# Negative Binomial GLM variables significant in full model- but not on their own

I'm not sure of the best way to ask this but I'll try:

I'm using negative binomial models to explore habitat relationships with species abundance. I found 4 significant factors in my full model. I then wanted to investigate relationship between abundance and each factor individually- both statistically and by plotting the relationships. The general form of my model is:

Model.full = glm.nb( SpeciesCounts ~ X1 + X2 + X3 + X4, data = dataset)


First Question: When variables X1-X4 are included in my full Model, all are significant (P < .05), but when I run the sub models with each variable independently, some are no longer significant. Is there a theoretical explanation as to why all four variables would be significant together, but on their own, not significant?

Below I'm looking at the variable DistShelf in particular:

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 19.4004609  7.9092487   2.453 0.014172 *
Slope       -0.0522097  0.0140412  -3.718 0.000201 ***
RugosityM    1.0193253  0.2121319   4.805 1.55e-06 ***
RugosityH    1.0412758  0.2891510   3.601 0.000317 ***
DistShelf   -0.0002885  0.0001066  -2.706 0.006817 **
Lat         -0.6546487  0.2163872  -3.025 0.002483 **


And on its own:

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -4.563e+00  1.286e-01 -35.469   <2e-16 ***
DistShelf   -1.130e-04  9.794e-05  -1.154    0.249


I was expecting the opposite - that DistShelf would have a stronger relationship on its own.

A second question is whether refitting a glm for each factor is a legitimate approach to investigate the relationship with each variable on its own e.g.

Model.full = glm.nb( SpeciesCounts ~ X1 + X2 + X3 + X4, data = dataset)

#Now can I follow up with how SpeciesCounts responds to each variable:

Model.2 = glm.nb( SpeciesCounts ~ X1 , data = dataset) # 1st variable
Model.3 = glm.nb( SpeciesCounts ~ X2 , data = dataset)  # 2nd variable etc....


Or should I use the coefficients calculated by Model.full to create individual plots of SpeciesCounts ~ X1, SpeciesCounts ~ X2 etc...

Thoughts?

• Do you have any clues about why the intercept changed so dramatically between the two models? exp(19) is about 200 million, exp(-4) is about 2/100. – mdewey Dec 16 '16 at 13:45
• I don't understand why the intercept is changing so much between the models other than that it seems to be my Latitude variable. (If I exclude Lat, the intercept is -4.5). Maybe my question above is an artifact of the particular data that I have, but I was curious if there is a reason why factors may have low p-values when included in a model together, but high p-values when individually included in models. Would it be helpful to see the structure of my data? – Kodiakflds Dec 20 '16 at 0:32