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

Question

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.

Answer 1

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).