[R-sig-Geo] CAR model related negative interaction variable and threshold

[Li, Han]

Dear list,

I am experiencing some difficulty with a negative interaction variable in my CAR models. Any insight will be highly appreciated. My CAR model result from R is:

Call: spautolm(formula = group ~ intersect * popu_dense, data = tabr, listw = tabr_neighbor_k1_max_wb, family = "CAR")

Residuals:
Min          1Q          Median   3Q         Max
-1.02934 -0.25278 -0.05858 0.29985 0.93158

Coefficients:
                                  Estimate      Std. Error       z value    Pr(>|z|)
(Intercept)                  5.9235e-02  3.7017e-02    1.6002    0.109552
intersect                     7.7819e-02  1.4418e-02    5.3974    6.762e-08
popu_dense                4.8762e-05  1.5287e-05    3.1898    0.001424
intersect:popu_dense -1.0898e-05  3.7201e-06   -2.9295    0.003395

Lambda: 0.025376 LR test value: 57.924 p-value: 2.72e-14

Log likelihood: -261.4561
ML residual variance (sigma squared): 0.15865, (sigma: 0.39831)
Number of observations: 516
Number of parameters estimated: 6
AIC: 534.91

In this model, the independent variable was the presence/absence of a bat species in urban environment (group:0/1). Two continuous explanatory variables were street intersection density (intersect, # of street intersections per area, range 0-10) and human population density (# of people per area, range 100-15000). I modeled these two variables as well as the interaction between them. From the result, both variables were significant positive variables. But the interaction term is significantly negative. Basically, when the population density increases, the positive effect of intersection density will decrease (or vice versa). As far as I understand, there should be a threshold for each variable. After passing the threshold, the positive effect of a variable would turn into a negative effect due to the interaction with the other variable.

My biggest question is (assuming my knowledge by far is correct) how to find out this threshold.
If this is a linear model, like z=a+b*x+c*y+m*x*y, I can calculate the thresholds for x as -m/c. The threshold for y is -m/b.
Can I apply these calculations to the coefficients in CAR model?

Another idea I had was to calculate the prediction value from CAR model and plot the prediction in a 3-D graph. So the threshold can be seen. But I read that it is not possible to get the prediction values from CAR model.

If anyone can suggest a way to either calculate the threshold or visually present the threshold in the prediction or just provide any thoughts/comments, thank you in advance.

Han

Han Li
Department of Biology
Baylor University
Waco, TX  76798-7388
Phone: (254) 710-2151
Fax: (254) 710-2969
han_li at baylor.edu<mailto:han_li at baylor.edu>