How to deal with Linear Regression model with some data aggregated Lets say I am trying to find a linear regression between Weight and Height of a person.
$W=b_0+b_1 H+e$
The data I have gathered from 8 people is like this:
# W(kg)  H(cm)
1. 68    168
2. 64    170
3. ?     160
4. ?     180
5. ?     145
6. ?     191
7. 69    185
8. 80    191

Where I know that the sum of ?'s is 280, but I do not know exact data of each of them (because, let's say, at the time, I only had scales, which had a minimum scale of $200$. Dumb reason, I know, but that's just for the sake of example). 
So, my question is this: how do I create my $W$ matrix so I can make computations (to find out $b_0$ and $b_1$ using least squares method)? :)
 A: The minimization of the mean square error can be regarded as a result of maximizing the likelihood that the data resulted from normally distributed errors. To deal with the aggregated data, we could integrate this likelihood over all values consistent with the constraint. The likelihood is
$$\prod_i\mathrm e^{-\beta(w_i-(b_1h_i+b_0))^2}\;,$$
and the integral over all values consistent with the constraint is
$$
\iiiint\mathrm dw_1\mathrm dw_2\mathrm dw_3\mathrm dw_4\delta(w_1+w_2+w_3+w_4-280)\prod_i\mathrm e^{-\beta(w_i-(b_1h_i+b_0))^2}
$$
(where I've numbered the unknown values $1$ to $4$ for convenience). Integrating out $w_4$ yields 
$$
\prod_{i\gt4}\mathrm e^{-\beta(w_i-(b_1h_i+b_0))^2}\iiiint\mathrm dw_1\mathrm dw_2\mathrm dw_3\mathrm e^{-\beta(w^\top Aw-2s^\top w+c)}
$$
with
$$
\begin{eqnarray}
A_{ij}&=&\delta_{ij}+1\;,\\
s_i&=&280+b_1(h_i-h_4)\;,\\
c&=&280^2+\sum_{k=1}^4{(b_1h_k+b_0)^2}-2\cdot280(b_1h_4+b_0)\;,
\end{eqnarray}
$$
where $i$ and $j$ run from $1$ to $3$. The integral is proportional to $\mathrm e^{\beta(s^\top A^{-1}s-c)}$.
With
$$\displaystyle A^{-1}=\frac14\pmatrix{3&-1&-1\\-1&3&-1\\-1&-1&3}\;,$$
this is $\mathrm e^{-4\beta(\overline w-(b_1\overline h+b_0))^2}$, with $\overline w=(w_1+w_2+w_3+w_4)/4$ and $\overline h=(h_1+h_2+h_3+h_4)/4$.
Thus, you should treat these four measurements as if you had made four measurements of the average weight $\overline w$ at the average height $\overline h$.
A: Just for the future generations I am going to put a link to a program that I used to diaggregate the data for me. ECOTRIM. Event it is an old program, it is working decently and diaggregated data for me very well. I did compare the real GDP data with diaggregated one, and it was shooting as close as not more than 5% away. Quite a decent tool.
