With these data, the only way to get around it is to drop cov (for this model and data). I did some more playing around, which I'll explain here. You can see the impact of the confounding by focusing on the Type I tests, and putting cov and animnum as the first two variables in the model, that way you can focus better on the implications of the link between these two, ignoring everything else. If you use cov and animnum in that order, you get very reasonable results for Type I tests. cov is significant with F(1,19)=66.4 (slope = 0.6), and animnum is significant with F(6,19)=4.13. The continuous cov variable provides some information on the response, but not all of the information since cov is capturing just the linear effect. Any nonlinear effect is left over, which is then captured by the animnum factor. I think the df_N is 6 and not 7 because some of the effect of animnum factor is already captured by cov (one more level is 0 in the solution).
Now, if you put animnum and cov as the first two variables (in that order), things are very different. animnum is significant with F(7,19)=13.03, but cov has no effect, with F(0,.)=missing (slope = 0). The entire effect of animnum/cov is already captured by the first term in the model.
You can even look at this in a different way, although this is just for demonstration purposes. Define cov as a factor. Then fitting the model with cov factor (and no animnum) gives the exact same fit (same -2LL) as the model with animnum factor (an no cov factor). Trying to put both in the model as factors blows things up real good.
It was interesting to think about all of this.
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