GLM with multiplicative coefficients

Let’s say I’ve got this dataset, where I’m interested in Weight Gain as a function of calories consumed at Breakfast, Lunch, Dinner.

4×6 DataFrame
 Row │ Participant  Day    B      L      D      WG      
     │ String       Int64  Int64  Int64  Int64  Float64 
   1 │ Bob              1     65    231    345      0.5
   2 │ Bob              2    300    777    674     -0.3
   3 │ Mary             1    100    856    321      2.0
   4 │ Mary             2    555    845    656      1.0


I know (for the sake of example) that calories consumed at breakfast might be more readily absorbed than at lunch, or dinner (not participant-specific), and I know that each Participant may have a different general Absorption rate a_p, which affects all intakes. So I’d like to fit something like this:

WG = a_p * (b * B + l * L + d * D)

(maybe with an intercept, but that looks mathematically irrelevant?)

For N participants, that is N + 3 coefficients to fit. How should I fit this in Julia / GLM? Since the equation is linear holding b, l, d constant, I can fit a_p. Likewise I can fit b, l, d holding a_p constant. So unless I’m mistaken I could alternatively fit both of these coefficient sets, until convergence. Doing this with \ looks like a hassle. @formula looks appealing, but I’m a total newbie with GLM.jl, and a bit overwhelmed. I can see how I can build a dataframe where I’ve done the A_p B multiplication in order to fit b, l, d, but urg, is there anything simpler? Any help / pointer / @formula code appreciated.

(Side note: I know that this problem is underspecified, so throw in a constraint that b+l+d=1 if that’s better)

Your dataset represents panel data on your subject matter; there are observations of the same entities at several times.
Typically this calls for regression with Fixed Effects regression or multi-level bayesian models. Some books can give you an introduction to these types of models, such as “Introduction to Econometrics” from James H Stock and Mark W. Watson or for Bayesian " Statistical Rethinking" by Richard McElreath, or “Doing Bayesian data analysis” by John Kruschke.
For the related Julia packages I would look into Fixed Effects (GitHub - FixedEffects/FixedEffectModels.jl: Fast Estimation of Linear Models with IV and High Dimensional Categorical Variables), and Turing (Turing.jl - Turing.jl).

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I suppose you can use workaround: make factor (for example ‘a’) with vector of ones, and use zero-intercept model, something like this: ‘0 + aF1 + aF2 + … a*Fn’ if you have correlated data - look at MixedModels.jl and Metida.jl