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I've run two different linear mixed effects models on the same data and got two different estimates for the gradient of the longitudinal variable. e.g.

model 1 has estimate 30 with standard error 5. model 2 has estimate 40 with standard error 4.

I'm interested in calculating a p value for the probability that the models are different, from the estimate and standard error. How do I do this? I'm aware that checking for overlap in the 95% confidence intervals is a bad idea, and that overlapping 83% CIs are a better test, but would like to be able to quantify this with a p value.

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  • $\begingroup$ AIC might be more appropriate for comparing the two models. $\endgroup$ – Dedekind Cuts Jul 8 at 14:48
  • $\begingroup$ AIC makes a very different kind of comparison to the kind of comparison I'm looking to make. Whether a test is appropriate is dependant on the question you're actually trying to answer with the test. I'm looking for a way to compare how different the estimates are, rather than compare model quality. $\endgroup$ – H. Green Jul 8 at 15:03
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    $\begingroup$ To clarify what you're attempting: When you say you've run two different models, do you mean that you have fitted two models with different terms, e.g. as a simple example for response variable y, and explanatory variables x, z and v, did you fit models y = a.x + b.z + c as opposed to y = l.x + m.v + e, and now you want to compare coefficients a and l? Or have you parameterised a model twice with the same terms (e.g. y = a.x + b.z + c but presumably something more complicated) using different fitting processes each time, and you now want to compare your two estimates of a? $\endgroup$ – Izy Jul 9 at 10:25
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    $\begingroup$ I suggest you try to formulate what null hypothesis you're actually trying to test. $\endgroup$ – Dedekind Cuts Jul 9 at 17:28
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    $\begingroup$ your model estimates are different, you know that. what do you want to test? $\endgroup$ – carlo Jul 14 at 9:27
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It is unclear from your question how these models relate to one another. However, the way you perform a statistical test for equivalence of models is to pose them as nested models and then perform a statistical test to see whether the additional terms in the more complex model are "zero". For example, suppose you have two linear models like this:

$$\begin{matrix} \text{Simple Model} & & y_i = \beta_0 + \beta_1 x_{1,i} + \cdots \beta_k x_{k,i} + \epsilon_i, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \\[6pt] \text{Complex Model} & & y_i = \beta_0 + \beta_1 x_{1,i} + \cdots \beta_k x_{k,i} + \beta_{k+1} x_{k+1,i} + \cdots \beta_{k+m} x_{k+m,i} + \varepsilon_i. \\[6pt] \end{matrix}$$

This is an example where your simple model is "nested" within your more complex model. You can test the equivalence by testing the hypotheses:

$$H_0: \beta_{k+1} = \cdots = \beta_{k+m} = 0 \quad \quad \quad \quad \quad H_A: H_0 \text{ is false}.$$

The null hypothesis here reduces the complex model to the simple model, which is equivalent to saying that the models are really the same. Using a classical hypothesis test we can see if there is evidence to reject the null hypothesis that both models are the same.

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