Dear Douglas and list,

Again, thanks for the thorough answers.

Regarding my last email: I'm mixing up my own data! Sorry for that.

1. plogis(x)

*SameSite=1* means that birds did use the SAME nest site, hence: they show
site-fidelity.

*SameSite=0* means that birds did CHANGE nest site, hence: no site-fidelity.

This means that site-fidelity of "bird" who failed breeding corresponds to a
probability of 44% and that bird who was successful corresponds to a
probability of 72% (see below).

Thus, with "plogis" of the Intercept and/or Intercept+Fixed Effect Estimate
I can calculate the probability of my observed event (R-help: plogis gives
the distribution function; being the inverse logit-function)?

That is: *plogis *of the* intercept* *estimate *is based upon *BreedSuc1=0,
SameSite=1 *and plogis of the *intercept+BreedSuc1 Estimate* is based
upon *BreedSuc1=1,
SameSite=1*?

And the "result" will always depend upon my coding (BreedSuc1=1 means
successful, SameSite=1 means site-fidelity). This is still a bit
confusing...

My question was:
What if both codings change? I.e. *BreedSuc1=0 *means successful and *
SameSite=0* means site-fidelity? Would it then still be  1-p instead of p?


2. Z-value:

I assume that z-value is the same as z-score? I've found the following link
(
http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref/spatial_statistics_toolbox/what_is_a_z_score_what_is_a_p_value.htm).


There, the z-score is defined as:

"The Z score is a test of statistical significance that helps you decide
whether or not to reject the null hypothesis".

"Z scores are measures of standard deviation. For example, if a tool returns
a Z score of +2.5 it is interpreted as "+2.5 standard deviations away from
the mean". P-values are probabilities. Both statistics are associated with
the standard normal distribution. This distribution relates standard
deviations with probabilities and allows significance and confidence to be
attached to Z scores and p-values".

Is this correct? Would make sense.

Thanks again.

Cheers, Julia






>
> #site-fidelity (SameSite=1)  of "bird"
> who was unsuccessful in breeding in the previous season (BreedSuc1=0),
> corresponds to a probability of 44%:
> plogis(-0.2135)
> [1] 0.4468268

Perhaps I misunderstood your original posting.  I thought that
SameSite=0 meant that the bird did not return to the same nest site.
That is, site fidelity corresponds to SameSite = 1.

In any case the probabilities constructed as you have done are the
probabilities for SameSite = 1.

> Further, site-fidelity (SameSite=0)  of "bird" who was successful in
> breeding in the previous season (BreedSuc1=1), corresponds to a
probability
> of 72%:
> plogis(-0.2135 + 1.1831)
> [1] 0.7250398
>
>
> Is my interpretation correct?


> So this output is always based on SameSite=0?

> (SameSite=0 means that birds did NOT change nest site, i.e. they show high
> site-fidelity). But what if SameSite=1 would mean high site-fidelity (no
> change of nests?)?

Then the 0's and 1's would be reversed for that response variable and
your probabilities would be the complement (i.e. 1-p instead of p) of
those calculated above.




2011/2/22 Douglas Bates <bates@stat.wisc.edu>

> On Mon, Feb 21, 2011 at 8:26 AM, Julia Sommerfeld
> <Julia.Sommerfeld@utas.edu.au> wrote:
> > Dear Douglas and list member,
> >
> > Thank you heaps for your answers. The interpretation of the summary
> output
> > (lmer) is becoming much clearer now. I have to admit I had a slightly
> (not
> > to say HUGE) different idea of the summary output.
> >
> > But a few questions still remain...
> >
> > I have tried out the suggestions with the following results:
> >
> > 1. I tested if "Sex" is an important factor in the model:
> >
> > fm<-lmer(SameSite~BreedSuc1+Sex+(1|Bird), family="binomial")
> > fm1<-lmer(SameSite~BreedSuc1+(1|Bird), family="binomial")
> > anova(fm1,fm)
> >
> > Data:
> > Models:
> > fm1: SameSite ~ BreedSuc1 + (1 | Bird)
> > fm: SameSite ~ BreedSuc1 + Sex + (1 | Bird)
> >     Df    AIC    BIC  logLik  Chisq Chi Df Pr(>Chisq)
> > fm1  3 75.518 81.485 -34.759
> > fm   4 77.379 85.335 -34.690 0.1387      1     0.7096
>
> So I would conclude that even though sex was recorded it turns out
> that it is not a significant predictor of the probability of site
> fidelity and I would omit that term from the model.  As Ben described,
> he would not be in favor of this approach because it verges on "data
> snooping".  One can argue either way and, in this case it wouldn't
> make much difference in the final conclusion whether or not the Sex
> term is included.
>
> > 2. Since Sex is not "important",  I compared fm1 with fm2 to test if
> > BreedSuc1 is an important factor:
> >
> > fm2<-lmer(SameSite ~ 1 + (1|Bird), family="binomial")
> > anova(fm2,fm1)
> >
> > Data:
> > Models:
> > fm2: SameSite ~ 1 + (1 | Bird)
> > fm1: SameSite ~ BreedSuc1 + (1 | Bird)
> >         Df    AIC        BIC      logLik      Chisq     Chi Df
> >  Pr(>Chisq)
> > fm2  2     77.617     81.595     -36.808
> > fm1  3     75.518      81.485     -34.759     4.0991          1
> >  0.04291 *
>
> So if I needed to quote a p-value for the BreedSuc factor this is what
> I would quote.
>
> > BUT: Ben (thanks for the input) disagreed here:
> > "I'm afraid I have to disagree with Doug here. This kind of model
> reduction,
> > while seemingly sensible (and not as abusive as large-scale, automated
> > stepwise regression), is a mild form of data snooping. Don't do it ..."
> >
> > If I don't do it, what could be an alternative? Simply rely on the output
> of
> > fm1 without looking at fm2?
>
> The alternative is to fit a model of the form
>
> fm2a <- lmer(SameSite ~ 1 + Sex + (1|Bird), family="binomial")
>
> and compare it to the original model, fm, using
>
> anova(fm2a, fm)
>
> The general idea of testing whether BreedSuc makes a significant
> contribution to predicting the probability of site fidelity is to fit
> a model with the term and then fit the model without the term and
> compare the quality of the fits.  To me the most sensible way to
> compare the quality of the fits is to consider the likelihood ratio.
> The model with the term will always do better than the one without the
> term - the question is, "Is it significantly better?".  One way to
> answer that question is to convert the likelihood ratio test (LRT)
> statistic to a probability or p-value using the result that, under the
> null hypothesis (that the term does not make a significant
> contribution) the LRT statistic has a chi-squared distribution with 1
> degree of freedom.  One can set up other criteria; for example AIC
> penalizes each parameter as 2 units on the deviance scale (negative
> twice the log-likelihood).  BIC is a bit more complicated in that the
> number of units of penalty per parameter on the deviance scale depends
> on the number of observations in the data set.
>
> I would claim that the LRT statistic is always a good way of
> evaluating the difference in the quality of fit for two models - it is
> how you convert it to a p-value that is not clear when you have small
> sample sizes.
>
> The difference between what I would advise and what Ben would advise
> regarding the LRT is what the null and alternative models are.  I
> would remove the Sex term from both.  He would retain the Sex term in
> both.  This will result in slightly different conclusions.
>
> This, by the way, emphasizes the point that a test statistic and its
> corresponding p-value is not a property of the BreedSuc term.  It
> results from comparing the quality of fits of two models - one with
> the term and one without.  When we quote t- or z-statistics, and
> p-values, in a coefficients table we are providing a summary of many
> different types of tests simultaneously.  Unfortunately the conclusion
> that is often drawn from the table is that the p-value is a property
> of the term itself, which is wrong.
>
>
> > 3.  From the output above, I conclude that BreedSuc1 has an effect on
> > SameSite:
> >
> > summary(fm1)
> >
> > Generalized linear mixed model fit by the Laplace approximation
> > Formula: SameSite ~ BreedSuc1 + (1 | Bird)
> >    AIC       BIC         logLik     deviance
> >  75.52     81.48     -34.76    69.52
> > Random effects:
> >  Groups Name            Variance     Std.Dev.
> >  Bird   (Intercept)         0.12332      0.35117
> > Number of obs: 54, groups: Bird, 46
> >
> > Fixed effects:
> >                     Estimate     Std. Error     z value     Pr(>|z|)
> > (Intercept)      -0.2135       0.3794          -0.563        0.5736
> > BreedSuc11    1.1831         0.5921           1.998        0.0457 *
> > ---
> > Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> >
> > Correlation of Fixed Effects:
> >            (Intr)
> > BreedSuc11 -0.638
> >
> >
> > Now, again the interpretation of the Fixed effects:
> >
> > BreedSuc1 has a significant effect on SameSite.
> >
> > #site-fidelity (SameSite=0)  of "bird" (bird(!), since I droppped "Sex"?)
> > who was unsuccessful in breeding in the previous season (BreedSuc1=0),
> > corresponds to a probability of 44%:
> > plogis(-0.2135)
> > [1] 0.4468268
>
> Perhaps I misunderstood your original posting.  I thought that
> SameSite=0 meant that the bird did not return to the same nest site.
> That is, site fidelity corresponds to SameSite = 1.
>
> In any case the probabilities constructed as you have done are the
> probabilities for SameSite = 1.
>
> > Further, site-fidelity (SameSite=0)  of "bird" who was successful in
> > breeding in the previous season (BreedSuc1=1), corresponds to a
> probability
> > of 72%:
> > plogis(-0.2135 + 1.1831)
> > [1] 0.7250398
> >
> >
> > Is my interpretation correct?
>
>
> > So this output is always based on SameSite=0?
>
> > (SameSite=0 means that birds did NOT change nest site, i.e. they show
> high
> > site-fidelity). But what if SameSite=1 would mean high site-fidelity (no
> > change of nests?)?
>
> Then the 0's and 1's would be reversed for that response variable and
> your probabilities would be the complement (i.e. 1-p instead of p) of
> those calculated above.
>
> > 4. if I don't drop the term "Sex":
> >
> > summary(fm)
> >
> > Generalized linear mixed model fit by the Laplace approximation
> > Formula: SameSite ~ BreedSuc1 + Sex + (1 | Bird)
> >    AIC   BIC logLik deviance
> >  77.38 85.34 -34.69    69.38
> > Random effects:
> >  Groups Name        Variance     Std.Dev.
> >  Bird   (Intercept)     0.14080       0.37524
> > Number of obs: 54, groups: Bird, 46
> >
> > Fixed effects:
> >                     Estimate         Std. Error     z value     Pr(>|z|)
> > (Intercept)     -0.3294         0.4890         -0.674     0.5006
> > BreedSuc11   1.1988          0.5957        2.012      0.0442 *
> > Sex    M             0.2215           0.5877           0.377      0.7062
> > ---
> > Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> >
> > Correlation of Fixed Effects:
> >            (Intr)             BrdS11
> > BreedSuc11     -0.536
> > SexM           -0.628      0.065
> >
> >
> > site fidelity for a female bird (i.e. SexM is 0) who was unsuccessful in
> > breeding the previous season (i.e.
> > BreedSuc1 is 0) is -0.3294, corresponding to a probability of about 42%
> >> plogis(-0.3294)
> > [1] 0.4183866
> >
> > The log-odds of site fidelity for a female bird who was successful in
> > breeding is -0.3294 + 1.1988, corresponding to a probability of about 70%
> >> plogis(-0.3294 + 1.1988)
> > [1] 0.7046208
> >
> >
> > 5. The z-value: Sorry, I still have some trouble with this value....
> >
> > In model fm (without "Sex") the z-value of BreedSuc1 corresponds to
> 1.998.
> >
> > In model fm1("Sex" included) the z-value corresponds to 2.012.
> >
> > Nearly the same value in both models. But what can someone conclude from:
> p<
> > 0.05, z=1.998 ??? Because this is what many people write in their result
> > section (I was told to do so the same....).
> >
> > "You should see a LRT test statistic close to, but not exactly the same
> as,
> > the square of the z value when you compare the models with and without
> that
> > term.
> > This is the sense in which the z-value is an approximation".
> >
> > I don't really understand how the z-value can be seen as an
> approximation?
> > Am I missing some background knowledge here?
>
> I may have been too terse in my explanations.  As mentioned above, I
> would claim that the LRT statistic is a reasonable way to compare the
> fits of two models, because it is based upon fitting the model with
> and without the term of interest.  In the case of a linear model
> without random effects it is not necessary to fit the model without
> the term just to discover what the LRT statistic would be.  You can
> tell from the model fit with the term what the maximum value for the
> likelihood of any sub-model will be.
>
> In the case of a linear mixed model or a generalized linear mixed
> model you can't decide on the basis of the one model fit what the
> likelihood for the other will be.  You can approximate but you you
> don't get an exact value.  When it took a very long time to do a model
> fit we just used the approximation.  Now that these fits can be done
> much more quickly, it makes sense to fit both with and without.
>
> The nature of the approximation is to take the parameter estimate for
> BreedSuc and divide it by its approximate standard error.  We call
> this the z-statistic because, when everything is working properly,
> this should have a distribution close to a standard normal, which we
> often write as Z.  The LRT statistic is the difference in the deviance
> of the model without and the model with the term.  To me, that is the
> quantity of interest and the fact that it should be approximately the
> square of the z-statistic is helpful in making rough decisions but I
> still want to calculate the difference in the deviance before making a
> final decision.
>
> In the summary of fm1 the z-statistic is 1.998 whereas the LRT
> statistic comparing fm2 to fm1 is 4.0991.  The square of the
> z-statistic will be close to, but not exactly the same as, the LRT
> statistic.
>
> > Again, thanks heaps for your help!!!!
>
> You're welcome.  Thanks for the question.
>
> > 2011/2/19 Douglas Bates <bates@stat.wisc.edu>
> >>
> >> Thank you for your questions and for transferring the discussion to
> >> the R-SIG-Mixed-Models mailing list, as we had discussed.  I have also
> >> copied the mailing list for a class on mixed-effects models that I am
> >> teaching.
> >>
> >> I particularly appreciate your desire to learn about the model instead
> >> of just quoting a p-value.  I often lament to my classes that
> >> statisticians have been far too successful in propagating the idea of
> >> p-values, to the extent that some researchers believe that is all that
> >> is needed to learn about an analysis.
> >>
> >> On Sat, Feb 19, 2011 at 3:05 AM, Julia Sommerfeld
> >> <Julia.Sommerfeld@utas.edu.au> wrote:
> >> > Dear Douglas and list members,
> >> >
> >> > Apologies in advance if you might consider my questions as too simple
> to
> >> > be asking the godfather of lme4 for an answer...thus, please feel free
> >> > to
> >> > ignore my email or to forward it to someone else.
> >> >
> >> > I'm a PhD student (Australia/Germany) working on tropical seabirds. As
> >> > many of my PhD-collegues, I'm having some difficulties with the
> analysis
> >> > of my data using lmer (family=binomial). While some say: What do you
> >> > care
> >> >
> >> > about all the other values as long as you've got a p-value... I do
> >> > believe
> >> > that it is essential to understand WHAT I'm doing here and WHAT all
> >> > these
> >> > numbers/values mean.
> >> >
> >> > I've read the Chapters (lme4 Book Chapters) and publications about the
> >> > use
> >> > of lmer and searched the forums - but I don't find a satisfying
> answer.
> >> > And I have the feeling that 1. the statistic lecture at my university
> >> > was
> >> > a joke (sad to say this) 2. that I need a huge
> statistical/mathematical
> >> > background to fully understand GLMMs.
> >> >
> >> >
> >> > One of the question I would like to answer is:
> >> > Does the previous breeding success influences nest site fidelity?
> >> >
> >> > I have binomial data:
> >> > SameSite=1 means birds use the same site
> >> >
> >> > SameSite=0 means birds change nest site
> >> >
> >> > BreedSuc1=1 Birds were successful in previous breeding season
> >> > BreedSuc1=0 Birds were not successful "     "        "
> >> >
> >> > Sex= male, female
> >> > Bird= Bird ID
> >> >
> >> > This is my model:
> >>
> >> > fm<-lmer(SameSite~BreedSuc1+Sex+(1|Bird), family="binomial")
> >>
> >> > where Bird is my random factor (same birds were sampled more than
> once)
> >>
> >> One thing to note is that there are 46 different birds in the 54
> >> observations.  Most birds will have just one observation so a random
> >> effect for bird may not be necessary.
> >>
> >> > summary(fm)
> >> >
> >> > Generalized linear mixed model fit by the Laplace approximation
> >> >
> >> > Formula: SameSite ~ BreedSuc1 + Sex + (1 | Bird)
> >> >   AIC   BIC logLik deviance
> >> >  77.38 85.34 -34.69    69.38
> >> > Random effects:
> >> >  Groups Name        Variance Std.Dev.
> >> >  Bird   (Intercept) 0.14080  0.37524
> >> > Number of obs: 54, groups: Bird, 46
> >> >
> >> > Fixed effects:
> >> >            Estimate Std. Error z value Pr(>|z|)
> >> > (Intercept)  -0.3294     0.4890  -0.674   0.5006
> >> > BreedSuc11    1.1988     0.5957   2.012   0.0442 *
> >> > SexM          0.2215     0.5877   0.377   0.7062
> >>
> >> this suggests that sex is not an important factor in the model.  The
> >> (Intercept) term is close to zero, relative to its standard error, but
> >> we would retain it in the model as explained below.
> >>
> >> > ---
> >> > Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> >> >
> >> > Correlation of Fixed Effects:
> >> >           (Intr) BrdS11
> >> > BreedSuc11 -0.536
> >> > SexM       -0.628  0.065
> >> >
> >> >
> >> > From this summary output I do understand that the Breeding Success has
> a
> >> > significant effect on nest-site fidelity (p<0.05).
> >>
> >> Yes, but ... this p-value should be used as a guide only.  As
> >> described below a p-value must be viewed in context.  It is not a
> >> property of the Breeding Success factor; it comes from a comparison of
> >> two models and we should bear in mind that these models are before
> >> interpreting this number.
> >>
> >> The interpretation of a p-value for a particular coefficient is that
> >> it is an approximation to the p-value we would get from comparing the
> >> model that has been fit to the mode fit without this particular
> >> coefficient.  In this case the coefficient corresponds to one of the
> >> terms in the model and I would advocate performing a likelihood ratio
> >> test comparing the two models
> >>
> >> fm <- glmer(SameSite~BreedSuc1+Sex+(1|Bird), family="binomial")
> >> fm0 <- glmer(SameSite~Sex+(1|Bird), family="binomial")  # the null
> >> hypothesis model
> >> anova(fm0, fm)
> >>
> >> Even though the function is called anova it will, in this case,
> >> perform a likelihood ratio test (LRT).  It also prints the values of
> >> AIC and BIC if you prefer to compare models according to one of those
> >> criteria but I prefer using the likelihood ratio for nested models.
> >>
> >> However, before doing that comparison you should ask yourself whether
> >> you want to compare models that have the, apparently unnecessary term
> >> for Sex in them.  The way I would approach the model building is first
> >> to reduce the model to
> >>
> >> fm1 <- lmer(SameSite~BreedSuc1+(1|Bird), family="binomial")
> >>
> >> You could then compare
> >>
> >> anova(fm1, fm)
> >>
> >> which I presume will give a large p-value for the LRT, so we prefer
> >> the simpler model, fm1.  After that, I would compare
> >>
> >> fm2 <- lmer(SameSite ~ 1 + (1|Bird), family="binomial")
> >> anova(fm2, fm1)
> >>
> >> to see if the BreedSuc1 factor is an important predictor in its own
> right.
> >>
> >> Note that we don't drop the implicit "(Intercept)" term, even though
> >> it has a high p-value in the coefficient table.  The reason is that
> >> the interpretation of the (Intercept) coefficient depends on the
> >> coding of BreedSuc1.
> >>
> >> In model fm, the log-odds of site fidelity for a female bird (i.e.
> >> SexM is 0) who was unsuccessful in breeding the previous season (i.e.
> >> BreedSuc1 is 0) is -0.3294, corresponding to a probability of about
> >> 42%
> >>
> >> > plogis(-0.3294)
> >> [1] 0.4183866
> >>
> >> The log-odds of site fidelity for a female bird who was successful in
> >> breeding is -0.3294 + 1.1988, corresponding to a probability of about
> >> 70%
> >>
> >> > plogis(-0.3294 + 1.1988)
> >> [1] 0.7046208
> >>
> >> If you had reversed the meaning of BreedSuc to BreedFail, where 0
> >> indicates no failure at breeding and 1 indicates failure, then the
> >> coefficient would change sign (i.e. the coefficient for BreedFail
> >> would be -1.1988) and the intercept would change to
> >>
> >> > -0.3294 + 1.1988
> >> [1] 0.8694
> >>
> >> because the reference level would now be a female bird who was
> >> successful in breeding.
> >>
> >> Because the interpretation of the intercept depends upon the coding of
> >> other factors, we retain it in the model whenever other terms are
> >> retained.
> >>
> >>
> >>
> >> > But what else can I conclude from this model?
> >> >
> >> > Questions:
> >> >
> >> > 1.Random effects: What does the Random Effect table - the Variance,
> Std.
> >> > Dev. and Intercept - tells me: Is there a random effect that my model
> >> > has
> >> > to account for?
> >>
> >> First I would remove the apparently unnecessary Sex term then,
> >> ideally, I would check by comparing the fit of the reduced model to
> >> that of a GLM without the random effect for Bird.  Unfortunately, I
> >> don't think the definition of deviance for a glm fit is compatible
> >> with that for a model fit by glmer.  This is something we will need to
> >> fix.  For the time being I would instead examine the "caterpillar
> >> plot" obtained with
> >>
> >> dotplot(ranef(fm1, postVar=TRUE))
> >>
> >> which represent the 95% prediction intervals for each of the birds.
> >> If these all overlap zero comfortably I would conclude that the random
> >> effect is not needed an fit a glm without a random effect for bird.
> >> > Random effects:
> >> >  Groups Name        Variance Std.Dev.
> >> >  Bird   (Intercept) 0.14080  0.37524
> >> > Number of obs: 54, groups: Bird, 46
> >>
> >> That estimated standard deviation is fairly large. We would expect a
> >> range of contributions on the log-odds scale of about +/- 2 sd which,
> >> at this point of the logistic curve corresponds to considerable
> >> variability in predicted probabilities for birds with the same
> >> characteristics.
> >>
> >> > 2. Fixed Effects: Again the Intercept? Not sure if I understand the
> >> > meaning of it (sorry, explanation in Chapter I also doesn't help much)
> >>
> >> Actually in this model it is a bit different from the models described
> >> in chapter 1.  I hope the explanation above makes sense.  Think of it
> >> as the log-odds of site fidelity for a bird in the "reference group"
> >> where reference group means that all the other variables are a the
> >> zero level.
> >>
> >> > Fixed effects:
> >> >            Estimate Std. Error z value Pr(>|z|)
> >> > (Intercept)  -0.3294     0.4890  -0.674   0.5006
> >> > BreedSuc11    1.1988     0.5957   2.012   0.0442 *
> >> >
> >> > SexM          0.2215     0.5877   0.377   0.7062
> >> >
> >> > 3. Meaning of the z-value? Why shall I mention it in te result
> section?
> >>
> >> I would regard the z-value as an approximation.  The quantity of
> >> interest is the likelihood ratio test statistic which has a
> >> chi-squared distribution under the null hypothesis (i.e. the term can
> >> be deleted from the model without getting a significantly worse fit).
> >> It happens that this would be a chi-squared distribution with 1 degree
> >> of freedom, which corresponds to the square of a standard normal
> >> distribution.  You should see a LRT test statistic close to, but not
> >> exactly the same as, the square of the z value when you compare the
> >> models with and without that term.  This is the sense in which the
> >> z-value is an approximation.  To me the LRT statistic is more reliable
> >> because it is based upon actually refitting the model.
> >>
> >> > 4. Estimate and Std. Error of the fixed effects? How can I tell from
> >> > these
> >> > values WHAT kind of effect (positiv, negativ?) these parameter have on
> >> > nest-site fidelity? Do birds that were successful during the previous
> >> > breeding success show a higher nest-site fidelity? Remember, I have
> >> > binomial data...
> >>
> >> That is described above.  If you want the estimate of the site
> >> fidelity for bird with certain characteristics you evaluate the
> >> corresponding combination of coefficients and apply plogis to the
> >> result.
> >> > I would highly appreciate your feedback and/or suggestions of
> >> >
> >> > papers/chapters I could read for a better understanding of the output.
> >> >
> >> > Best regards,
> >> >
> >> >
> >> > Julia
> >>
> >> I hope this helps.
> >
> >
> >
>



-- 
Julia Sommerfeld - PhD Candidate
Institute for Marine and Antarctic Studies
University of Tasmania
Private Bag 129, Hobart
TAS 7001

Phone: +61 458 247 348
Email: julia.somma@gmx.de
Julia.Sommerfeld@utas.edu.au

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