“Summing” linear regression models

I would like to combine regression models where each $Y_i$ is a subset of the total $Y$. The independent variable is the same but not from the same sample. Some illustrating example: say I am interested to estimate the tree density in a city, tree density is negatively related to population density. What I have is: $Y_1=b_1X$ where $X$ is the population density of some cities and $Y_1$ is the tree density of a certain tree species.

$Y_2=b_2X$ is a similar data set for but a different tree species from some other cities. What I would like to estimate is the total tree density $Y=Y_1+Y_2$. I think just summing the result of the two models would overestimate the total number of trees because the proportion between the two species varies among cities. In the most extreme case all trees in acity would be from the same species.

The problem seems not so uncommon, so any keywords that would guide some search would be helpful. It would be just fine if I could estimate the prediction bounds of the total tree density of cities with population density $X$, e.g by randomly drawing from a distribution of a weighting parameter that mixes different fractions of $Y_1$ and $Y_2$

• Why not use multilevel model? – Tim Apr 1 '17 at 5:25
• What I have just learned about multilevel models ist that they can be used for impoving the prediction at lower level by incorporating higher level information i.e. as in my case the climate setting of a city. I could not find an example where this direction is reversed which I think is the problem in this example: making predictions from low-level - trees species for a higher level - all trees. – robbenhundt Apr 3 '17 at 20:35
• First of all: it would let you build a single model for all your data instead assuming that every subsample is totally independent of every other one. – Tim Apr 3 '17 at 21:03
• @robbenhundt Multilevel models are frequently used with post-stratification to generate better higher level forecasts. – John Apr 3 '17 at 21:40
• Thank you for providing the key word 'post-stratification' which really helps to dive further into the topic. – robbenhundt Apr 4 '17 at 7:13

It sounds like a mixed effects/hierarchical regression problem. If $Y_{ij}$ is the number of trees of $i$-th kind in $j$-th city, $X_j$ is the population density of $j$-th city, then the basic model you can think of is

$$Y_{ij} = \mu + \gamma_i + \beta_{j} X_{ij} + \varepsilon_{ij}$$

so the intercept $\mu$ is the base population, $\gamma_i$ are intercepts for each of the kinds of trees, and $\beta_j$ are (random) slopes for the cities. This basic model can and possibly should be extended, at least to include interactions as you notice that there are cities that differ in the species of trees that grow in them (so you have effects for kinds of trees nested in cities), or to include other covariates not mentioned by you.

Putting everything in a single model would let you not to worry about arbitrary weighting to sum your estimates since the optimal weighting would be estimated by your model.

If you are not familiar with this kind of models you can check the great introductory book:

Gelman, A. & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge.

• Thanks for the reference and the clarification. I guess I am on the right track now. – robbenhundt Apr 4 '17 at 7:14