Term is significant linearly, quadratically and in interaction?

Question

I used deletion tests to identify ecological factors that relate to the number of parasites on rodents. There is one factor that is significant linearly, quadratically and in interaction. However, the linear term explains more variance than the quadratic term, and if I remove the quadratic term from the minimal model then the interaction is no longer significant. What does this mean?

Please note that during the deletion tests of nested models that contained all explanatory variables measured, the quadratic term was more significant than the linear one. However, in the minimal model obtained the linear term is more significant. Should I remove the quadratic term?

I made a plot of the interaction and it shows a very interesting pattern that also makes biological sense.

EDIT To clarify: the interaction I describe is between the linear term and another explanatory variable.

My model is as follows:

model <- glmmadmb(parasites~y+x+z+(y)²+(y*x)+(treatment|id), family="nbinom")

It has repetitive measures of the same individuals, so I used a generalized linear mixed model with id and treatment as random effects. Since the model was overdispersed with a Poisson distribution I used the glmmadmb library and negative binomial to account for overdispersion.

model <- glmmadmb(parasites~y+x+z+(y)²+(y*x)+(treatment|id), family="nbinom")
summary(model)

AIC: 1050 
Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
(Intercept)    1.122164   0.513620    2.60   0.0092
z             -0.005472   0.002636   -2.50   0.0112 
x              0.007541   0.004547    1.67   0.0936  
y              0.579946   0.141240    4.17   2.0e-05 
y2            -0.020436   0.009167   -2.12   0.0009 
x:y           -0.002060   0.000754   -2.69   0.0065  # SIGNIFICANT 

Number of observations: total=133, onest=47 
Random effect variance(s):
Group=onest

Negative binomial dispersion parameter: 1.6365 (std. err.: 0.20241)
Log-likelihood: -513.988 

model <- glmmadmb(parasites~y+x+z+(y*x)+(treatment|id), family="nbinom")
summary(model)

AIC: 1057.9 
Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
(Intercept)    1.49767    0.510770    4.67  2.2e-06 
z             -0.007274   0.001202   -2.70   0.0069 
x              0.004936   0.005075    0.97   0.1219    
y              0.153317   0.057712    2.67   0.0076  
x:y           -0.001124   0.000745   -1.45   0.1475  # NOT SIGNIFICANT    

Number of observations: total=133, onest=47 
Random effect variance(s):
Group=onest

Negative binomial dispersion parameter: 1.6101 (std. err.: 0.20962)
Log-likelihood: -517.966 

Answer 1

If you have a y and a y2, and you want to form an interaction, you need to add both x:y and x:y2 (see here, but moreso the links therein). You may also be interested in reading this great answer about interactions and curvilinear terms: Either quadratic or interaction term is significant in isolation, but neither are together.

To test the interaction, drop all (i.e., both) the terms and perform a nested model test (I discuss that here in the context of a linear model--I don't know if there would be extra complications with glmmadmb).

Lastly, there is also collinearity induced in creating power and product terms (see, perhaps, here), which expands their standard errors--making the tests underpowered. This is hinted by the fact that the reduced model has a worse AIC. This, in conjunction with the danger of dropping terms based on tests of the same data (noted in my comment above), it is probably best not to drop the term.