[R-sig-eco] Unbalanced data and random effects

[Chris Howden]

Thanks Krzysztof,

Your explanation makes a lot of sense.

Chris Howden B.Sc. (Hons) GStat.
Founding Partner
Evidence Based Strategic Development, IP Commercialisation and Innovation,
Data Analysis, Modelling and Training
(mobile) 0410 689 945
(skype) chris.howden
chris at trickysolutions.com.au




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-----Original Message-----
From: Krzysztof Sakrejda [mailto:krzysztof.sakrejda at gmail.com]
Sent: Thursday, 17 October 2013 11:57 AM
To: Chris Howden
Cc: r-sig-ecology at r-project.org
Subject: Re: [R-sig-eco] Unbalanced data and random effects

On Wed, Oct 16, 2013 at 6:20 PM, Chris Howden
<chris at trickysolutions.com.au> wrote:
> Hi Krysztof,
>
> Did you have a specific section of Zuur et als book in mind? I've
> pulled it off my shelf and tried looking up shrinkage, unbalanced
> design, design, etc in the index but couldn't find anything relevant.
> I'm sure it's in there, but it’s a rather large book to read in 1 go!!

I was thinking "Mixed Effects Models and Extensions in Ecology with R",
but now that I search through Zuur in Google books there appears to be no
mention of either "partial pooling" or "shrinkage" (I don't have the book
on hand).  It's not mentioned in the index either...  I recommended  Zuur
because I know a lot of ecologists use it and shrinkage is such a basic
and useful topic I expected it to be covered.

It's page 477 in Gelman and Hill.  Since I stuck my foot in it by
recommending Zuur without checking: the basic idea is that if you have a
data set with unbalanced group sizes and you just call everything one
group, you could get an estimated group mean, MU.  If you use fixed
effects and you estimate one mean per group(mu_1, mu_2, ..., mu_k), and
the means for the small groups will be poorly estimated (large standard
errors).  If you use a random effects model, you estimate one mean per
group but you also constrain the group means (mu*_1, mu*_2, ..., mu*_k) to
come from a normal distribution (with an estimated mean, MU*, and
variance) which has two
effects important for interpretation: 1) groups with fewer observations
will mostly be represented by the overall mean (mu*_1 is closer to MU*
than mu_1 is to MU, and the effect is more extreme for groups with small
sample size); and 2) this effect is even more pronounced in groups with
large deviations from MU*.

You can get a feel for how much this matters by simulating/fitting some
data similar to your data in R (Kery's "Introduction to WinBUGS for
ecologists"
does a lot of this
kind simulation).  The terms used to describe these effects are
"shrinkage" and "partial pooling", since complete pooling is what you get
when you disregard the divisions.  You can also calculate how much pooling
is being done directly (Gelman and Hill, pg. 477):

mu*_j = w_j x MU* + (1-w_j) * mean(observations in group j)

w_j = 1- (estimated variance of random effect) / (estimated variance of
random effect + within-group variance/group size)

Where w_j tells you how much that groups estimate is pooled towards the
mean.

That's the short and sloppy version, but the discussion in Gelman is good,
sorry for the confusion, maybe somebody else knows for sure where/if Zuur
discusses this?

Krzysztof

>
> Chris Howden B.Sc. (Hons) GStat.
> Founding Partner
> Evidence Based Strategic Development, IP Commercialisation and
> Innovation, Data Analysis, Modelling and Training
> (mobile) 0410 689 945
> (skype) chris.howden
> chris at trickysolutions.com.au
>
>
>
>
> Disclaimer: The information in this email and any attachments to it
> are confidential and may contain legally privileged information. If
> you are not the named or intended recipient, please delete this
> communication and contact us immediately. Please note you are not
> authorised to copy, use or disclose this communication or any
attachments without our consent.
> Although this email has been checked by anti-virus software, there is
> a risk that email messages may be corrupted or infected by viruses or
> other interferences. No responsibility is accepted for such
> interference. Unless expressly stated, the views of the writer are not
those of the company.
> Tricky Solutions always does our best to provide accurate forecasts
> and analyses based on the data supplied, however it is possible that
> some important predictors were not included in the data sent to us.
> Information provided by us should not be solely relied upon when
> making decisions and clients should use their own judgement.
>
>
> -----Original Message-----
> From: r-sig-ecology-bounces at r-project.org
> [mailto:r-sig-ecology-bounces at r-project.org] On Behalf Of Krzysztof
> Sakrejda
> Sent: Thursday, 17 October 2013 12:29 AM
> Cc: r-sig-ecology at r-project.org
> Subject: Re: [R-sig-eco] Unbalanced data and random effects
>
> On Wed, Oct 16, 2013 at 6:41 AM,  <v_coudrain at voila.fr> wrote:
>> Dear all,
>>
>> I performed a census of insects at different sites and measured there
>> size. I would like to know if size is related to an environmental
>> factor. I modelled the size as a fonction of the factor with site as
>> a
> random variable to account for within-site variability. However I have
> strong unbalanced data with some sites having only two individuals and
> others up to 100. Is having site as a random factor sufficient to deal
> with this strong data unbalance?
>
> I'm not sure what you mean by "deal with", but reading about shrinkage
> in random effects models in any decent source would probably be a fine
> start for you, either here:
>
> http://www.amazon.com/Effects-Extensions-Ecology-Statistics-Biology/dp
> /038
> 7874577/ref=la_B001JRWU88_1_2/192-3027843-3405263?s=books&ie=UTF8&qid=
> 1381
> 929893&sr=1-2
>
> or here:
>
> http://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Mode
> ls/d
> p/052168689X/ref=sr_1_3?s=books&ie=UTF8&qid=1381929942&sr=1-3&keywords
> =gel
> man+bayesian
>
> The short answer is that the site effect will shrink toward the
> average site effect for sites with few individuals.
>
> Krzysztof
>
>> The residual fit of the data is quite bad, certainly because of the
>> strong difference in variance among sites.
>
>> Would anybody have some advice? Thank you!
>> ___________________________________________________________
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>
>
> --
>
> Krzysztof Sakrejda
>
> Organismic and Evolutionary Biology
> University of Massachusetts, Amherst
> 319 Morrill Science Center South
> 611 N. Pleasant Street
> Amherst, MA 01003
>
> work #: 413-325-6555
> email: sakrejda at cns.umass.edu
>
> _______________________________________________
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> R-sig-ecology at r-project.org
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--

Krzysztof Sakrejda

Organismic and Evolutionary Biology
University of Massachusetts, Amherst
319 Morrill Science Center South
611 N. Pleasant Street
Amherst, MA 01003

work #: 413-325-6555
email: sakrejda at cns.umass.edu