Solved – How is ARMA/ARIMA related to mixed effects modeling

arimamixed modelmultilevel-analysistime series

In panel data analysis, I have used multi-level models with random/mixed effects to deal with auto-correlation issues (i.e., observations are clustered within individuals over time) with other parameters added to adjust for some specification of time and shocks of interest. ARMA/ARIMA seem designed to address similar issues.

The resources I've found online discuss either times series (ARMA/ARIMA) or mixed effect models but beyond being build on regression, I don't understand the relationship between the two. Might one want to use ARMA/ARIMA from within a multilevel model? Is there a sense in which the two are equivalent or redundant?

Answers or pointers to resources that discuss this would be great.

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I think the simplest way to look at it is to note that ARMA and similar models are designed to do different things than multi-level models, and use different data.

Time series analysis usually has long time series (possibly of hundreds or even thousands of time points) and the primary goal is to look at how a single variable changes over time. There are sophisticated methods to deal with many problems - not just autocorrelation, but seasonality and other periodic changes and so on.

Multilevel models are extensions from regression. They usually have relatively few time points (although they can have many) and the primary goal is to examine the relationship between a dependent variable and several independent variables. These models are not as good at dealing with complex relationships between a variable and time, partly because they usually have fewer time points (it's hard to look at seasonality if you don't have multiple data for each season).

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Solved – Including (and interpreting!) random intercepts and/or slopes in linear mixed models

I think you're specifying a great deal of random effects which are better modeled as fixed effects since you'll directly estimate the odds ratio for their association with outcome. For instance, the question of whether partisanship modifies the relationship of discount and product is simply done as a test of interaction:

m1 <- glmer(chose ~ partisanship + iipm*discount + product + (1|id) + (1|time), data=long1, 
    family="binomial")
m2 <- glmer(chose ~ partisanship*(iipm*discount + product) + (1|id) + (1|time), data=long1, 
    family="binomial")

lrtest(m2, m1)

Under the null hypothesis, the interaction effects have a log odds ratio of 0. Note we have a LOT of such effects: partisianship:iipm, partisianship:discount, partisianship:product and the three way effect partisianship:iipm:discount. Another important modification to the null model is that I included partisanship as a fixed effect in the null model. This is because when testing for effect modification, you need to ensure that the interaction parameters have the interpretation as a difference in odds ratios in the main effect for a unit difference in the modifier. I think "Regression Methods in Biostatistics" from Vittinghoff, ch. 7 addresses this.

While the meaning of the p-value in the interaction analysis is difficult to interpret, I guarantee you it's much easier than the test of random effect you included above. Worse, the variance component test doesn't address the question you stated.

Lastly,

If there's no improvement in fit (p>.05), I (very tentatively) interpret this as support for my hypothesis that demographics and ideology don't matter.

This is not the correct interpretation of a $p$-value. Basic NHST says there was insufficient evidence to suggest otherwise. Given the complexity of this model, I'd be very dubious of the interpretation you provided.

Solved – Time varying predictors at higher aggregation levels in multilevel survival analysis

I think I found a solution. I read two book chapters about multilevel event history models (Courgeau, 2007; Goldstein, 2011), which discuss similar cases and suggest using a three-level structure such as time (level-1) nested within households (level-2), which are in turn nested within municipalities (level-3). Goldstein (2011, p. 221) explicitly states for this structure that “The exploratory variables can be defined at any level. They may also vary over time, allowing so-called time varying covariates.”

So here is a quick explanation why I think that such a three-level model is able to correctly incorporate time-varying predictors at the municipality-level (level-3), such as the environmental variable “Env1”. Because Env1 varies across time, the model automatically treats it as a level-1 variable. It does not know that at each time step (e.g., year 1990), the values for Env1 are the same for all households located in a particular municipality. However, I don’t think that this biases the standard errors for the Env1 variable because I have household random effects (level-2) included in the model, which estimate a separate intercept for each household. Moreover, I also include an additional variance component at level-2 that allows the slope of Env1 to vary randomly across households. In this way the effect of Env1 is uniquely computed for each household.

References:

Courgeau, D. (2007). Multilevel synthesis: From the group to the individual. Dordrecht, The Netherlands: Springer.

Goldstein, H. (2011). Multilevel statistical models (4th ed.). Chichester, U.K.: John Wiley & Sons.

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Solved – Including (and interpreting!) random intercepts and/or slopes in linear mixed models

Solved – Time varying predictors at higher aggregation levels in multilevel survival analysis

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