Time-varying Predictors in Multilevel Modeling

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It seems to me that time-varying predictors are ususally included in the level-1 model (individual level or repeated measurement/occation). Do you know whether the multilevel modeling software can support the modeling of the effect of time-varying predictor at the group level (level-2 or above)?

I want to build a three-level model for a longitudinal dataset. The level-1 model has Time and time-varying predictors for repeated measurement, the level-2 model captures the effect of time-invarying predictor of individuals, and the level-3 model includes the effect of predictors at the group level (such as census tract level). The predictors at the group level are time-varying. If HLM can not help build such a multilevel model, what is the alternative solution to my modeling idea?

Thank you for your reply in advance.

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关键词：Predictors Multilevel predictor Modeling varying individual software captures included repeated

Multilevel models usually assume that the effect at level 1 and the effect at level 2 of a level-1 variable are equal.  If you want to allow for different level 1 and level 2 (and, potentially, level 3) effects then you can include the group mean of the level-1 variable as you would any other higher level variable.  I don't use HLM but I assume this is relatively doable.

See Snijders and Bosker 2012 p56-60, and also this working paper:
http://polmeth.wustl.edu/mediaDetail.php?docId=1324

Hope that helps!

Thank you for your quick response. The references you suggest are very useful for accounting for the between effect of level-1 time-varying predictor/covariate by adding the higher-level mean into the model of the higher level.

My question is more about how to deal with the effect of level-2 or higher-level time-varying variables.  Usually, the level-2 or higher-level variable is time-invariant (e.g. students within schools, the number of enrollment in school is regarded as constant over time, and it is easy to incorporate its effect in the level-2 model). However, those variables could vary over time. For example, given a hierarchical structure like households within neighborhoods, the characteristics of neighborhoods are dynamic and can change over time. It seems not appropriate to add those variables in the level-2 or 3 (neighborhood level); but if we add them to the level-1 (repeated measures for households), we also need to include dummy variables for the neighborhood entity.

In my mind, the combined model would be formulated as follows.
Ytij = a0 + a1*X'tij + a2*Z'tj + Utj + Etij

t  (occasions) - represent the level-1; i (individuals) for level-2, j (neighborhood) for level-3;
X'  is level-2 time-varying predictor,
Z' is level-3 time-varying predictor;

Is there any multilevel model specification that can incorporate the trajectory of individuals as well as the trajectory of place-based groups such as neighborhoods or census tracts? Could you provide a reference explaining or examplifying such a model?

Assuming that your time points are in the level-one portion of the model you can include the time-varying covariates there regardless of their association with a higher level characteristic.

Yes, my time points are in the level-one portion of the model. But both of lower and higher level characteristics change over time. I want to add the time-varying covariates that represent higher level characteristic into the multilevel model. Can I achieve it? If yes, how?

If I understand correctly, I think what you need is a four-level model, where occasions (t) are cross-classified in individuals (i) and in area-waves (w), and area-waves are in turn nested in areas (j). Individuals may also be nested in areas, if they never move from one area to another.

Per Andy's suggestion, you could mean-center both the individual- and area-level variables that are time-varying, yielding mean components indexed i and j, and mean-centered (or de-meaned) components t and w.

I'm not aware of any applications that have used such an approach, though there may be something out there. I know that R's lme4 and MCMCglmm packages, and MLwiN can handle four-level models of this kind -- whether other packages can I don't know. However, no matter the software, this is getting to be quite a complicated model to interpret, and will make heavy demands on the data.

Aside from the papers Andy suggested, you can also refer http://seis.bris.ac.uk/~ggmhf/MHF.MLM-longit.2013.pdf.

This addresses the analysis of repeated cross-sectional survey data, where repeated surveys are made of the same areas (e.g., countries) over time, but the individuals change across waves. That's a three-level situation. In your case, as mentioned above, you have an additional level, because individual people are observed multiple times.

Dear member,
That the time can be a factor in a another level of the the estimation (second or higher) is not a technical problem. In fact, any application such HLM or SPSS or Stata, Mlwin and others) can support any variation in any levels. The problem is the interpretation of estimates. In any longitudinal structure of data, the effect of time is usually treated in the first level and the other levels will be define to explain the unobserved variation that is due to heterogeneity of indivividuals with respect to other nested consideration. I advise you to traet the time variability in the first level of estimation, particulary if you are in the economic and social field.
Hope that helps