# When do you consider a variable is a latent variable?

The problem is to define when a variable might be considered as a latent variable. I am interested in how to describe a latent variable, and what are the properties of latent variables.

My twofold question is:

• When you try to explain what a latent variable is, what do you consider as the main differences between a manifest and a latent variable?
• When does factor analysis or item response modeling seem more appropriate ?

Example. On the one hand, if you want to measure fish weight without any instrument, you can devise items to measure it. In this case, do you rely on a latent variable model? On the other hand, social level is sometimes measured directly through answers to a survey with a linear model (or other models) applied to items such as highest diploma, numbers of books at home, numbers of electronics devices, but not by considering a latent variable. But why can't we use a latent variable model in this case?

• That seems rather a general question - would you like to take a look at en.wikipedia.org/wiki/Latent_variable and see if that helps, or if not, which bits you don't follow? – onestop Dec 4 '10 at 12:17
• Can you fix this question? It gets answers because of the topic, but the content seems a pure word clutter to me. – user88 Dec 4 '10 at 15:48
• i fix questions with examples... – pbneau Dec 4 '10 at 18:54

The relevant section of the classical typology distinguishes between (observed) variables, latent variables, and parameters.

Regular variables are observed and have a distribution. Latent variables are not observed and have a distribution. Parameters are not observed and do not have a distribution.

Parameters vs latent variables is indeed a modelling decision. Consider a set of survey questions that tap an underlying scale. If you expect that learning about one subject's position on the scale is potentially informative about another subject's position and you wish to be able to generalise to new subjects then you should treat position as a latent variable. If not, you may as well treat it like a parameter.

Bringing up FA and IRT is a bit confusing because some measurement models aim to estimate subject parameters e.g. Rasch models, and some aim to estimate subject latent variables e.g. FA and IRT models. All types of model have parameters in addition, associated with the items.

For a survey context there are also indexes, constructed by combining several indicators (which are observed variables). You should probably think of these as non-parametric estimators of latent variables, for when you don't feel happy with measurement model parametric assumptions. (Although personally I've never been particularly sure about their status)

• Am I correct in assuming that when you talked of a parameter in place of one's location on a given latent trait you are indeed saying that subjects abilities are modeled without assuming any prior distribution (as is done in the marginal likelihood approach for estimating IRT model from the Rasch family)? – chl Dec 5 '10 at 20:27
• @chl Yes and no. Yes, although they probably do come from some population we have chosen to ignore that in the model. If we make a 'prior' assumption about the population then we can ask for a posterior over the individual's location but not otherwise. It's the panel model fixed effects/random effects distinction all over again. But no, it's not like Marginal Maximum Likelihood (as in Bock and Aitkin) for IRT models. This does assume a prior; that's what the quadrature machinery is busy integrating over. Joint ML is all parameters, but is inconsistent as you add subjects (Anderson 70) – conjugateprior Dec 6 '10 at 14:01
• (+1) Thanks for clarifying that point. I was unsure about (a) the use of 'parameters' because we often speak of person (resp. item) parameters when estimating IRT models (I don't speak of person-specific covariates, just locations on the latent trait), and (b) if there was any suggestion about differences between the marginal approach (where we assume that the $\theta_i$'s are random, with $\mathcal{N}(0;1)$ for prior distribution) and the conditional approach (where subject and item parameters are estimated separately, in line with Rasch's specific objectivity). – chl Dec 6 '10 at 14:16

That is a modeling decision. One way to look at it can be illustrated by the following example. A couple of hundreds electrodes are attached to the head to measure brain activity. Electricity, blood flow, whatever and you get lots of signals. These measurements that you get are observables. They are mixed in probably very non-linear way and are not useful.
Latent or also hidden variables are modeling the individual variables that are responsible for their generations. They are supposed to be more pure, more interpretable. How to extract the signal that is causing the eye to blink, or to open the mouth, or emotions and many more complicated signals. Hope it helps to understand the intuition.

• Thanks i see what you mean... But my goal is to have statistics rules. The complexity you talk about can be seen in econometrics problems which use classical approach. – pbneau Dec 4 '10 at 18:56