Comparing changes in a response variable over multiple proportions at different sites

Say you have a variable, like average house cost. You have values of this variable at multiple sites in a city, and want to see how it fluctuates with respect to surrounding business type proportions (multiple classes). For example, at one site, the average home cost is 10 and the surrounding proportions of business types is 30% restaurants, 60% government, and 10% agriculture. What is/are the/any recommended ways to analyze how the housing cost changes with respect to different proportions (over sites)? I am looking for statistical procedures, and have so far come across Redundancy Analysis. However, I'm having trouble figuring out if this is exactly what I'm looking for or if it is optimal. I've also looked into canonical analysis and am having the same confusion.

Further, I have different proportions over different spatial scales. For example, around each site I have drawn a circle and calculated the business proportions within the circle. Then I can make the circle a little bigger, and follow the same approach. Finally, I also have data over many years, thus am hoping to add a temporal analysis aspect to this work. Obviously I need to start with a single year and circle (proportion) to keep things simple first, but am looking for guidance in sifting through these layers in the most efficient/ methodological way.

• Do you have the proportions of business types measured for each circle/site? Commented May 5, 2020 at 15:45
• Yes; for each site, I draw multiple circles (of increasing size) around that site and calculate busines type proportion within that single site at the circle of that size. Thus, for each site I have multiple sets of proportions (one per circle) Commented May 5, 2020 at 16:19

I think that you can use regression to analyze your data. In essence, you have described a regression with a single outcome, y (home price), with three predictor variables x1-x3 (proportion restaurant, proportion government, and proportion agriculture, respectively). You further have multiple home price observations in each of your sites, necessitating some sort of correction for the likely correlation of home prices in the same site. Thus, a multilevel or mixed effects regression model could work well for these purposes. One model given the data described is as follows...

Level 1 (within-site):

$$y_{ij} = \beta_{0j} + e_{ij}$$

$$e_{ij}$$ ~ $$N(0, \sigma^2_{e})$$

Level 2 (between sites):

$$\beta_{0j} = y_{00} + y_{01}*x(prop.rest)_j + y_{02}*x(prop.gov)_j + y_{03}*x(prop.ag)_j + u_{0j}$$

$$u_{0j}$$ ~ $$N(0, \sigma^2_{u0})$$

You can substitute the terms to get the mixed model formulation, but the key idea is that your outcome, home price, has within- and between-site variation. The multilevel model allows you to model the variation at each level as a function of predictors. The predictors you presently describe are all site-level predictors and so show up in the model for the site-level intercept. In other words, they explain between site differences in home prices. To explain within-site differences in home prices, you would need a predictor that varies within sites and is linked to each home sale you have. An example would be something like square feet or number of bedrooms. If you had such a variable, you can easily augment this model by adding it to the level 1 equation:

$$y_{ij} = \beta_{0j} + \beta_1*x(covariate)_{ij} + e_{ij}$$

The second issue, having different sized sites/circles, is a bit trickier. You could run multiple versions of this model for various site sizes. But how do you make sense of the results? Perhaps you are looking for consistency across the different site sizes. You would need to be careful that your circles from adjacent areas do not overlap. Otherwise, you are violating an assumption of the model that the level 2 sites are independent of one another.

Edit: Based on the comments to my response, I think a MLM is still a viable approach for this analysis. It's not the only approach, for sure, but my suggestion is to treat site as the level 2 variable and then the different-sized circles become the level 1 repeated measure. The predictors (% restaurants, % government, % agriculture) vary depending on the circle size, and are thus level 1 predictors. The key is to make sure that you do not include circles that overlap with circles from other sites. This would violate the assumption that the random intercepts are uncorrelated across clusters (and uncorrelated with the level 1 residuals).

• This definitely helps, but I'm really not interested in variation within a single site. I'm just considering the average housing value at a each single site on its own as the predictor values and how they change relative to their respective surrounding type proportions, and if there is any scaling relation with findings as I increase the proportion radius/ the circle in which I'm considering housing type proportions Commented May 8, 2020 at 15:44
• As in I don't actually have multiple home price observations in each of my sites, I only have the average. Thus I think this is even more simple than that Commented May 8, 2020 at 15:45
• Ok. What about the issue of overlap? Is site A always site A no matter the size of the circle? Or is there a circle size at which site A and B overlap? Commented May 8, 2020 at 18:36
• I still think you can and should do a multilevel model. The difference would be that site would be your cluster/grouping variable and the different circle sizes are your level-1 identifiers within site. Then the predictors for % restaurant, government, and agriculture are level-1 predictors. The key is to keep the circles from overlapping. Does that make sense? Commented May 8, 2020 at 21:41
• Admittedly, it's not a textbook version of a multilevel model, so you will probably find most readings on this to be a bit more cut and dried (e.g., students within classrooms, measurement occasions within persons, persons within neighborhood, etc.). I have taken that and generalized it, if you will. I would suggest you also include circle size in your model as a level 1 predictor. In terms of readings,see ourcodingclub.github.io/tutorials/mixed-models and quantdev.ssri.psu.edu/tutorials/… Commented May 11, 2020 at 20:33