# Can I test hypothesis based on estimated parameters?

I estimated the mean and variance of two latent variables through two groups of data. I can't use the data to do hypothesis testing, because I am interested in the latent variable. Is there a way to test the whether the two latent variables are significantly different?

• Do you have a measure for how latent they are? Apr 19 '14 at 19:03
• My data follows Negative Binomial distribution, Negative Binomial is a mixture of Gamma and Poisson. The underlying Gamma distribution is what I am interested in, which is also what I meant by latent variable
– Q_Li
Apr 19 '14 at 19:06
• @Q_Li the latent variables or the manifest variables are negbin? Dec 1 '16 at 19:41

The difficulty of your quest seems to lie in the uncertainty of prior distribution of the latent parameters. For instance, in paired T-test for Gaussian distributions, the prior distribution of $\mu$ is irrelevant, since the Gaussian family is translation invariant. In the case of Gamma, the mean matters a lot. Perhaps the best way is to estimate the gamma parameters for one of the two samples as you did in your linked question, and compute p-value of the other sample under this distribution.

• Could you please explain why Gamma is not translation invariant.
– Q_Li
Apr 19 '14 at 23:49
• Let $G(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ be a Gaussian density, then $G(x-\mu)$ is also a Gaussian density. The Gamma density does not have this property because it's always supported on $[0,\infty)$ exactly. Apr 20 '14 at 1:06

If you've estimated your latent variables through something like confirmatory factor analysis, item response theory, or some other kind of structural equation model, you should be able to estimate factor scores for individuals. Factor scores are usually interval data. These can be plugged into most hypothesis tests without any particularly unusual problems AFAIK. McArdle (2009) also describes some ways to do tests of latent mean differences within structural equation models, though he presents it in a longitudinal context.

Reference
McArdle, J. J. (2009). Latent variable modeling of differences and changes with longitudinal data. Annual Review of Psychology, 60, 577-605.

• Than you for you response. I am not sure if I explained what I did clearly. My data follows Negative Binomial distribution, Negative Binomial is a mixture of Gamma and Poisson. The underlying Gamma distribution is what I am interested in, which is also what I meant by latent variable. My another post explained how I got the estimate of mean and variance of the Gamma distribution. math.stackexchange.com/questions/752964/…
– Q_Li
Apr 19 '14 at 20:14