# Why covariates are used in Latent class analysis (LCA)

I am trying to learn this relatively new statistical technique LCA. In literature i saw that for some LCA models, covariates are used.

I am having a difficulty to understand why additional covariates are using for this model. Is it due to the violation of independence assumption ?

Based on the things i understood, the predictors of the latent class model is categorical and the response which is unobserved categorical variable. So then why additional predictors called "covariates" are used in this model? Why cant we treated this "covariates" as regular predictors ?

Thank you

## What are indicators of the latent class?

First, let's distinguish between indicators and predictors with a heuristic example. I know for a fact that many latent class analysis models have been applied to depressive symptoms. The DSM-V criteria are based on these symptoms:

1. Depressed mood
2. Diminished interest or pleasure in usual activities
3. Appetite disturbance, or weight change
4. Sleep disturbance
5. Psychomotor agitation or retardation
6. Fatigue
7. Feelings of worthlessness
8. Diminished concentration
9. Suicide ideation

Per the DSM-V, you need one of the first two, and at least 5 total in the preceding two weeks.

Now, if you thought that maybe some people experience atypical patterns of depression, you could fit a latent class analysis to the above items and examine the response patterns among the latent classes. For example, maybe some people don't endorse either of the first two items but endorse the rest. (Note: this is a made up example.)

Here, the DSM-V questions (aka items) are the indicators of the latent class. We assume that a latent (unordered) categorical variable causes responses to the items. We don't observe the latent class directly, because we can't, and we can only infer it from responses to the items.

## What are covariates?

Now, imagine you think that maybe older adults were more likely to fall into the atypical latent classes. Maybe it's because they aren't willing to say they're depressed or unhappy or that they feel worthless, but they're more able to say that they experience somatic symptoms.

This is the sort of scenario where you'd conduct a latent class regression - your latent class is a multinomial variable, and it's conceptually identical to a multinomial regression on an observed variable. (It's not the same in mechanical terms, because we don't observe the latent class directly.)

It wouldn't make sense to treat age as an indicator of the latent class in this example. The latent class doesn't cause your age - it causes responses to the questions. You're saying that you think older adults are more likely to be in the atypical class, so age is a predictor of the latent class.

## Some math

For a latent class model without covariates, this is the math that describes the probability of being in each latent class. The $$\gamma$$s denote the multinomial intercepts. C and k denote the latent classes, however many of them are present.

$$P(C = k) = \frac{exp(\gamma_k)}{\sum_{j=1}^K exp(\gamma_j)}$$

Had you fit a latent class regression with one covariate, the multinomial equation now looks like this:

$$P(c_i = k|x_i) = \frac{exp(\gamma_{0k} + \gamma_{1k}x_i)}{\sum_{j=1}^K exp(\gamma_{0j} + \gamma_{1j}x_i)}$$

Here, i indexes people. You've still got one set of $$\gamma_0$$s for each class, but now you add a $$\gamma_1$$ for each class to show the effect of the covariate in question. Needless to say, this approach generalizes to as many predictors as you want or the model can accommodate.

## Misc issues

You said that (emphasis mine):

Based on the things i understood, the predictors of the latent class model is categorical and the response which is unobserved categorical variable.

I assume that you meant that the indicators of the latent class are categorical. I hope I've illustrated the distinction between indicators of class membership, and predictors of which latent class you're in.

Note that actually, the indicators of the latent class do not have to be categorical. They can be continuous - usually, people will call the model latent profile analysis in this case. They can even be mixed continuous and categorical. Typically, the continuous indicators are modeled as Gaussian, but they do not have to be modeled thus; Stata's gsem command and the R package flexmix both can model indicators with any distribution supported by standardd GLMs (e.g. Poisson, negative binomial, Gaussian with a log link). There have got to be other R packages that can do this, I just don't know of them.

This may go without saying, but the predictors of the latent class don't have to be categorical either.

One side note: contrast the LCA approach with item response theory. In IRT, we have a bunch of items, and we assume that a latent continuous variable causes responses to the items.

Another side note: we may model the latent class as unordered categorical, but that doesn't mean it is necessarily unordered. If the latent variable were really continuous and unidimensional (as if it were IRT data), and you fit a latent class model, then chances are your latent classes would look something like low depression, medium depression, and high depression. I'm aware that a number of researchers have fit LCA models to depression questionnaire data, and a lot of them seem to find classes that are along those lines.