If X_{ti} is the observed value for individual i at time t, and X{.i} is the mean of X_{ti} over i, then
(X_{ti} - X_{.i}) depends on outcomes for time later than t. This applies of course to the dependent as well as explanatory variables.
This is why in the example in Chapter 15 (longitudinal multilevel modeling) of Snijders & Bosker ("Multilevel Analyais", 2nd ed.), we use the initial value of the changing covariate, rather than its person mean, as the level-two variable, to distinguish between within-person and between-person regression coefficients; see page 258-259.


Best,
Tom Snijders

This is awesome information; thank you!

If including the initial value of my time-varying covariate at the level 2 of analysis and not centering the time-varying covariates at the level 1 of analysis, would this be a more accurate examination of changes in my time-varying covariate? Can I still include the initial value at the level 1, or should I instead start with time 2?

Thank you!

Dear Serge and others,

Well, "more accurate" ... that kind of statements is impossible to make in general. It all depends on the design of the study, the purposes of analysis, etc.
The principle to which I referred is that for the variables affecting the dependent variable at time t, you shouldn't use any variables that, perhaps due to transformations, include information about things after time t. This means that if you use the initial ("time 1") value of explanatory variables as the level-2 variable with a fixed effect, replacing effectively the group mean (where group={level 2 unit}=individual in this case), then you can use all the variables starting from time 1 onward just as usual.
The principle seems pretty general to me. The elaboration is not a general rule, it should be considered in each given case whether this meaningful.

Best regards,

Tom Snijders

Thanks for this discussion. You've raised an interesting issue (at least to me), and I'm trying to think this through. But it's making my head spin. I see your point, but I think a lot rides on what you meant (in your previous post) by "depends".

The mean-centring approach yields the "within" estimator, which is precisely equivalent to the output of a "fixed effects" model. Your comment therefore implies a rather fundamental criticism of what is a pretty dominant methodology in some fields. Do you see what I mean?

One concern is that under the mean-centring approach the two resulting components are perfectly uncorrelated by construction, and at least the de-meaned component of X is guaranteed to be uncorrelated with any time-invariant error (i.e., with the unit random intercepts). Using your differenced-from-time-zero approach, on the other hand, the resulting components may be correlated. And it follows that they may *both* be correlated with any time-invariant random error. (In practice, in a given application, this correlation may well be small and unimportant, but I don't believe this can be tested.)

I guess, as you say, it may well depend on the specifics of a given study. I'd be interested to hear any further thoughts.

Malcolm

Dear Tom,

Thank you very much for your explanation. I see your point, but in addition to the concerns raised by Malcolm I also wonder one more thing: Wouldn’t this same argument also be an argument against estimating a random intercept at the person level (for the dependent variable) in a longitudinal design? Because estimating a random intercept leaves for level 1 the within-person deviations from the mean, i.e. more or less the same thing as you would get when mean-centering?

Just to add my 2 cents - my view of mean centering (and using mean variables generally) is that when measuring the mean you are trying to measure the effect of latent variables for which the mean acts as a proxy, rather than the effect of the mean per se.  So, to take the original example, the mean stress of an individual is really a proxy for the many individual characteristics (personality traits, etc) that are related to stress.  In measuring this proxy variable you want as much information as possible, and therefore using future information is no bad thing since that information is relevant to the time invariant characteristics of the individual.

So for me, it is less saying the future influences the past (which it probably doesn't), than that the future is relevant to the individual's time-invariant characteristics (which it probably is).  Therefore, if you are interested in deviations from an individuals 'norm', you would get potentially 'better' estimates by mean centering than centering on t=1 (given that t=1 may have been exceptional and not representative of an individuals t-invariant characteristics).

Hope that is helpful - would also be interested to know peoples thoughts on this.

Best wishes
Andrew Bell

No, because my argument applies only to observed variables. The random intercept is a latent variable.

Best regards,

Tom Snijders

Dear Tom,

Interesting, in that case perhaps the best option is to use Muthén’s Latent Covariate Model, in which a random intercept is estimated for the independent variable?

Lüdtke, O., Marsh, H. W., Robitzsch, A., Trautwein, U., Asparouhov, T., & Muthén, B. (2008). The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual studies. Psychological Methods, 13(3), 203–229. doi:10.1037/a0012869, 10.1037/a0012869.supp (Supplemental)

Muthén, B., & Asparouhov, T. (2011). Beyond multilevel regression modeling: Multilevel analysis in a general latent variable framework. In J. J. Hox & J. K. Roberts (Eds.), Handbook for advanced multilevel analysis. (pp. 15–40). New York, NY US: Routledge/Taylor & Francis Group. Retrieved from http://search.ebscohost.com/logi ... amp;site=ehost-live