to be “untied”!) MDS seems to have faded considerably in popularity since the late 20th C so feel free to ignore this!!!

A further, typical, bit of complication/small print is that both “factor” and “component” have other meanings in statistics outside psychometrics. Sorry.

#### Try also #

- Confirmatory factor analysis
- Factor analysis
- Multidimensional scaling (MDS)
- Orthogonal
- Principal component analysis
- Stevensâ€™ levels of measurement

#### Chapters #

Doesn’t come up in the OMbook.

#### Online resources #

None yet nor likely? Hm, does need more explanation!

#### Dates #

First created 3.viii.24.