[R] Repeatability and lme

Andrew Robinson A.Robinson at ms.unimelb.edu.au
Wed May 17 09:59:23 CEST 2006

Dear Roger,

I think that there is a problem with your strategy.  The problem is
that, because you have included random slopes within your model, the
quantity of variance explained by the random effects varies as a
function of Age.  Therefore it is not possible to pin down a single
repeatability, as I understand it.   Had you included only random
intercepts then you'd be on safe ground.

See, for example, Partitioning Variation in Multilevel Models.   By:
Goldstein, Harvey; Browne, William; Rasbash, Jon. Understanding
Statistics, 2002, Vol. 1 Issue 4, p223, 9p; (AN 8655390)



On Wed, May 17, 2006 at 09:14:20AM +0200, Roger Sch?rch wrote:
> Dear Spencer Graves
> First I would like to thank you very much for answering to my mail. Then I
> would like to clarify some points, so that I would eventually find a
> solution to my problem.
> ---
> SG:  I have not done a serious literature search of "repeatability", but I
> would not assume that it is defined in exactly the same way by all sources
> that use that term.
> ---
> Well, as stated in the introduction, I am following Lessells & Boag (1987),
> who define (in words): "REPEATABILITY is a measure used in quantitative
> genetics to describe the proportion of variance in a character that occurs
> among rather than within individuals."
> So, I would like to know, how consistent my fish behave, whether the
> variance is rather between the individuals that I have observed or within
> the individuals.
> I could use an anova, but I'd rather stick to mixed effects models, as it
> seems to be common sense to use that with longitudinal data (though it seems
> not to be widely used in zoological/behavioural research ...).
> ---
> SG:  What "slope" are you describing here?  Consider the following
> modification of one of the standard 'lme' examples:
> [...]
>  >      (fm1.1 <- lme(distance ~ age,
> +         random=~age|Subject, data = Orthodont)) # random is ~ age
> [...]
> >      (fm1.0 <- lme(distance ~ age,
> +         random=~1|Subject, data = Orthodont)) # random is ~ age
> The first model estimates a "slope" for "age" as a fixed effect AND a 
> variation in that for each Subject.  The second assumes this slope is 
> constant between Subjects, and only the "(Intercept)" varies between 
> subjects.
> ---
> I allow a different slope for every subject, so my model is similar to the
> first model. Additionally I have a fixed effect for "sex":
> myLme <- lme(fixed = explorationScore ~ Sex*I(Sequence - 1), data = myData,
> random = ~ I(Sequence - 1) | Fish, method = "ML")
> Sequence = 1st to 6th measurement (each measurement 30 d apart; do I have to
> specify that it is not a continuous variable??)
> Fish = Subject
> ---
> SG:  I would encourage you to first think carefully about the problem(s) 
> you want to solve.  What would people want to do with the results of 
> your study?  After you've answered that question, if some definition of 
> "repeatability" (carefully defined with an appropriate citation) seems 
> to provide some insight, I'd try to explain why it does, then give the 
> quantitative answer with my interpretation and with appropriate 
> citations to show that my logic here is not completely original.  If 
> however, "repeatability" did NOT seem to support my main message, then I 
> would likely ignore it.
> ---
> So, here are my questions for that particular model that I am investigating:
> 1. Do the fish change their behaviour during ontogeny?
> 2. Do the sexes differ in their behaviour?
> 3. Do my fish behave consistently (when one accounts for the change over
> time (see 1. point))?
> Let us consider you example again, but add "Sex" as a fixed effect, so that
> it is more similar to my analysis:
> > fm1.1 <- lme(distance ~ age*Sex,random=~age|Subject, data = Orthodont)
> And then have a look at the variance components:
> > VarCorr(fm1.1)
> Subject = pdLogChol(age) 
>             Variance   StdDev    Corr  
> (Intercept) 5.78842347 2.4059143 (Intr)
> age         0.03255509 0.1804303 -0.668
> Residual    1.71611214 1.3100046
> We see that an individual's true intercept is deviating from the mean
> intercept considerably, but the differences in rate of change seem to be
> rather small. Then there is some residual variance that is not accounted for
> by fitting a change trajectory for every individual, a scatter around an
> individual's true change trajectory.
> Now, if I am interested in the ratio (among_variance / (within_variance +
> among_variance)), how is that computed? For our example here I would
> suggest:
> > (as.numeric(VarCorr(fm1.1)[1])+as.numeric(VarCorr(fm1.1)[2])) /
> (as.numeric(VarCorr(fm1.1)[1])+as.numeric(VarCorr(fm1.1)[2])+as.numeric(VarC
> orr(fm1.1)[3]))
> [1] 0.772311
> To my understanding, this would be the repeatability of the character
> "distance", and would therefore result in a statement like this: most of the
> variance in distance is found between subjects, rather than within (r =
> 0.772311).
> The only question I would have liked to pose to the r-help list, is, whether
> I compute the among_variance from the R output correctly or whether I am
> lacking something that is not directly available from the standard output R
> produces.
> Repeatability is widely used in zoological literature, but it is usually
> assumed that there are no time effects. That is why I would like to use lme
> AND provide the readers with a familiar number ... Furthermore, it provides
> a guess for the maximal heritability that I can expect for that particular
> trait, which might prove very useful for further studies, and it would be
> comparable with other behaviours, e.g. in a table.
> This has become a somewhat lengthy e-mail, so here are my apologies. But I
> am still not sure whether I could make myself understood. Thanks anyway for
> your kind help!
> Yours
> Roger
> P.S.: There is a study that has done similar things (M?ller & Schrader,
> 2005, Behaviour 142, 1289--1306), but it rather confused me, and the
> corresponding author does not seem willing to correspond ... But perhaps it
> helps in understanding my problem.
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Andrew Robinson  
Department of Mathematics and Statistics            Tel: +61-3-8344-9763
University of Melbourne, VIC 3010 Australia         Fax: +61-3-8344-4599
Email: a.robinson at ms.unimelb.edu.au         http://www.ms.unimelb.edu.au

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