[R-sig-ME] Perfectly correlated random effects (when they shouldn't be)

[Jake Westfall]

Hi Steve,

I think the issue is that estimating 3 variances and 3 covariances for regions is quite ambitious given that there are only 6 regions. I think it's not surprising that the model has a hard time getting good estimates of those parameters.

Jake

> Date: Tue, 14 Jul 2015 20:53:01 -0400
> From: steven.v.miller at gmail.com
> To: r-sig-mixed-models at r-project.org
> Subject: [R-sig-ME] Perfectly correlated random effects (when they shouldn't	be)
> 
> Hi all,
> 
> I'm a long-time reader and wanted to raise a question I've seen asked here
> before about correlated random effects. Past answers I have encountered on
> this listserv explain that perfectly correlated random effects suggest
> model overfitting and variances of random effects that are effectively zero
> and can be omitted for a simpler model. In my case, I don't think that's
> what is happening here, though I could well be fitting a poor model in
> glmer.
> 
> I'll describe the nature of the data first. I'm modeling individual-level
> survey data for countries across multiple waves and am estimating the
> region of the globe as a random effect as well. I have three random effects
> (country, country-wave, and region). In the region random effect, I am
> allowing country-wave-level predictors to have varying slopes. My inquiry
> is whether some country-wave-level contextual indicator can have an overall
> effect (as a fixed effect), the effect of which can vary by region. In
> other words: is the effect of some country-level indicator (e.g.
> unemployment) in a given year different for countries in Western Europe
> than for countries in Africa even if, on average, there is a positive or
> negative association at the individual-level? These country-wave-level
> predictors that I allow to vary by region are the ones reporting perfect
> correlation and I'm unsure how to interpret that (or if I'm estimating the
> model correctly).
> 
> I should also add that I have individual-level predictors as well as
> country-wave-level predictors, though it's the latter that concerns me.
> Further, every non-binary indicator in the model is standardized by two
> standard deviations.
> 
> For those interested, I have a reproducible (if rather large) example
> below. Dropbox link to the data is here:
> https://www.dropbox.com/s/t29jfwm98tsdr71/correlated-random-effects.csv?dl=0
> 
> In this reproducible example, y is the outcome variable and x1 and x2 are
> two country-wave-level predictors I allow to vary by region. Both x1 and x2
> are interval-level predictors that I standardized to have a mean of zero
> and a standard deviation of .5 (per Gelman's (2008) recommendation).
> 
> I estimate the following model.
> 
> summary(M1 <- glmer(y ~ x1 + x2 + (1 | country) + (1 | country:wave) + (1 +
> x1 + x2 | region), data=subset(Data), family=binomial(link="logit")))
> 
> The results are theoretically intuitive. I think they make sense. However,
> I get a report of perfect correlation for the varying slopes of the region
> random effect.
> 
> Random effects:
>  Groups       Name        Variance Std.Dev. Corr
>  country:wave (Intercept) 0.15915  0.3989
>  country      (Intercept) 0.32945  0.5740
>  region       (Intercept) 0.01646  0.1283
>               x1          0.02366  0.1538    1.00
>               x2          0.13994  0.3741   -1.00 -1.00
> Number of obs: 212570, groups:  country:wave, 143; country, 82; region, 6
> 
> What should I make of this and am I estimating this model wrong? For what
> it's worth, the dotplot of the region random effect (with conditional
> variance) makes sense and is theoretically intuitive, given my data. (
> http://i.imgur.com/mrnaJ77.png)
> 
> Any help would be greatly appreciated.
> 
> Best regards,
> Steve
> 
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