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Suppose there are four models:

Model 1: $y = ax$

Model 2: $y = ax^2$

Model 3: $y = a\sqrt{x}$

Model 4: $y = ax^\theta$

Model 4 is the most complex model with two parameters (the others have one parameter). If we do model selection (e.g., based on AIC or LRT), when the estimated $\theta$ is close to 1, it is likely (of course not necessarily but) that Model 1 is selected. However, suppose estimated $\theta$ is very close to 1.3 and Model 4 is selected, we do not say, e.g., Model 5 ($y=ax^{1.3}$) is selected.

My questions is when is it ok to assume a specific value in a model? In the above example, what is so special about 1, 2, and 0.5? Does this depend the specific questions?

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  • $\begingroup$ Note that you can only do an LRT of model 4 against each of the three other models, not of, e.g., model 1 against model 2. The reason is that models 1–3 are nested in model 4, but model 1 is not nested in model 2 (you can’t get from model 2 to model 1 by setting one parameter to a certain value, or vice versa). That said, if you have enough observations to reliably estimate the parameters, and you know that model 4 is adequate, I would go for that model. $\endgroup$ Mar 28 '15 at 9:54
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In cases when we know the power exactly (i. e. physical laws), the power is most often 1. When it is not 1, it is often 2 or 0.5. Therefore, the prior probability that the power is 1, 2, or 0.5 is very high. Therefore, it is reasonable to assume that $\theta$ is probably 1, 2, or 0.5 if we know that $\theta$ is close to one of these values, and the difference is not statistically significant.

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