# Regression for multinomial distribution (negative and positive correlation coefficients)

I have a dependent variable, which is a multinomial distribution resulting from positive and negative correlation coefficients. This is a study from the field of Dendrochronology. Those correlation coefficients represent the relationships between tree-growths (i.e. standardized tree-ring widths) and sums of precipitation. For example: There is tree-ring width chronology (TRWi from site 1) and there are summer precipitation in mm (for site 1). An example table is below. Calculated correlation coefficient is between TRWi and Precipitation variable. Longitude, latitude and altitude are constants for this specific site. Now, there are approximately 1000 different sites and in the first step, correlation coefficients are calculated for each site. At some sites, trees respond positively to higher precipitation sums, while for others this relationship is negative. After 1000 calculations, regression table looks like one below. Now, it is time for step 2 and I would like to model this variable as a function of longitudes, latitudes and altitudes.

What model would you suggest?

• Please expand the question to include more detail of where these correlation coefficients came from. – Robert Long Apr 24 '19 at 7:24
• I added additional information. I hope it is more clear now. – JerryTheForester Apr 24 '19 at 7:47
• Is there a reason why you don't model the ring widths as a response variable, with precipitation, longitude, latitude and altitude as explanatory variables ? – Robert Long Apr 24 '19 at 8:03
• This is actually very interesting suggestion. If you look more closely, how each correlation coefficient is calculated, you will note that latitudes, longitudes and altitudes are constant values in this equation (see Expanded question). – JerryTheForester Apr 24 '19 at 8:25
• I am not sure what you mean by "constant values". Can you post a link to the actual data ? What is your research question ? – Robert Long Apr 24 '19 at 8:37

$$Width=\beta_0+\beta_1Precipation+\beta_2altitude+\beta_3latitude+\beta_4longitude+\beta_5Precipation*altitude+\beta_6Precipation*latitude+\beta_7Precipation*longitude$$
If the relation between amount of rainfall and growth changes with say altitude that would be picked up by $$\beta_5$$.