p value for difference in model outcomes 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.
 A: 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.
