Is there a statistical model for modelling variables that are measured in varying amounts and in different time points per individual?

Question

I have been trying to model a dataset of variables where each individual is measured a different number of times, and at a different point in time. Most of my variables are counts, but some are not (the target variable is a count). Furthermore there is a bunch of zeros over the count variable.

The closest I was able to find was a Zero-Inflated Poisson model, but as far as I've seen this model assumes that I measured each individual the same amount of times and over the same period. Neither of the two assumptions is true in my case.

I saw a formula (without name), where a regression can be performed such that:

$Y_i$$_t$ = $S_1$$_i$*$a_i$$_j$ + $\epsilon_i$$_t$ + $I_0$$_i$

$I_0$$_i$ = $\alpha_0$ + $\sigma_0$$_i$

$S_1$$_i$ = $\alpha_1$ + $\sigma_1$$_i$

where $i$ indicates individual and $t$ time, around the internet, but the slide simplified $a$ by removing the individual. Furthermore I am not certain if this time would allow different times for each individual.

Since time here is measured in seconds and the amount of measures for an individual can vary from 5 times to over 100 times across years, plotting many constant variables would most likely to a explosion of points as well.

Can statistics address this? I have been having the same issue with data mining and machine learning algorithms as well due to the time constraint. The regression is multivariate and predictors are not necessarily counts.

I appreciate any pointers, books, papers on any substantive field.

Answer 1

The model that you wrote is (or appears to be) a mixed-effects model. In your specific case, with a count DV, it would usually be called a generalized linear mixed model (GLMM). These mixed models can handle the type of data that you mentioned. There are many variants of these mixed models, although many (not all) of the terms floating around are just different names for the same model... these names include:

• Mixed-effects model
• (Generalized) linear mixed model
• Random-coefficient model
• Hierarchical linear model
• Multilevel model
• Growth curve model

There are a variety of high-quality sources for learning about mixed models. Knowing a little about your needs and background would help to narrow down suggestions. Some of my personal favorites are:

Books

• Pinheiro, J. C., & Bates, D. M. (2000). Mixed effects models in S and S-PLUS. Springer Verlag.
• Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.
• Snijders, T. A., & Bosker, R. J. (2011). Multilevel analysis: An introduction to basic and advanced multilevel modeling. Sage Publications Limited.
• Zuur, A. F., et al. (2009). Mixed effects models and extensions in ecology with R. Springer.

Web pages

• http://glmm.wikidot.com/faq