# Different p-values for fixed effects in summary() of glmer() and likelihood ratio test comparison in R

I'm using glmer() with a binomial response variable. My optimal model has two fixed effects (flow and DNA) which in summary() show a non-significant p value but when I remove each fixed effect in turn from the model the likelihood ratio test comparing the two models shows a significant p value. I'm struggling to understand (1) if this is normal, and (2) how to report the results if the explanatory variables "flow" and "DNA" are important but their p values in the model are well above 0.05?

Optimal model:

a25 <- glmer(Status_qpcr~(1|Root)+Flow+DNA,
family=binomial, data=spore)
summary(a25)

Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) ['glmerMod']
Family: binomial  ( logit )
Formula: Status_qpcr ~ (1 | Root) + Flow + DNA
Data: spore
AIC      BIC   logLik deviance df.resid
72.9     81.0    -32.4     64.9       52

Scaled residuals:
Min      1Q  Median      3Q     Max
-2.9318 -0.8163  0.4435  0.6848  1.6133

Random effects:
Groups Name        Variance Std.Dev.
Root   (Intercept) 0.3842   0.6199
Number of obs: 56, groups:  Root, 9

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.97752    0.79252  -1.233    0.217
Flow         3.82779    2.27165   1.685    0.092 .
DNA          0.01616    0.01039   1.556    0.120
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr) Flow   Flow -0.775
DNA    -0.576  0.227


Likelihood ratio test:

a26 <- update(a25,~.-DNA)
anova(a25,a26)

Data: spore
Models:
a26: Status_qpcr ~ (1 | Root) + Flow
a25: Status_qpcr ~ (1 | Root) + Flow + DNA
Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)
a26  3 74.802 80.878 -34.401   68.802
a25  4 72.897 80.998 -32.448   64.897 3.9049      1    0.04815 *

a27 <- update(a25,~.-Flow)
anova(a25,a27)

Data: spore
Models:
a27: Status_qpcr ~ (1 | Root) + DNA
a25: Status_qpcr ~ (1 | Root) + Flow + DNA
Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)
a27  3 78.440 84.723 -36.220   72.440
a25  4 72.897 80.998 -32.448   64.897 7.5427      1   0.006025 **


• take a look at tpr <- profile(a25,which="beta_"); lattice::xyplot(tpr). You should see that the lines are far from straight (straight lines would indicate a log-quadratic likelihood surface, which is what's assumed by Wald p-values)
• compare the results of confint(a25,which="beta_") (likelihood ratio intervals) and confint(a25,which="beta_",method="Wald"); they should be quite different.
LRT CI/p-values are essentially always better than the Wald equivalents (but much slower to compute, which is why Wald p-values are the default in summary()).