Differing p values (significant and insignificant) for the same predictor variable in different AIC models I'm looking at a multinomial logistic regression analysis of deer behavioural responses to camera traps. The levels of the response variable are no reaction, reaction and strong reaction. I've selected a number of models based on their AIC values. However, I've found that the p values gain and lose significance within differing models. For instance, with only season as a predictor:
Season               Summer  Winter

Coefficient r         0.8     0.03
Coefficient sr        0.4     0.5
P value     r         0.041   0.925
P value     sr        0.18*   0.024

AIC  1005.023
* P value of interest

With species, camera model, and season as predictors:
Species, Camera model and Season

                    Muntjac  Roe   Sika  Bushnellb   Reconyx  Summer Winter

Coefficients  r       0.94   .43  -0.56    0.76        0.71    1.6    0.5
Coefficients  sr      1.11   .19  -0.19    0.28        0.77    0.95   0.8
P value       r       0.1    0.4   0.4     0.13        0.1     0.008  0.3
P value       sr      0.005  0.6   0.6     0.4         0.01    0.03*  0.02

AIC 1006.618
* P value of interest

Am I right to assume that this may be due to collinearity inflating the significance of this p value, and therefore when analysing the probability of the predictor variable summer influencing the response variable sr, using the p value from the first model? Is it correct to assume that the p value from the 2nd model is due to collinearity and thus not representative of the actual significance of this variable (or level?) summer?
Thanks!
 A: Strong multicollinearity tends to inflate standard errors, so no, that's probably not why the summer coefficient is more significant when controlling other variables. Response variance due to differences in species or camera model may be suppressing the relationship between season and responses. Controlling that noise may give you a clearer sense of how season would relate in isolation, but that may not represent the reality of the broader scenario in which species and camera model may vary (and possibly interact) freely.
It's important that you decide which hypothesis you want to test. Do you want to control the effects of species and camera model when describing the predictive relationship of season, or would you rather leave all other variables uncontrolled? Some other variables will always be uncontrolled, so in my view, controlling them is optional, but may be helpful depending on what you're really trying to learn from your analysis.
P.S. Again, ordinal logistic regression would've been a better analytic choice here. If you have the option, I'd redo the analysis and focus on interpreting those results instead.
A: I seriously doubt that even your best selected model will be able to successfully deal with so many variable and model problems (collinearity, normality assumption, independent,..). If you really need a functional model, not just something that impressive on paper, construct a hypothetical data set with strong, neutral and negative responses in a spreadsheet. Maintain the number of variables and total number of data points. You know the underlying true values and the selected amount of noise. Does your model work at all in recovering the known underlying values? My experience is seldom. I would recommend reduce the dimensionality by using one variable, and weight the one variable models together. Try taking two variables and similarly combining the models. Are the results much better now in getting back to the known underlying?
Trying to understand why the models in higher dimensions work poorly doesn't help one in getting meaningfully accurate results if that is one true intent. Violating underlying model assumptions (seldom discussed or even revealed) and sensitivity to numerical analysis issues (examples, taking the inverse of a matrix with a determinant value that is small) can be very detrimental to the analysis.
