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I am running a multi-level model looking at factors that explain attainment.

There are pupil- and school-level predictors, and the school the pupil attends is modelled as a random effect.

I have run a Durbin-Watson test which suggests there is serial correlation in the residuals from this regression.

I'm wondering whether this is an issue, given there is not a natural order to the observations?

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    $\begingroup$ "there is not a natural order to the observations" Does it mean if you change the order of the observations in the dataset, you will get the different results on Durbin-Watson test? $\endgroup$
    – user158565
    Commented Oct 9, 2018 at 17:32
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    $\begingroup$ I agree with the (implicit) comment of a_statistician, that is, the concept of "auto-correlation" and the use of the Durbin-Watson tests implies that there is a natural ordering of the observations. If there is not, then I don't think that this test is that meaningful. Do you have from each pupil one measurement of multiple? $\endgroup$
    – Dimitris Rizopoulos
    Commented Oct 9, 2018 at 19:19
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    $\begingroup$ The description of data is not clear enough. I feel good choice is to fit a mixed model (linear or generalized linear depending on response variable) with school specific intercepts as random effect. $\endgroup$
    – user158565
    Commented Oct 10, 2018 at 1:54
  • $\begingroup$ The model is a linear mixed model, with a continuous outcome variable, pupil-level characteristics and school-level characteristics as fixed effects, and then a school-specific intercept as a random effect. Each pupil has one measurement, so ordering by pupil ID doesn't make that much sense. Ordering by school ID would be one 'natural ordering', but given there are school specific intercepts as random effects it feels like this shouldn't matter. Yes changing the order gives different DW test results, some where we reject the null and some where we fail to reject the null. $\endgroup$
    – Mark Francis
    Commented Oct 10, 2018 at 8:33

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If data are autocorrelated and there is individual heterogeneity, then Hamel, Yoccoz, and Gaillard (2012) show that parameter estimates are biased if there is no correction. I suggest constraining the error terms to follow an AR(1) process (Kwok etc al, 2008). You could alternativly introduce an autoregressive independent variable, but this is a biased parameter estimate as well. However, this latter method allows you to use a dynamic multilevel model (Hamaker and Wichers, 2017).

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    $\begingroup$ Uh, just for clarity, the references are: Hamel, S., Yoccoz, N.G. and Gaillard, J.-M, "Statistical evaluation of parameters estimating autocorrelation and individual heterogeneity in longitudinal studies" (2012) ...and Hamaker & Wichers's "No Time Like the Present: Discovering the Hidden Dynamics in Intensive Longitudinal Data" (2017)? $\endgroup$
    – Alex Nelson
    Commented Aug 2, 2019 at 15:11

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