# separate models vs joint model

My goal is to estimate the association between children BMI and distance to the nearest fast food restaurants. The hypothesis is that children BMI increases with increasing proximity of fast food restaurant. Children BMI is higher when the fast food restaurants are very close to school compared to BMI in children where the fast food restaurants are far away from school.

$$i: 1,2,3...n$$ represents the kids

$$k: 1,2,3...k$$ represents the fast food restaurant: k=1 KFC, k=2 Wendy, k=3 Burger King so on..

$$Y_{ik}$$ : represents BMI of $$i$$th kid near $$k$$th fast food restaurant.

$$X_{ik}$$ : represents $$k$$th fast food restaurant closest to $$i$$th kid.

Approach1:

The marginal association between outcome $$Y_{ik}$$ and each predictor $$X_{ik}$$, can be estimated based on separate regression

$$E[Y_{i}|X_{ik}] = \beta_{0k} + \beta_{1k}X_{ik}$$, where $$\beta_{0k}, \beta_{1k}$$ are the intercept and slope parameters in the $$k$$th regression , $$k=1,....K$$.

Approach2:

The joint estimation of model parameters can be estimated by re-structuring the data as

Each kid is treated as an independent cluster with $$K$$ repeated measures . Assuming an identity link, constant variance and a working independence correlation matrix.

So my question is what is the difference between estimating model parameters $$\beta_{0k}, \beta_{1k}$$ seperately based on Approach1, vs jointly as outlined in Approach2. If assume constant variance and a working independence correlation matrix, wont the results from both Approach1 & Approach2 be the same ?

Also, if I am not wrong, it looks like the joint modeling approach is advocating imputing 0s in Xs where there are not measurements. Is this acceptable ?