# p-value of multivariate response in partykit::ctree

I wonder if anyone can help me to understand the two questions regarding partykit::ctree:

1. what's the difference between "quadratic" and "maximum" in ctree_control(teststat = c("quadratic", "maximum"))? When should I use one vs the other?

2. For multivariate response, what's the meaning of p-value at each splitting node, separately for regression and classification? Here is one example for classification:

library(partykit)
airq <- subset(airquality, !is.na(Ozone))
airct2 <- ctree(Ozone + Temp ~ ., data = airq)
airct2
plot(airct2)

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As for the question "quadratic" vs. "maximum"`: In the case with two numeric response variables, the former will typically be more powerful if the mean of both response variables differs for each split found in the tree. In contrast, if there are many splits where only one or the other mean of the responses changes but the other mean remains constant, the maximum test may have somewhat better power. Of course, in practice it is typically unkown (before fitting the tree) which situation you are in.