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For concreteness:

library( mgcv )
set.seed( 1 )
RawData <- data.frame( y = rbinom( 1000, 1, 0.5 ), x1 = rnorm( 1000 ),
                      x2 = as.factor( rbinom( 1000, 1, 0.5 ) ), x3 = rnorm( 1000 ),
                      x4 = as.factor( rbinom( 1000, 1, 0.5 ) ) )
fit <- gam( y ~ s( x1 ) + x2 + s( x3, by = x2 ) + x4, data = RawData, 
            family = nb( link = log ) )

How to measure the importance of these four variables?

I understand that "variable importance" is not a well-defined concept, so I am looking for the most straightforward way, such as an explained variance approach.

The ANOVA table seems to be a natural choice, however, as explained in this answer, it is not working: for the smooth terms in GAM models they do not have an explained variance interpretation.

What is the sound approach then?

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  • $\begingroup$ What is importance here? Statistical significance? $\endgroup$
    – AdamO
    Commented Nov 15, 2017 at 22:53
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    $\begingroup$ Many of these techniques in statistical learning use other metrics for variable importance; change in MSE when included in a random forest model bag sample, R2, or GCV score in MARS models. As there isn't a clear statistic that measures this that is routinely output by mgcv, it would help to think how might you otherwise measure variable importance, even conceptually or at a high level. This might indicate how to proceed with computing something you can use practically to answer your question. $\endgroup$ Commented Nov 17, 2017 at 18:14
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    $\begingroup$ @GavinSimpson Sure, I happily share you my original problem that motivated this question. Let x3 be the age of a patient, x2 be the sex, x1 be his/her blood pressure, x4 be whether he/she received a certain drug, and y be the number of times a certain event happened to him/her. The question the doctors ask: "OK, I understand blood pressure is significant, but we have a very high sample size, so it doesn't mean a lot, I'd be more interested to see how it compares to the other predictors in explaining y". $\endgroup$ Commented Nov 18, 2017 at 14:59
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    $\begingroup$ Yeah, I get that, but for you, what would be a good a good way to measure "explaining y"? Would a loss function work, and you could compare the Poisson loss of models with and without a particular covariate? You'd need to decide whether to reevaluate the other smooths or keep them fixed at their smoothness parameter estimates from the full model? $\endgroup$ Commented Nov 18, 2017 at 16:33
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    $\begingroup$ @Gavin Simpson : Very, very good questions. I'd really appreciate to read a review of the possibilities, as honestly, I've no idea what would work, and what decisions are meaningful in the scenario I described. $\endgroup$ Commented Nov 18, 2017 at 22:52

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