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I am building a non-linear model aiming to describe the mechanistic process of resource allocation. There several terms, and what makes the model non-linear is competition between lines that are facing each other pair by pair.

I tested the significance of terms through AIC comparisons, deleting non significant terms (it is not really hierarchical, I explored a big set of pathways).

I have the feeling that sometime, removing one term can change a lot the value of parameters for other terms, so that the values of parameters seem to be completely inter-dependent.

I have 2 questions:

  1. How can I formally assess whether this is true (that parameter values change a lot when removing terms)? Please note I have a huge number of parameters, so comparing them one by one is very difficult.
  2. If it is true, is it due to the intrinsic nature of non-linear model, or is it due to some kind of colinearity in my terms? Standard error are very small, but when comparing confidence intervals of parameters for two models differing by only one term, these CI do not overlap at all.

Thanks!

Aurelie
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  • Often the parameter values are highly interdependent, yes. In many cases, a reparameterization may improve that somewhat. – Glen_b Jun 02 '14 at 00:53
  • I do not understand how it is possible to reparameterize. If I have a model whith e.g. the following shape: a X + b Y + c (d Z + e W) What would be possible a reparameterization? Or perhaps do you have a reference I could read? – Aurelie Jun 02 '14 at 10:19

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If I have a model with e.g. the following shape: $a X + b Y + c (d Z + e W)$, what would be possible a reparameterization?

I can see the cause of the problem in that specification - if $a$ to $e$ are all parameters, your model is unidentifiable.

Even tiny changes in situation could lead to arbitrary changes in the resulting estimates. Note that for any $(c,d,e)$, you can get an identical fit with $(c/k,kd,ke)$ (for any $k\neq 0$).

The estimates you get for $(c,d,e)$ from fitting the model won't mean anything.

If that's typical of the kind of model you're fitting, you must first reduce it to an identifiable model before attempting any inference on it.

Glen_b
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    [This question](http://stats.stackexchange.com/q/21843/1036) has some similarities. The stuff in the parenthesis I would associate with *formative measurement models* in structural equation modelling. For $c(dZ + eW)$ to be identified, imagine replacing $(dZ + eW)$ with a latent variable $\lambda$. If you are able to identify $\lambda$ through some alternate equation, then you can insert it into the parenthesis to estimate $c$. There are a host of interpretation problems with these types of formative models though. – Andy W Jun 02 '14 at 11:54
  • @AndyW +1 ... Indeed, estimating such a $\lambda$ separately would be one of the ways to 'reduce it to an identifiable model'. – Glen_b Jun 02 '14 at 11:57
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    I think it also stands to be said that sometimes people want to estimate a model of the form listed here not because they are really interested in identifying $d$ or $e$ or $c$, but because $Z$ and $W$ have some functional relationship that makes it nonsensical to interpret them separately. In that case one can estimate the usual model of $(cd)Z + (ce)W$ and then draw marginal effect plots over reasonable values of $Z$ and $W$ to visualize the effects of $Z$ and $W$ jointly. – Andy W Jun 02 '14 at 12:43
  • Thanks! Ok, my model is much more complex than the example I gave, I will see if it is identifiable. From what I understand, the "linear" terms of the model should not meet this issue, and they are those that are dropped with model selection. I hope my model selection is not biased with this issue... – Aurelie Jun 02 '14 at 12:58
  • Ok, after investigation, my model is identifiable. What happened was that I have interactions, and they were "absorbing" part of the main effects, and because the model is not linear, this made figures that were very difficult to compare from a model to another. When I remove the main effects from the model to leave only the interaction (which is significant), all what was in the main effects goes to the interaction term, and estimates are much easier to interpret and very similar from a model to another. Then, problem solved! – Aurelie Jun 23 '14 at 14:42