# Chapter 5: Latent variable models with categorical indicators

### One-factor model for binary items: Factor scores

The general idea of calculating factor scores, i.e. predicted values of a latent factor given the observed items, is the same for latent trait models as it was for factor analysis. Most commonly, the factor score for an individual respondent is calculated as the expected value of the factor given the observed values of the items. These scores are calculated by any computer software (such as Stata) which can be used to fit latent trait models.

To get a sense of what the factor scores tell us, it is useful to consider a simpler version of them, the “component score” *y*_{1}λ̂_{1} + *y*_{2}λ̂_{2} + ... + *y*_{p}λ̂_{p}. Since the values of the binary items *y*_{j} are coded as 0 or 1, this is simply the sum of the estimated loading parameters λ̂_{j} for those items for which a respondent’s value is 1 (e.g. all the items that the respondent agrees with, if the coding is 1 for “Agree” and 0 for “Disagree”). The largest possible value of this score is then obtained by a respondent who has the value 1 for all the items which have positive estimated loadings and 0 for all the items which have negative loadings. This score is not exactly equal to the expected value that standard software use as the factor score, but it carries essentially the same information; in particular, the component scores and factor scores give exactly the same ranking of individuals in terms of the values of the latent factor. This result, and component scores and factor scores more generally, are discussed by [Bar08] and [Bar11]

Factor scores derived from a latent trait model may be used as observed variables in other analyses. This approach is discussed further later in this chapter, and an illustration of it is given in Example 2.

#### References

- [Bar08] Bartholomew, D. J., Steele, F., Moustaki, I. and Galbraith, J. G. (2008). Analysis of multivariate social science data (Second edition). Chapman & Hall/CRC.
- [Bar11] Bartholomew, D. J., Knott, M. and Moustaki, I. (2011). Latent Variable Models and Factor Analysis: a unified approach (Third edition). Wiley.