[R-sig-ME] MCMCglmm with multinomial models
jessica comley
je@@|ecom|ey44 @end|ng |rom gm@||@com
Wed Jul 27 02:20:36 CEST 2022
Dear Jarrod,
Thank you so much for your help, I greatly appreciate it!
All the best,
Jess
On Wed, Jul 27, 2022 at 3:32 AM Jarrod Hadfield <j.hadfield using ed.ac.uk> wrote:
> Hi Jess,
>
> Section should definitely not be left out, but I would imagine it is going
> to be very difficult to separate culling, predator and Section effects - I
> would expect the credible intervals to be large.
>
> As mentioned in my previous post you can test for an effect of culling by
> fitting the model
>
> ~trait+trait:culling+trait:predator
>
> And then fitting a Wald test to the three terms with 'culling' in. The
> effect of predator can be tested similarly but with the 3 terms with
> 'predator' in.
>
> Since your covariates do not vary within Section it will be much easier to
> aggregate the counts at the Section level (i.e have a data frame with 14
> rows and 1 column for each activity with the number observed for each
> activity) and fit family="multinomial". You can then get rid of the random
> formula as the Section effects are now effectively the residuals. Given the
> lack of replication I would advise using the idv formula that I suggested
> previously and hope the model isn't too misspecified:
>
> prior=list(R=list(V=1, nu=0.002))
>
> m1<-MCMCglmm(cbind(dawn, diurnal, dusk,
> nocturnal)~trait+trait:culling+trait:predator,
> rcov=~idv(units+trait:units), prior=prior, ...)
>
> Note this models is identical to the original model, it's just
> parameterised in a more efficient way.
>
> Cheers,
>
> Jarrod
>
>
>
>
> On 25 Jul 2022, at 03:52, jessica comley <jessiecomley44 using gmail.com> wrote:
>
> This email was sent to you by someone outside the University.
> You should only click on links or attachments if you are certain that the
> email is genuine and the content is safe.
> Dear Jarrod and Walid,
>
> Thank you for your replies, it is greatly appreciated.
>
> The predator and culling factors do not vary within sites. As shown in the
> example data in one of my previous emails, Bucklands only has culling as
> lethal and predator as low, whereas Colchester only has predator as high
> and culling as none.
>
> We are trying to submit a paper on black-backed jackal and caracal
> activity in the presence of different culling practices and
> predator presence. The reviewers want us to try a GLMM approach to
> determine whether culling or predators have an effect on black-backed
> jackal or caracal activity.
>
> Therefore, in your opinion how could be go about this given our data?
> Would it be advisable to leave out the random effect of Section?
>
> All the best,
> Jess
>
> On Wed, Jul 20, 2022 at 3:06 PM Jarrod Hadfield <j.hadfield using ed.ac.uk>
> wrote:
>
>> Hi Jess
>>
>> In multinomial models the linear model is set up as a (logit) difference
>> in probability between an outcome and some base-line outcome. Often, as
>> here, the base-line outcome is arbitrary, and so the idh structure is a
>> little odd. For example, if A is the base line category, idh assumes
>> COV(B-A, C-A) = 0 which therefore assumes
>> COV(B,C)+VAR(A) =COV(A,B)+COV(C,A). It's not clear why this would be the
>> case. Perhaps a more reasonable, but less parameter rich, option would be
>> to have:
>>
>> ~idv(Section+trait:Section)
>>
>> which parameterises the Section covariance matrix by a single parameter
>> (rather than 6). The term idv(Section+trait:Section) fits a 3x3 covariance
>> matrix of the form v*(I+J) where v is the estimated variance. This assumes
>> i) Sections are repeatable in outcome, but knowing that a Section has an
>> increased 'preference' for A doesn’t tell you whether it also has an
>> increased preference for one of the other categories and ii) the
>> repeatability for each outcome within sites is the same (on the latent
>> scale).
>>
>> To test groups of effects (in your case the 3 culling:trait effects), I
>> usually use a Wald test and the posterior covariances (see here
>> https://stat.ethz.ch/pipermail/r-sig-mixed-models/2017q3/025930.html).
>> It's far from correct and so Walid's suggestions may be better, but
>> small-scale simulations suggests it has good frequentist properties.
>>
>> To add predator presence you can just add a predator:trait effect into
>> the linear model. If the culling and predator factors do not vary within
>> sites then you probably don't have enough information to reliably estimate
>> these effects.
>>
>> Cheers,
>>
>> Jarrod
>>
>>
>>
>>
>>
>>
>> > On 19 Jul 2022, at 18:17, Walid Mawass <walidmawass10 using gmail.com> wrote:
>> >
>> > This email was sent to you by someone outside the University.
>> > You should only click on links or attachments if you are certain that
>> the email is genuine and the content is safe.
>> >
>> > Hey Jess,
>> >
>> > 1) Yes that is correct
>> >
>> > 2) To my knowledge there is a rule of thumb, where you set the nitt (#
>> of
>> > iterations) to a large number that includes the burnin amount, then you
>> > choose your thinning interval (sampling of the chain). For example,
>> this is
>> > what I would use: nitt= 150000, burnin=50000, thin=100. This will give
>> you
>> > a decent burnin and a final sample of 1000 saved iterations. Note
>> however
>> > that this does not have to increase the effective sample size for
>> certain
>> > variables, but it might do the trick.
>> >
>> > 3) hmm...I think one way to do it is to make predictions using the above
>> > model and interpret the patterns you see for each relationship you are
>> > interested in. Another way to compare effect size would be to use
>> bayesian
>> > posterior indices. I suggest these two papers by Makowski et al. (2019a
>> &
>> > b) that present both interesting posterior indices to use with Bayesian
>> > statistical analysis and an associated R package that does the job of
>> > computing these indices, *bayestestR*.
>> >
>> > Good luck
>> > --
>> > Walid Mawass
>> > Ph.D. candidate in Evolutionary Biology - UQTR
>> > *Currently* Postdoctoral Research Associate
>> > Masel Lab - University of Arizona
>> >
>> >
>> > On Sun, Jul 17, 2022 at 11:32 PM jessica comley <
>> jessiecomley44 using gmail.com>
>> > wrote:
>> >
>> >> Hi Walid,
>> >>
>> >> Thank you for your reply, I greatly appreciate it. I have a few more
>> >> questions and if you could help that would be great.
>> >>
>> >> I tested for correlation between activities and the 14 Sections and the
>> >> correlation comes out as low. Therefore I have changed my code to use
>> idh()
>> >> instead of us as suggested:
>> >>
>> >> test1c.5b <- MCMCglmm(activity ~ -1 + at.level(culling,1):trait +
>> >> at.level(culling, 2):trait, random=~idh(trait):Section, rcov =
>> >> ~idh(trait):units, data = caracal, family = "categorical", prior =
>> prior,
>> >> burnin=5000, nitt=80000)
>> >>
>> >> 1) Is this correct?
>> >>
>> >> 2) Increasing the number of interactions increases the effective sample
>> >> size, therefore is there a general rule of thumb as to how large your
>> >> effective sample size should be?
>> >>
>> >> 3) I understand how to use and interpret the results of HPDinterval
>> (i.e.
>> >> if intervals do not overlap 0 then relationship is strong), but how am
>> I
>> >> able to test the relationship between all four activities and fixed
>> effects
>> >> and not just have the three categories (i.e. diurnal, dusk, nocturnal)
>> >> compared to the base category (dawn)? For example, I am also
>> interested in
>> >> whether there is a significant/strong relationship between activities
>> of
>> >> caracal at dusk with culling(Lethal)/no culling(none) compared to
>> >> activities of caracal at diurnal with culling(Lethal)/no culling(none).
>> >>
>> >> Below is an example of our dataset:
>> >> Camera Section CameraID Animal predator culling activity
>> >> 1a Bucklands Bucklands1a Caracal low Lethal diurnal
>> >> 1a Bucklands Bucklands1a Caracal low Lethal dawn
>> >> 2a Bucklands Bucklands2a Caracal low Lethal dusk
>> >> 2a Bucklands Bucklands2a Caracal low Lethal diurnal
>> >> 3a Bucklands Bucklands3a Caracal low Lethal dawn
>> >> Cam 1 Colchester ColchesterCam 1 Caracal high none diurnal
>> >> Cam 1 Colchester ColchesterCam 1 Caracal high none diurnal
>> >> Cam 1 Colchester ColchesterCam 1 Caracal high none diurnal
>> >> Cam 1 Colchester ColchesterCam 1 Caracal high none diurnal
>> >> Cam 2 Colchester ColchesterCam 2 Caracal high none diurnal
>> >> Cam 2 Colchester ColchesterCam 2 Caracal high none diurnal
>> >> Cam 3 Colchester ColchesterCam 3 Caracal high none diurnal
>> >> Cam 3 Colchester ColchesterCam 3 Caracal high none diurnal
>> >> Cam 3 Colchester ColchesterCam 3 Caracal high none diurnal
>> >> Cam 4 Colchester ColchesterCam 4 Caracal high none diurnal
>> >> Cam 4 Colchester ColchesterCam 4 Caracal high none diurnal
>> >> Cam 4 Colchester ColchesterCam 4 Caracal high none nocturnal
>> >> 1a Connaught Connaught1a Caracal low Lethal nocturnal
>> >> 1a Connaught Connaught1a Caracal low Lethal nocturnal
>> >> 1d Connaught Connaught1d Caracal low Lethal diurnal
>> >> 3B Connaught Connaught3B Caracal low Lethal diurnal
>> >> 3B Connaught Connaught3B Caracal low Lethal diurnal
>> >> 4a Connaught Connaught4a Caracal low Lethal nocturnal
>> >> 4a Connaught Connaught4a Caracal low Lethal nocturnal
>> >> 4b Connaught Connaught4b Caracal low Lethal diurnal
>> >> 6a Connaught Connaught6a Caracal low Lethal nocturnal
>> >> 6b Connaught Connaught6b Caracal low Lethal diurnal
>> >> 7a Connaught Connaught7a Caracal low Lethal nocturnal
>> >> 9a Connaught Connaught9a Caracal low Lethal nocturnal
>> >> 9d Connaught Connaught9d Caracal low Lethal nocturnal
>> >> 9d Connaught Connaught9d Caracal low Lethal dusk
>> >> 7d Diepdam Diepdam7d Caracal absent Lethal dusk
>> >> 8d Diepdam Diepdam8d Caracal absent Lethal diurnal
>> >> 9c Diepdam Diepdam9c Caracal absent Lethal nocturnal
>> >>
>> >> All the best,
>> >> Jess
>> >>
>> >>
>> >> On Fri, Jul 15, 2022 at 11:37 PM Walid Mawass <walidmawass10 using gmail.com
>> >
>> >> wrote:
>> >>
>> >>> Hello,
>> >>>
>> >>> I don't think I can specifically help you with some of your inquiries.
>> >>> However, I do want to comment on a few things that might need some
>> >>> attention.
>> >>>
>> >>> First, MCMCglmm is based on a Bayesian implementation and does not
>> >>> compute p-values to compare. What you need to compare are the
>> posterior
>> >>> distributions of your effect sizes. This can be done visually using
>> the
>> >>> base plot function in R. Or by comparing the HPD intervals and the
>> mode (or
>> >>> mean) of the posterior distributions.
>> >>>
>> >>> Second, I have no idea what your data structure looks like (which
>> makes
>> >>> it hard to interpret model results), but the effective sample size
>> (from
>> >>> the 5500 saved iterations sample) for your random variable Section is
>> very
>> >>> low (the same applies for your fixed effects). You should consider
>> this
>> >>> issue and look again at your assumption of correlation between
>> >>> activities for the 14 sections you have in your dataset. If you do not
>> >>> expect among activity correlations then you can use the idh() function
>> >>> instead of us().
>> >>>
>> >>> Hopefully this helps and in hope that people on this list with more
>> >>> knowledge of these models will help out.
>> >>>
>> >>> Best,
>> >>> --
>> >>> Walid Mawass
>> >>> Ph.D. candidate in Evolutionary Biology - UQTR
>> >>> *Currently* Postdoctoral Research Associate
>> >>> Masel Lab - University of Arizona
>> >>>
>> >>>
>> >>> On Fri, Jul 15, 2022 at 8:49 AM jessica comley <
>> jessiecomley44 using gmail.com>
>> >>> wrote:
>> >>>
>> >>>> Dear all,
>> >>>>
>> >>>> I am hoping that someone will be able to help me with conducting
>> MCMCglmm
>> >>>> multinomial models.
>> >>>>
>> >>>> The data I am working with is for black-backed jackal (bbj) and
>> carcal.
>> >>>> For
>> >>>> each species we have a multinomial response variable called activity
>> >>>> which
>> >>>> has four categories (dawn, diurnal, dusk, nocturnal). We have two
>> >>>> categorical fixed effects which are 1) culling (none, lethal) and 2)
>> >>>> predator presence (absent, high, low). We also have a categorical
>> >>>> variable
>> >>>> called Section (made up of 14 different reserves/ farms where the
>> >>>> activity
>> >>>> of caracal and bbj were recorded). There are 273 observations for
>> caracal
>> >>>> and 4399 for bbj. We are wanting to test the effects of culling and
>> >>>> predators on caracal and bbj activity separately.
>> >>>>
>> >>>> I have been working through Jarrod Hadfields course notes,
>> particularly
>> >>>> with regards to Chapter 5.2. The chi-square analyses reveal that the
>> >>>> frequencies of culling and predators differ as do activities.
>> >>>>
>> >>>> I have managed to work out the specific probabilities for the culling
>> >>>> none
>> >>>> vs culling lethal for each activity (dawn, diurnal, dusk, nocturnal)
>> for
>> >>>> caracal, but I'm confused as to how to determine p-values to
>> determine
>> >>>> which activities culling none vs culling lethal are affecting?
>> >>>>
>> >>>> Myy code and outcomes are pasted below with questions stated in bold.
>> >>>>
>> >>>> caracal2 <- read.csv("caracal_new.csv", header=T)
>> >>>> caracal <- as.data.frame(unclass(caracal2), stringsAsFactors = TRUE)
>> >>>>
>> >>>> #Chi-squared tests
>> >>>> Ctable1 <- table(caracal$activity, caracal$culling)
>> >>>> chisq.test(rowSums(Ctable1)) #strongly suggests activities differ
>> >>>> chisq.test(Ctable1)#strongly suggests culling category differs
>> >>>>
>> >>>> Ctable2 <- table(caracal$activity, caracal$predator)
>> >>>> chisq.test(rowSums(Ctable2))#strongly suggests activities differ
>> >>>> chisq.test(Ctable2)#strongly suggests predator category differs
>> >>>>
>> >>>> prior = list(R = list(fix=1, V=(1/k) * (I + J)), G = list(G1=list(V =
>> >>>> diag(k-1), nu=1)))
>> >>>> test1c.5 <- MCMCglmm(activity ~ -1 + at.level(culling,1):trait +
>> >>>> at.level(culling, 2):trait, random=~us(trait):Section, rcov =
>> >>>> ~us(trait):units, data = caracal, family = "categorical", prior =
>> prior,
>> >>>> burnin=5000, nitt=60000)
>> >>>> *##I'm not sure how to add the three predator levels to this model
>> or if
>> >>>> it
>> >>>> would be appropriate?*
>> >>>>
>> >>>>
>> >>>> k <- length(levels(caracal$activity))
>> >>>> I <- diag(k-1)
>> >>>> J <- matrix(rep(1, (k-1)^2), c(k-1, k-1))
>> >>>> IJ <- (1/k) *(diag(k-1) + matrix(1,k-1, k-1))
>> >>>>
>> >>>> contrasts(caracal$activity)
>> >>>>
>> >>>> #culling lethal
>> >>>> Delta <- cbind(c(0,1,0,0), c(0,0,1,0), c(0,0,0,1))
>> >>>> c2 <- (16 * sqrt(3)/(15 * pi))^2
>> >>>> D <- ginv(Delta %*% t(Delta)) %*% Delta
>> >>>> Int <- t(apply(test1c.5$Sol[,1:3],1, function(x) + D %*% (x/sqrt(1 +
>> c2 *
>> >>>> diag(IJ)))))
>> >>>> summary(mcmc(exp(Int)/rowSums(exp(Int))))
>> >>>>
>> >>>> prop.table(Ctable1[,1])
>> >>>>
>> >>>> #culling none
>> >>>> Delta <- cbind(c(0,1,0,0), c(0,0,1,0), c(0,0,0,1))
>> >>>> c2 <- (16 * sqrt(3)/(15 * pi))^2
>> >>>> D <- ginv(Delta %*% t(Delta)) %*% Delta
>> >>>> Int <- t(apply(test1c.5$Sol[,4:6],1, function(x) + D %*% (x/sqrt(1 +
>> c2 *
>> >>>> diag(IJ)))))
>> >>>> summary(mcmc(exp(Int)/rowSums(exp(Int))))
>> >>>>
>> >>>> prop.table((Ctable1[,2]))
>> >>>>
>> >>>> HPDinterval(test1c.5$Sol)
>> >>>>
>> >>>> #model summary
>> >>>>> summary(test1c.5)
>> >>>>
>> >>>> Iterations = 5001:59991
>> >>>> Thinning interval = 10
>> >>>> Sample size = 5500
>> >>>>
>> >>>> DIC: 699.7014
>> >>>>
>> >>>> G-structure: ~us(trait):Section
>> >>>>
>> >>>> post.mean
>> l-95%
>> >>>> CI
>> >>>> u-95% CI eff.samp
>> >>>> traitactivity.diurnal:traitactivity.diurnal.Section 1.8124
>> >>>> 0.09784
>> >>>> 5.665 77.01
>> >>>> traitactivity.dusk:traitactivity.diurnal.Section 0.8450
>> >>>> -0.83585
>> >>>> 3.856 64.17
>> >>>> traitactivity.nocturnal:traitactivity.diurnal.Section 1.3621
>> >>>> -1.19129
>> >>>> 6.157 58.48
>> >>>> traitactivity.diurnal:traitactivity.dusk.Section 0.8450
>> >>>> -0.83585
>> >>>> 3.856 64.17
>> >>>> traitactivity.dusk:traitactivity.dusk.Section 1.2034
>> >>>> 0.07090
>> >>>> 3.681 102.16
>> >>>> traitactivity.nocturnal:traitactivity.dusk.Section 0.7505
>> >>>> -1.77113
>> >>>> 4.524 43.53
>> >>>> traitactivity.diurnal:traitactivity.nocturnal.Section 1.3621
>> >>>> -1.19129
>> >>>> 6.157 58.48
>> >>>> traitactivity.dusk:traitactivity.nocturnal.Section 0.7505
>> >>>> -1.77113
>> >>>> 4.524 43.53
>> >>>> traitactivity.nocturnal:traitactivity.nocturnal.Section 2.7148
>> >>>> 0.09401
>> >>>> 8.397 76.59
>> >>>>
>> >>>> R-structure: ~us(trait):units
>> >>>>
>> >>>> post.mean l-95%
>> CI
>> >>>> u-95% CI eff.samp
>> >>>> traitactivity.diurnal:traitactivity.diurnal.units 0.50
>> 0.50
>> >>>> 0.50 0
>> >>>> traitactivity.dusk:traitactivity.diurnal.units 0.25
>> 0.25
>> >>>> 0.25 0
>> >>>> traitactivity.nocturnal:traitactivity.diurnal.units 0.25
>> 0.25
>> >>>> 0.25 0
>> >>>> traitactivity.diurnal:traitactivity.dusk.units 0.25
>> 0.25
>> >>>> 0.25 0
>> >>>> traitactivity.dusk:traitactivity.dusk.units 0.50
>> 0.50
>> >>>> 0.50 0
>> >>>> traitactivity.nocturnal:traitactivity.dusk.units 0.25
>> 0.25
>> >>>> 0.25 0
>> >>>> traitactivity.diurnal:traitactivity.nocturnal.units 0.25
>> 0.25
>> >>>> 0.25 0
>> >>>> traitactivity.dusk:traitactivity.nocturnal.units 0.25
>> 0.25
>> >>>> 0.25 0
>> >>>> traitactivity.nocturnal:traitactivity.nocturnal.units 0.50
>> 0.50
>> >>>> 0.50 0
>> >>>>
>> >>>> Location effects: activity ~ -1 + at.level(culling, 1):trait +
>> >>>> at.level(culling, 2):trait
>> >>>>
>> >>>> post.mean l-95% CI u-95%
>> CI
>> >>>> eff.samp pMCMC
>> >>>> at.level(culling, 1):traitactivity.diurnal 1.2306 -0.0533
>> 2.6793
>> >>>> 145.29 0.0418 *
>> >>>> at.level(culling, 1):traitactivity.dusk 0.6605 -0.6006
>> 2.0761
>> >>>> 92.91 0.2840
>> >>>> at.level(culling, 1):traitactivity.nocturnal 1.6090 0.0914
>> 3.1356
>> >>>> 151.02 0.0265 *
>> >>>> traitactivity.diurnal:at.level(culling, 2) 1.2664 -0.1552
>> 2.7750
>> >>>> 226.40 0.0604 .
>> >>>> traitactivity.dusk:at.level(culling, 2) 0.3533 -0.9898
>> 1.5218
>> >>>> 148.44 0.5447
>> >>>> traitactivity.nocturnal:at.level(culling, 2) 1.0447 -0.6405
>> 2.8354
>> >>>> 346.40 0.1618
>> >>>> ---
>> >>>> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>> >>>>
>> >>>> *##So for the model summary I get that lethal culling at activity
>> diurnal
>> >>>> is significantly different from lethal culling at dawn (its the base
>> >>>> reference), but I'm also interested in whether lethal culling at
>> activity
>> >>>> diurnal is different from lethal culling at dusk for example. Is this
>> >>>> possible? *
>> >>>>
>> >>>> #outcomes culling lethal
>> >>>>> summary(mcmc(exp(Int)/rowSums(exp(Int))))
>> >>>>
>> >>>> Iterations = 1:5500
>> >>>> Thinning interval = 1
>> >>>> Number of chains = 1
>> >>>> Sample size per chain = 5500
>> >>>>
>> >>>> 1. Empirical mean and standard deviation for each variable,
>> >>>> plus standard error of the mean:
>> >>>>
>> >>>> Mean SD Naive SE Time-series SE
>> >>>> [1,] 0.1253 0.05565 0.0007504 0.002484
>> >>>> [2,] 0.3748 0.10497 0.0014155 0.003204
>> >>>> [3,] 0.1757 0.06640 0.0008954 0.002515
>> >>>> [4,] 0.3242 0.11939 0.0016099 0.003514
>> >>>>
>> >>>> 2. Quantiles for each variable:
>> >>>>
>> >>>> 2.5% 25% 50% 75% 97.5%
>> >>>> var1 0.03641 0.08695 0.1198 0.1554 0.2553
>> >>>> var2 0.17298 0.30580 0.3704 0.4431 0.5896
>> >>>> var3 0.06166 0.12913 0.1705 0.2161 0.3215
>> >>>> var4 0.12610 0.23999 0.3090 0.3901 0.6045
>> >>>>
>> >>>>> prop.table(Ctable1[,1])
>> >>>> dawn diurnal dusk nocturnal
>> >>>> 0.1250000 0.2812500 0.1770833 0.4166667
>> >>>>
>> >>>>
>> >>>> #outcomes culling none
>> >>>>> summary(mcmc(exp(Int)/rowSums(exp(Int))))
>> >>>>
>> >>>> Iterations = 1:5500
>> >>>> Thinning interval = 1
>> >>>> Number of chains = 1
>> >>>> Sample size per chain = 5500
>> >>>>
>> >>>> 1. Empirical mean and standard deviation for each variable,
>> >>>> plus standard error of the mean:
>> >>>>
>> >>>> Mean SD Naive SE Time-series SE
>> >>>> [1,] 0.1288 0.06141 0.0008280 0.002787
>> >>>> [2,] 0.3804 0.10406 0.0014032 0.002662
>> >>>> [3,] 0.1710 0.06844 0.0009228 0.002592
>> >>>> [4,] 0.3198 0.11812 0.0015928 0.002956
>> >>>>
>> >>>> 2. Quantiles for each variable:
>> >>>>
>> >>>> 2.5% 25% 50% 75% 97.5%
>> >>>> var1 0.02891 0.08896 0.1220 0.1594 0.2685
>> >>>> var2 0.18007 0.31094 0.3783 0.4474 0.5965
>> >>>> var3 0.05840 0.12425 0.1634 0.2083 0.3250
>> >>>> var4 0.12430 0.23921 0.3077 0.3862 0.5964
>> >>>>
>> >>>>> prop.table((Ctable1[,2]))
>> >>>> dawn diurnal dusk nocturnal
>> >>>> 0.1306818 0.4375000 0.1875000 0.2443182
>> >>>>
>> >>>> Any help or guidance will be greatly appreciated.
>> >>>>
>> >>>> All the best,
>> >>>> Jess
>> >>>>
>> >>>> --
>> >>>> Jessica Comley (PhD)
>> >>>> Research Scientist
>> >>>>
>> >>>> [[alternative HTML version deleted]]
>> >>>>
>> >>>> _______________________________________________
>> >>>> R-sig-mixed-models using r-project.org mailing list
>> >>>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>> >>>>
>> >>>
>> >>
>> >> --
>> >> Jessica Comley (PhD)
>> >> Research Scientist
>> >>
>> >>
>> >
>> > [[alternative HTML version deleted]]
>> >
>> > _______________________________________________
>> > R-sig-mixed-models using r-project.org mailing list
>> > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>
>> The University of Edinburgh is a charitable body, registered in Scotland,
>> with registration number SC005336. Is e buidheann carthannais a th’ ann an
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>>
>
>
> --
> Jessica Comley (PhD)
> Research Scientist
>
>
>
--
Dr Jessica Comley
Lecturer: Environmental and Life Sciences
Faculty of Science
Universiti Brunei Darussalam
Email: jessica.comley using ubd.edu.bn
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