Good day R.sig.ecology members.
We seem to have detected highly significant fixed effects from a GLMM fit using
glmer, despite having a fairly small sample size (30 observations, 4 groups in
the random factor). We are hoping that group members can either verify that we
are interpreting these results correctly or tell us where we've gone wrong. We
are working in R for Windows version 3.0.2 and using the latest version of lme4.
We sought to simultaneously test for (1) effects of annual variation in natural
food availability for black bears (measured using a rank bear food index; bfi)
on the number of occurrences of human-bear conflict (oc), and (2) a temporal
trend (across years) in oc. oc and bfi were measured annually in different
administrative districts (dist) within the larger region of interest. Not all
districts provided data in all years.
Our data look like this:
dist yr lyr bfi oc
1 ban 2004 1 2.30 852
2 pemb 2004 1 2.70 105
3 ps 2004 1 1.10 969
4 ban 2005 2 2.05 657
5 pemb 2005 2 1.86 269
6 ps 2005 2 1.69 1119
7 ban 2006 3 2.57 275
8 mid 2006 3 2.66 271
9 pemb 2006 3 2.17 57
10 ps 2006 3 2.08 429
11 ban 2007 4 1.75 530
12 mid 2007 4 2.17 606
13 pemb 2007 4 2.03 189
14 ps 2007 4 1.23 1567
15 ban 2008 5 2.86 363
16 mid 2008 5 3.00 467
17 pemb 2008 5 3.13 95
18 ps 2008 5 2.32 770
19 ban 2009 6 2.19 672
20 mid 2009 6 1.47 653
21 pemb 2009 6 2.03 471
22 ps 2009 6 1.90 1522
23 ban 2010 7 2.00 409
24 mid 2010 7 2.10 640
25 pemb 2010 7 2.10 176
26 ps 2010 7 1.90 1218
27 ban 2011 8 2.50 466
28 mid 2011 8 2.40 462
29 pemb 2011 8 2.40 117
30 ps 2011 8 2.00 963
We used a GLMM to test for effects of bfi and lyr (a dummy variable
representing the # of years since the beginning of the study), with district as
the only random effect. The log-link function improved linearity.
Code for fitting the model, and the summary of the model, appear below:
> oc.sr.bfi.yr = glmer(oc ~ bfi + lyr + (1|dist), data = dat, family =
> poisson(link = log))
> summary(oc.sr.bfi.yr)
Generalized linear mixed model fit by maximum likelihood ['glmerMod']
Family: poisson ( log )
Formula: oc ~ bfi + lyr + (1 | dist)
Data: dat
AIC BIC logLik deviance
1960.3134 1965.9182 -976.1567 1952.3134
Random effects:
Groups Name Variance Std.Dev.
dist (Intercept) 0.3006 0.5483
Number of obs: 30, groups: dist, 4
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 7.040105 0.277215 25.396 <2e-16 ***
bfi -0.486053 0.019922 -24.398 <2e-16 ***
lyr 0.035638 0.003598 9.905 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) bfi
bfi -0.132
lyr -0.019 -0.295
The results indicate a negative effect of bfi (more conflict when natural foods
are scarce, as expected) and a slight increasing trend across years. However,
we were surprised by the reported p-values for the fixed effects, given our
relatively small sample size.
We found the "Getting p-values for fitted models" help page for the lme4
package.
We fitted parameter-reduced models for comparison:
> oc.sr.null = glmer(oc ~ 1 + (1|dist), data = dat, family = poisson(link =
> log))
> oc.sr.bfi = glmer(oc ~ bfi + (1|dist), data = dat, family = poisson(link =
> log))
AIC and BIC values were greater by 100 when we excluded the effect of year, and
were greater by 500 when we also excluded the effect of the bfi. These results
seem to agree with the summary of the more general model, but they also seem a
bit extreme given our small sample size.
Model comparisons using PBmodcomp from the pbkrtest package also seem to
support the inclusion of both fixed effects.
> mc.oc.bfi.nobfi = PBmodcomp(oc.sr.bfi, oc.sr.null, nsim = 100)
> mc.oc.yr.noyr = PBmodcomp(oc.sr.bfi.yr, oc.sr.bfi, nsim = 100)
> mc.oc.bfi.nobfi
Parametric bootstrap test; time: 26.13 sec; samples: 100 extremes: 0;
large : oc ~ bfi + (1 | dist)
small : oc ~ 1 + (1 | dist)
stat df p.value
LRT 513.43 1 < 2.2e-16 ***
PBtest 513.43 0.009901 **
---
> mc.oc.yr.noyr
Parametric bootstrap test; time: 39.83 sec; samples: 100 extremes: 0;
large : oc ~ bfi + lyr + (1 | dist)
small : oc ~ bfi + (1 | dist)
stat df p.value
LRT 98.323 1 < 2.2e-16 ***
PBtest 98.323 0.009901 **
---
We calculated bootstrap confidence intervals using "confint". They seem to
indicate that the estimated effects were unambiguous, but we were unsure
whether our sample size was adequate to apply this method.
> confint (oc.sr.bfi.yr, method = c("boot"))
Computing bootstrap confidence intervals ...
2.5 % 97.5 %
sd_(Intercept)|dist 0.13871500 0.83969172
(Intercept) 6.54440740 7.56234905
bfi -0.53389761 -0.44979598
lyr 0.02835891 0.04325712
Simple diagnostic plots (residuals vs fitted values and a normal QQ plot of
residuals) of the GLMM with effects of both bfi and lyr did not reveal
violations of assumptions. We would be happy to report significant effects of
food availability and a significant trend across years, but are concerned that
we may not be interpreting these results correctly. Any help or suggestions
would be greatly appreciated.
Sincerely,
Eric
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