Use LCA memberships as independent variables in a linear regression model I performed LCA using the "depmixS4" library in R and got a three cluster solution for my data; as such for each record in my data, I have three LCA memberships (probabilities) for the three clusters which add up to 1.
I was trying to fit a linear regression model using these three memberships (X1, X2 and X3, say) to predict an outcome measure (Y, say) for each of the records. 
When I fit a regression model, I get NA as the coefficient for X3. 
This is happening because X1, X2 and X3 are linearly related (X1, X2 and X3 are probabilities and they add upto 1, always).
Is there another way to see how each cluster affects the outcome variable? 
 A: The answer is latent class regression. More generally, let's call $X$ the latent class. Recall that the latent class is an un-ordered categorical variable that you can't directly observe. You inferred its existence from whatever indicators you specified.
Latent class regression models $P(Y = y | X)$ or $\bar{Y} | X$. In the links below, you'll see me make some reference to latent class with distal outcomes in addition to latent class regression. Latent class with distal outcomes asks $P(X = x | Y)$.
In the links, do note that my notation differs from yours. I adopt the standard notation where the $Y_i$s are indicators of the latent class and $K$ is the latent class itself.
Why covariates are used in Latent class analysis (LCA)
How can you implement latent class analysis with distal outcomes in R?
A: The logic of LCA is that the underlying latent variable is a nominal categorical variable, like race ("white, black, asian, other") or region ("north, south, east, west"). So just treat the latent classes like you would any other nominal variable in a regression: include two dummy variables for "X1 vs anything else" and "X2 vs anything else" and leave X3 as the reference category.
However, you should know that if you (or R) assigned observations to classes using "modal assignment" (each observation gets assigned to the class they have the highest probability of belonging to) then you WILL end up underestimating the size of the relationship between the classes and your dependent variable. Check out this paper for a more detailed discussion of the issue and some approaches for addressing the bias.
