VARIABLES seems to be in reverse. Double check how you define X and Y. I did a crude plotting of the U in the attached spread sheet. (i) X = GeoDisp and (ii) Y = geoDisp. Both shows the same shape, but the variables are in reverse.

Y = a - bX2

Yi = (y1, y2, ..., yn) and Xi = (x1, x2, ..., xn). Constant = a. The attached plot in the spread sheet uses your Model-3 to run.

ATTACHED is an article on U-shape modeling. Cheers.

All the variables in Table 2 are Xs, and all of them explain the variable Y, which is named "Acquired" (see Table 1).

Therefore, the model is something like this: Acquired =B0+B1*GeoDisp+B2*GeoDisp^2+B3*GeodDisp*TechDiversification+B4*TechDiversification+B5*FirmSize+....+ Bn*FirmAge. The only thing is thatI did not use an OLS model but random-effects negative binomial one.

Does this help?

PS. Thak you very much for your answer.

All the variables in Table 2 are Xs, and all of them explain the variable Y, which is named "Acquired" (see Table 1).

Therefore, the model is something like this: Acquired =B0+B1*GeoDisp+B2*GeoDisp^2+B3*GeodDisp*TechDiversification+B4*TechDiversification+B5*FirmSize+....+ Bn*FirmAge. The only thing is thatI did not use an OLS model but random-effects negative binomial one.

Does this help?

PS. Thak you very much for your answer.

There is an example of Poisson/ Negative binomial with random effects in this

https://www.researchgate.net/publication/268331676_Vol2Traingmanual2014withminorcorrections

Developing multilevel models for analysing contextuality, heterogeneity and change using MLwiN, Volume 2Aug 2013 by Kelvyn Jones VS Subramanian

Data Vol2Traingmanual2014withminorcorrections

 Dear Kelvyn,

Thank you for the interesting document, I found it really useful. However, I still have doubts on how to plot interactions (Figure 4 in you document). I cannot understand  how to calculate the right values of the negative binomial distribution (based on my results) that can let me plot the graphs.

Lorenzo

I am sorry but I am not understanding what the problem is. I think that you are interested in fixed effects and so the NBD contribution will be zero for the subject- specific effect so the random effects can be ignored.

Consequently to make a plot - in a spreadsheet

1 create a set of data for  x1 and x2 like

x1 x2

1  1

2   1

3 1

4 1

1 2

2 2

3 2

4 3

etc

chose say the deciles of the real X1 and X2 values so you get 100 values in all

2 create the interaction of X1 times X2

3 for these 100 data points put in a typical value - say the median - for each and every other modelled predictor variable - and also a 1 for the Constant - so that you are making predictions at typical values

4 use the estimated fixed parameters to predict Yhat for these X variables - forget the random effect - that is zero

5 Take the log of Yhat and plot it against X1 while grouping (that is a separate line) on X2- )this will give a plot of the estimated count; it sometimes is easier to see if you plot against X2 and group on X1 

 This is all done fairly automatically in MLwin's customized predictions facility see Manual Supplement - this will also allow you to predict the population average value 

https://www.researchgate.net/publication/234015221_Manual_supplement_for_MLwiN_Version_2.14

Data Manual supplement for MLwiN Version 2.14