Preferential sampling with both random and fixed effects

[Scott Foster]

Hi,

I'm trying to implement a model of preferential sampling.  The information I can find on-line is very helpful but I think falls a little short of where I would like to be.  I'm happy with the marked-point-process (two likelihoods) and I'm happy with using f(.,copy='i', .), but I'd also like to "copy" the fixed effects and have them share a common preference parameter.

This could be a little vague, let me try to firm things up a bit.

Let Y be the marks at sites X.  Model the intensity of the LGCP for X as:

log( E(X|u)) = \eta = X\tau+u

and the model for the marks as

g(E(Y|X,u)) = \alpha + \beta * \eta

                    =\alpha + \beta * X\tau + \beta * u

Currently, the examples online (Book https://becarioprecario.bitbucket.io/spde-gitbook/ch-lcox.html#model-fitting-under-preferential-sampling) show how to add the final term, but not the term with the fixed effects.

I'd settle for a model with a separate parameter for fixed and random contributions, viz

g(E(Y|X,u)) = \alpha + \beta_f * X\tau + \beta_r * u

with \beta_f != \beta_r, but it would be nice to have the option.

Any advice would be greatly appreciated.  I'm a bit lost right now.

Thanks in advance,

Scott

[Helpdesk (Haavard Rue)]

you need to zero out contributions you do not like when you define the
joint (or union) of the two models, either by 0's in the covariates, or
with idx[]=NA in f(idx,...)

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[Scott Foster]

Thanks for the response.  It is greatly appreciated.

However, I don't understand how to use it in relation to the proposed model.  I think that there might have been ambiguous communication about the model.  I'll try again (and then try to explain my confusion).

Consider a marked point process and let Y be the marks measured at sites X.  The conditional model for the marks is

g( E(Y|u)) = \eta_Y 

             = W_Y\tau + u_Y

where W_Y is a design matrix for the covariates at the sites X, \tau is a vector of coefficients and u_Y is a vector containing realisations of a Gaussian process at sites X.  This is 'just' a standard geostatistical model for the marks

The model for the point process allows for possible preferential sampling. It is 

log( E(X|u)) = \alpha + \beta*\eta 

             = \alpha + \beta*W_X\tau + \beta*u

where W_X is a design matrix (dimension M_X \times p) that depends upon the approximation to the Poisson point process (on- or off-grid, for example), \tau and u are as before, \alpha is a constant and \beta is a preference parameter.  If \beta>0 then the sites are associated with higher values of E(Y|u).

Please note that, since last email, I have swapped \beta and \alpha so that they now affect the locations and not the marks.  The same problem persists though (this is just a rescaling to make the preference more obvious).

Note that in the INLA call, the design matrix will be of dimension (|X|+M_X) \times p, to allow for the two outcomes and will consist of the two W matrices above being stacked.

Adding in the scalar parameter \alpha is easy, as it involves only adding a column of |X| zeros and |M_X| ones to the design matrix fed into INLA.

Adding \beta is where my confusion lies.  If it were known a priori then I could simply multiply the relevant part of the design matrix.  If it were additive, then I could add columns to the design matrix (like \alpha).  In the model for the locations, the design matrix retains the p columns but the coefficients are now \beta*\tau -- a 'compound' parameter (my made-up term).

So, I don't think that the model can be specified by altering the design matrix -- the terms of interest are 'compound' parameters, not additive.  The \beta scalar parameter also spans many covariates (columns of the design matrix).

I read the helpfile for f() to indicate that it applies only to single covariates, not multiple ones as needed here.

Am I missing something? Is this even possible as it is a non-linear model (just)?

Thanks again,

Scott

[Elias T. Krainski]

Hi,

In order to copy the entire linear predictor, you can define a "fake zero" observations, so that

 0 = W\tau + u - v

where v ~ N(0, bigVariance)

then you copy 'v' instead of 'u' for the LGCP part. Please check the section 3.3 in the book.

Elias

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