Slope index of inequality (SII) calculation

[Lode van der Velde]

Hi INLA community,

I am working in health inequality research and have recently fitted a spatio temporal model using R_INLA. I would like to calculation the slope index of inequality (SII) using the same model. This requires a continuous exposure variable and more importantly an identity link for the poisson family. The documentation indicates that this option doesn't exist (yet). 

SII <- inla(Y ~ 1 + midpoint + year_id +
                      f(area_id, model = "bym2", graph = nb_graph) +
                      f(area_id1, year_id, model = "iid"),
                    family = "poisson", data = df_SII, E = E,
                    control.family = list(control.link=list(model= "identity")))

Does anyone have experience with the SII in R-INLA, or fitting a model using an identity link?

Any help is much appreciated!

Kind regards, 

Lode van der Velde

Much appreciated!

[Finn Lindgren]

Hi,

from what I can tell from a chick googling, the SII measure isn't very clearly defined, and _clearly_ whoever came up with it didn't consider that a linear effect can be negative, so it's not clear how to define it in a way that makes sense.

A Poisson model cannot have an identity link when the coefficient distributions are Gaussian, as that would lead to a positive probability that the expectation of the Poisson model would be negative.

In a purely point-estimation frequentist context, it's enough if the point estimate doesn't lead to impossible results such as that, but in the Bayesian context it needs to be explicit, hence the need for the log-link, or some other non-negative (inverse)link function.

So you may need to consider what interpretation of SII you really want to compute. Also, assuming that "year" is the variable with respect you want to compute SII, the value will depend on the location, so perhaps one option is to do a linear basic least squares fit to the estimated predictor expression evaluated separately for each location (perhaps with some extra work to include the posterior uncertainty; posterior sampling, and linear regression on each sample would achieve that).

The effect of this approach would be to treat SII as the "average linear slope", which is defined also for nonlinear functions, such as the exp(linear) output from the Poisson model with log-link.

Whether this is a sensible approach depends on context and how variable/nonlinear the data are...

Finn

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Finn Lindgren
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