Why Is It Important To Simulate Data From A Statistical Model? Within Statistics, I have seen the following applications of simulation:

*

*Bayesian Sampling: When the Posterior Distribution can not be analytically integrated, we use MCMC algorithms (e.g. Metropolis-Hastings) to "simulate" points from the Posterior Distribution

*Model Misspecification: Suppose I create a statistical model where I assume that the response variable has a certain distribution (e.g. Exponential Distribution) - I can simulate data from different distributions (e.g. Gamma Distribution) and see how "robust" my model is to conditions outside of what the model was created under

However, I keep hearing that it can be important to simulate data from the same model itself - for example, I fit a specific regression model on a specific dataset, I might need to simulate data from this regression model itself. I am trying to understand why this might be important and useful.
As an example, suppose there is a school with 100 students. I assume that the heights of these students follow a Normal Distribution. I then measure the height of all students, calculate the average height (e.g. "a") and the standard deviation (e.g. "b"). I now have a Normal Distribution(a,b).
If I simulate many samples from this Normal(a,b) - the average value of all these simulations should equal to "a". The way I see it, simulating numbers from this Normal(a,b) has not really provided me with any new information .
This is where my confusion is:

*

*I can understand why its important to simulate data from a different model to test the robustness of your own model. As an example, if I design a bridge that I believe is well suited to design weights of "x KG" and windspeeds of "y KM/h" - I might be interested in seeing how this bridge behaves under different conditions, such as with weights of " x + z KG" and windspeeds of "y + w KM/h".

*But why is it important to simulate data from the same regression model that you just fit - how can this result any new information that the you can't analytically infer from the model itself?
To reiterate my confusion one more time -  I can simulate 1000 random samples from a Normal(a,b) and find out that the average value of these samples is close to "a".... but I could have just taken the Expected value of Normal(a,b) and determined that the average of many samples from this distribution should be close to "a". Thus, what is the importance of simulating from the same model itself?
Thanks!
Note:

*

*Additional Reference:
"When conducting simulation studies to evaluate the performance of new and existing statistical methods for analyzing survival data, one is required to simulate event times under a known data generating model. Similarly, one may need to simulate event times for the purpose of power calculations when designing new studies." (https://www.jstatsoft.org/article/view/v097i03)

*However, when reading this quotation - it is not immediately clear to me why simulating data from the same model is required for goals such as  "power calculations"


Simulating from the same model that you use to analyze the data is useful for understanding the model and its limitations. To some extent it's a sanity check that the model does not have some serious issues, because a model that does not work very well when its assumptions are met is problematic. You may say:"But how could this ever be the case?" And the answer includes, but is not limited to:

*

*You have messed up and something you assumed/derived/implemented is just not right.

*If you simulate a huge amount of data, you should hope to get something close to the parameter values you simulated with back. If you don't there's a hint the first bullet might apply.

*Parameter identifiability: I.e. the data you plan to collect/the experiment you do does not inform you at all - or at least not very well - about the parameter you are interested in. Example: randomized experiment with a placebo group and a single dose of a drug (let's say 50 mg), the outcome is wonderfully normally distributed and you fit a non-linear sigmoid Emax model assuming that $Y_i \sim \beta_0 + \beta_1 \text{dose}^h / (\text{dose}^h + \text{ED50}^h) + \epsilon_i$ with i.i.d. $\epsilon_i \sim N(0, \sigma^2)$. All assumption for your model are met, it is even the true data generating model, but with just one active dose the parameters are not identified. This may sound like a very specific situation (and it is), it's just pointing out that you may occasionally be in that kind of situation (and not realize it, e.g. you specify multiple levels of variability that cannot be distinguished).

*Poor small sample performance (where "small sample" could occasionally be pretty large) e.g. due to using asymptotics. Example: Logistic regression fit using maximum likelihood with asymptotic standard errors (vs. exact logistic regression or similar) in case of 0 (or 1) events out of 100 (group 1) vs. 5 events out of 100 (group 2). You might not realize this is an issue, until you evaluate systematically what your model/its implementation does in these situations.

*Some particular issues keep occurring that cause issues in defining what the maximum likelihood estimate would be (like complete separation in logistic regression).

*Numeric optimization, convergence or sampling problems (that might be particularly common in some data situations).

*Other numerical problems (e.g. covariates that are on very different scales cause issues, finite difference approximation makes a MCMC sampler not work when proper auto-differentiation would have worked etc.)

