Mixed model interpretation with interaction

If you have multiple (repeated) measurements from both bears and feeding
site, you may even have a nested or cross classified design. In such case,
bears might be nested within feeding sites, and both bear and feeding site
might be modelled as random intercept. Here?s a very short gist showing the
difference between nested and cross classified design and how to write this
in lme4-notation:




http://htmlpreview.github.io/?https://github.com/strengejacke/mixed-models-snippets/blob/master/nested_fully-crossed_cross-classified_models.html



Best

Daniel



*Von:* Ren? <bimonosom at gmail.com>
*Gesendet:* Sonntag, 9. Juni 2019 17:29
*An:* d.luedecke at uke.de
*Cc:* Patricia Graf <patricia.graf03 at gmail.com>; r-sig-mixed-models <
r-sig-mixed-models at r-project.org>
*Betreff:* Re: [R-sig-ME] Mixed model interpretation with interaction



Ps:



I also agree with Daniel to take care of repeated measurements of the same
bears coming to the sites in both years.

However, the main problem I guess, will be that not every bear comes back
in the second year. This means, having random slopes for bears that were
observed only once, will bias the effect estimate (i.e. the random slopes
for year will not be separable from the fixed effect of year).

A solution to this, however would be, to use an extra variable (lets call
it 'repeat') that codes, whether a bear has there in both years (=1) or not
(=0; numeric coding - not factored). Then you the following should work:



model<-(y~site*year+(0+repeat*year| bearID))

Which will estimate random slopes for year for all bears that were there
at least twice, but not for others, where the term before the | becomes 0
(and nothing happens)).



Best, Ren?



Pps: You can tell your colleagues that:

the Model-intercept is the only direct mean that the model estimates
directly (i.e. the reference cell) and all other deviations (including
other means) are linear combinations from that intercept (for any factor)
...  And ... of course, the parameters still can be interpreted this way
(as illustrated above) ... but you need to know some details how to do so
:)) (you can impress them now...)



The easiest way to let a function reconstruct the model outputs is using
emmeans()

e.g.

emmeans(model1, ~Site) should give the marginal estimates of the - site
main effect - (site 1 and site 2 means) on the log scale

and

emmeans(model1, ~Site, type = "response") will give the estimates on the
actual response (probability) scale. I find this often very helpful (also
for plotting).







Am So., 9. Juni 2019 um 16:57 Uhr schrieb Ren? <bimonosom at gmail.com>:

Hi,



I don't know if this adds anything new but the most direct answers that
come into my mind would be.

1) It seems you use dummy coding, and this defines the interpretation of
the estimated coefficients, which would be different from (often preferred
because more easy to interpret)  effect / or contrast coding (dummy coding
has some 'fitting' advantages which are mainly discussed with respected to
centered vs. non-centered likelihood (or least-square-mean) estimation
processes, which might be insightful for you to look up in the internet);
but any coding design you use will eventually simply try to estimate
cell-means (in your case on a log scale), and you need to check how to get
these cell means out of your coefficients (via back-transformation). One
way of doing this is by using marginal predictions, as Daniel points out.



2) For another (technical) illustration: a test-design matrix as yours
with (e.g.) 2 feeding sites and 2 years, then it would be a 2(site 1 vs.
site 2) by 2(year 1 vs year 2) independent measures design; or 2 x 2 for
short, which could be simply expressed by 4 probabilities or by using means
on a log scale, one mean for each of the design-cells, which would be the
"centered" variant of estimation; but usually dummy coding implies a
non-centered (but mathematically equivalent  - standard) coding:

If the model is:

y = site+year (ignoring random effects now), then

cellmean(site1:year1) = Model_Intercept

cellmean(site1:year2) = Model_Intercept + year2

cellmean(site2:year1) = Model_Intercept + site2

cellmean(site2:year2) = Model_Intercept + site2 + year2



mean(site1) = (2*Model_intercept + year2)/2

mean(site2) = ( 2(Model_intercept + site2)+year2))/2

and so on...

(Where intercept in most estimation methods is by default is defined in
reference to the first level of the first predictor in the equation; thus
site1 (+year1, which is 0 in this type of coding); but the reference point
can be changed manually)



If the model is:

y=site+year+site:year, then

cellmean(site1:year1) = Model_Intercept

cellmean(site1:year2) = Model_Intercept + year2

cellmean(site2:year1) = Model_Intercept + site2

cellmean(site2:year2) = Model_Intercept+site2+ year2 +   site2:year2



Where only the fourth equation changes, which nontheless can have a huge
impact on the estimation of the other parameters



(usually R outputs the reference levels for the intercept and the
coefficients, which you can easily identify)

In case there are more sites than two... e.g.. 4 of them, then:

cellmean(site1:year1) = Model_Intercept

cellmean(site2:year1) = Model_Intercept + site2

cellmean(site3:year1) = Model_Intercept + site3

cellmean(site4:year1) = Model_Intercept + site4



You might get the gist :)



Finally, if you actually want to test for an overall interaction in this
way (or main effects), looking at these coefficients is not meaningful,
which you can tell by just looking at the formulas above...,  So you might
want to do it differently (correctly), namely by using likelihood ratio
tests:

(in R like coding)



Model1<- y=site+year+site:year

vs

Model2<- y=site+year



with

anova(Model1,Model2)  (I think aov() should work as well)

If the interaction of both variables is significant (i.e. the anova()
output gives a * for the comparison between Model 1 and Model 2... :)))
then the interaction effect explains some 'significant' amount of variance.
(If there is no *, you can consider the models as equal in terms of
explained variance). Same for other effects (e.g. full model vs. model a
specific main effect).

Maybe Check whether the "afex::mixed" function which does this for you in
a sensible way (there are different ways of doing LRT tests...)

;))



Having done this in the first place, is often viewed as prerequisite for
'digging' into the model estimates (as discussed above) to find out, what
significant then actually means in terms of 'mean-changes' :)



Hope this helps,

Best, Ren?







Am So., 9. Juni 2019 um 12:45 Uhr schrieb <d.luedecke at uke.de>:

Dear Patricia,

when you include an interaction, your assumption is that the relationship
between an independent X1 and the dependent variable Y varies *depending on
the values of another independent variable X2*. Indeed, for logistic
regression models (as well as for many models with non-Gaussian families),
the interpretation of interaction terms can be tricky. In such cases, I
would recommend to compute (at least additionally) marginal effects, which
give you an intuitive output of your results.

You can do so e.g. with the "ggeffects" package (
https://strengejacke.github.io/ggeffects/), and there is also an example
for a logistic mixed effects model (
https://strengejacke.github.io/ggeffects/articles/practical_logisticmixedmodel.html),
which might help you.

In your case, the code would be
ggpredict(M1, c("feed", "year")) for the model with interaction. If you
want to plot the results, simply call
me <- ggpredict(M1, c("feed", "year"))
plot(me)

A comment on your model: I'm not sure, but if you compare subjects (or
feeding sites) at two time points, you might want to model the
auto-correlation of subjects / feeding site ("repeated measure") using your
time variable as random slope:

M1 <- glmer((bear_pres ~  feed * year + (1 + year | Feeding.site), family
= binomial, data = df10)

Computing marginal effects than would be the same function call:
ggpredict(M1, c("feed", "year"))


Best
Daniel


-----Urspr?ngliche Nachricht-----
Von: R-sig-mixed-models <r-sig-mixed-models-bounces at r-project.org> Im
Auftrag von Patricia Graf
Gesendet: Sonntag, 9. Juni 2019 09:17
An: r-sig-mixed-models at r-project.org
Betreff: [R-sig-ME] Mixed model interpretation with interaction

Hello,



I have a few questions concering the interpretation of a GLMM output table
when the model includes an interaction.

We want to analyse bear presence at feeding sites (bear_pres) related to
the year (two years: 2016, 2017) and the feed supplied at feeding sites
(carrion, maize). So the response is binary (0 = no bear present, 1 = bear
present within 5-min intervals over the whole day) and both predictors are
categorical, we include feeding site ID as random factor.



The model includes some other variables too but for simplicity I just use
those two variables for explanation.



1) As I understand, in a model without interaction, the interpretation of
the results would be as follows:



M1 <- glmer((bear_pres ~  feed + year + (1|Feeding.site), family=binomial,
data=df10)

Fixed effects:

           Estimate Std. Error z value Pr(>|z|)

(Intercept) -4.58524    0.08529 -53.76   <2e-16 ***the intercept is bear
presence at maize sites in 2016

feedcarrion    0.39178    0.02139   18.32  <2e-16 ***bear presence at
feeding sites in 2017 compared to 2016

year2017    0.23027    0.01978   11.64  <2e-16 ***bear presence at carrion
feeding sites compared to maize feeding sites



Is this interpretation right?





2) To my knowledge, the output changes when you include an interaction:



M2<- glmer(bear_pres ~  year*feed + (1|Feeding.site), family=binomial,
data=df10)

Fixed effects:

                  Estimate Std. Error z value Pr(>|z|)

(Intercept)       -4.36413    0.10730 -40.67  < 2e-16 ***the intercept is
bear presence at maize sites in 2016 (baseline)

year2017          -0.18010    0.05119  -3.52 0.000434 ***difference in bear
presence in 2017 compared to 2016 for maize

feedcarrion          -0.02933    0.05318  -0.55 0.581222    difference in
bear presence at carrion sites compared to maize sites in 2016

year2017:feedcarrion  0.85275   0.09953    8.57  < 2e-16 ***difference in
bear presence at carrion sites 2017 and the sum of ?0+ ?1+ ?2



So to my questions: Is this interpretation right? What is the coding of the
model so it does produce this output, e.g. why is the year not comparing
2016 to 2017 anymore as in the model without the interaction? Or why
doesn?t the model still use the two food types for comparison?



As I understand, when you include an intercation between the two binary
dummy-coded categorical variables, the interpretation of what was main
effects before (year, carrion) changes, and so do the betas (these are
called ?simple effects? afterwards).



In my group, there is a strong believe that in M2, the year still compares
the two years (and so does feed), it?s just the coefficient cannot be
interpreted anymore. Also, there is a believe that the interaction term
compares to feedmaize in the year 2016.



If my interpreation is correct, I need some background on how the algorithm
works, how simple effects evolve and why the interaction should be
interpreted as in the output table of M2.



Thank you for your help in advance!

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