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I am trying to figure out the inner workings of the mob function in the party package. I can't figure out how the splitting variable is selected when it is a categorical variable.

In the publications by the authors, they say that it is done by a chisquare test of association between the residual deviances and the categorical variable. I can't see how this works.

In the toy example below, the mob function works fine. The variable X is the splitting variable.

b <- (1:1000) / 100 - 5
a <- c(b,-b)
b <- c(b, b)
X <- as.factor(c(rep(FALSE, 1000), rep(TRUE, 1000)))
mob(a ~ b |  X)

However, if I fit a linear model without splitting, there is (by construction) no association between the residuals and the values of the splitting variable:

res <- sign(residuals(lm(a ~ b)))
table(res, X)

The resulting table is:

X
res  FALSE TRUE 
-1   499  500
1    501  500

This table does not show that the X variable is important, still the algorithm can figure it out. How is that done?

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The parameter stability tests in mob() are based not only on residuals but the entire score (aka gradient aka estimating function). For a linear regression model with an intercept, the residual is the first component of the score but there are additional components for each regressor (= residual * regressor). In your example, the second score has the information because only the slope but not the intercept is affected by the change:

library("sandwich")
res <- sign(estfun(lm(a ~ b))[,2])
table(res, X)

which yields

    X
res  FALSE TRUE
  -1     0  999
  0      1    1
  1    999    0

Two further comments: (1) The test not only uses the sign of the score, though, but a full chi-squared statistics. (2) A more flexible and improved reimplementation of mob() is available in package partykit which also provides dedicated functions lmtree() and glmtree().

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