extend_family function

Extend a family

This function adds some internally required elements to an object of class family (see, e.g., family()). It is called internally by init_refmodel(), so you will rarely need to call it yourself.

extend_family(
  family,
  latent = FALSE,
  latent_y_unqs = NULL,
  latent_ilink = NULL,
  latent_ll_oscale = NULL,
  latent_ppd_oscale = NULL,
  augdat_y_unqs = NULL,
  augdat_link = NULL,
  augdat_ilink = NULL,
  augdat_args_link = list(),
  augdat_args_ilink = list(),
  ...
)

Arguments

• family: An object of class family.

• latent: A single logical value indicating whether to use the latent projection (TRUE) or not (FALSE). Note that setting latent = TRUE

  causes all arguments starting with augdat_ to be ignored.

• latent_y_unqs: Only relevant for a latent projection where the original response space has finite support (i.e., the original response values may be regarded as categories), in which case this needs to be the character vector of unique response values (which will be assigned to family$cats

  internally) or may be left at NULL (so that projpred will try to infer it from family$cats). See also section "Latent projection" below.

• latent_ilink: Only relevant for the latent projection, in which case this needs to be the inverse-link function. If the original response family was the binomial() or the poisson() family, then latent_ilink can be NULL, in which case an internal default will be used. Can also be NULL

  in all other cases, but then an internal default based on family$linkinv

  will be used which might not work for all families. See also section "Latent projection" below.

• latent_ll_oscale: Only relevant for the latent projection, in which case this needs to be the function computing response-scale (not latent-scale) log-likelihood values. If !is.null(family$cats) (after taking latent_y_unqs into account) or if the original response family was the binomial() or the poisson() family, then latent_ll_oscale can be NULL, in which case an internal default will be used. Can also be NULL

  in all other cases, but then downstream functions will have limited functionality (a message thrown by extend_family() will state what exactly won't be available). See also section "Latent projection" below.

• latent_ppd_oscale: Only relevant for the latent projection, in which case this needs to be the function sampling response values given latent predictors that have been transformed to response scale using latent_ilink. If !is.null(family$cats) (after taking latent_y_unqs

  into account) or if the original response family was the binomial() or the poisson() family, then latent_ppd_oscale can be NULL, in which case an internal default will be used. Can also be NULL in all other cases, but then downstream functions will have limited functionality (a message thrown by extend_family() will state what exactly won't be available). See also section "Latent projection" below. Note that although this function has the abbreviation "PPD" in its name (which stands for "posterior predictive distribution"), projpred currently only uses it in proj_predict(), i.e., for sampling from what would better be termed posterior-projection predictive distribution (PPPD).

• augdat_y_unqs: Only relevant for augmented-data projection, in which case this needs to be the character vector of unique response values (which will be assigned to family$cats internally) or may be left at NULL if family$cats is already non-NULL. See also section "Augmented-data projection" below.

• augdat_link: Only relevant for augmented-data projection, in which case this needs to be the link function. Use NULL for the traditional projection. See also section "Augmented-data projection" below.

• augdat_ilink: Only relevant for augmented-data projection, in which case this needs to be the inverse-link function. Use NULL for the traditional projection. See also section "Augmented-data projection" below.

• augdat_args_link: Only relevant for augmented-data projection, in which case this may be a named list of arguments to pass to the function supplied to augdat_link.

• augdat_args_ilink: Only relevant for augmented-data projection, in which case this may be a named list of arguments to pass to the function supplied to augdat_ilink.

• ...: Ignored (exists only to swallow up further arguments which might be passed to this function).

Returns

The family object extended in the way needed by projpred.

Details

In the following, $N$, $C_cat$, $C_lat$, $S_ref$, and $S_prj$ from help topic refmodel-init-get are used. Note that $N$ does not necessarily denote the number of original observations; it can also refer to new observations. Furthermore, let $S$

denote either $S_ref$ or $S_prj$, whichever is appropriate in the context where it is used.

Augmented-data projection

As their first input, the functions supplied to arguments augdat_link and augdat_ilink have to accept:

• For augdat_link: an c("$S x N x\n$", "$C_cat$") array containing the probabilities for the response categories. The order of the response categories is the same as in family$cats (see argument augdat_y_unqs).
• For augdat_ilink: an c("$S x N x\n$", "$C_lat$") array containing the linear predictors.

The return value of these functions needs to be:

• For augdat_link: an c("$S x N x\n$", "$C_lat$") array containing the linear predictors.
• For augdat_ilink: an c("$S x N x\n$", "$C_cat$") array containing the probabilities for the response categories. The order of the response categories has to be the same as in family$cats (see argument augdat_y_unqs).

For the augmented-data projection, the response vector resulting from extract_model_data (see init_refmodel()) is coerced to a factor (using as.factor()) at multiple places throughout this package. Inside of init_refmodel(), the levels of this factor have to be identical to family$cats (after applying extend_family() inside of init_refmodel()). Everywhere else, these levels have to be a subset of <refmodel>$family$cats (where <refmodel> is an object resulting from init_refmodel()). See argument augdat_y_unqs for how to control family$cats.

For ordinal brms families, be aware that the submodels (onto which the reference model is projected) currently have the following restrictions:

• The discrimination parameter disc is not supported (i.e., it is a constant with value 1).
• The thresholds are "flexible" (see brms::brmsfamily()).
• The thresholds do not vary across the levels of a factor-like variable (see argument gr of brms::resp_thres()).
• The "probit_approx" link is replaced by "probit".

For the brms::categorical() family, be aware that:

• For multilevel submodels, the group-level effects are allowed to be correlated between different response categories.
• For multilevel submodels, mclogit versions < 0.9.4 may throw the error 'a' (\<number\> x 1) must be square. Updating mclogit to a version >= 0.9.4 should fix this.

Latent projection

The function supplied to argument latent_ilink needs to have the prototype

latent_ilink(lpreds, cl_ref, wdraws_ref = rep(1, length(cl_ref)))

where:

• lpreds accepts an $S x N$ matrix containing the linear predictors.
• cl_ref accepts a numeric vector of length $S_ref$, containing projpred's internal cluster indices for these draws.
• wdraws_ref accepts a numeric vector of length $S_ref$, containing weights for these draws. These weights should be treated as not being normalized (i.e., they don't necessarily sum to 1).

The return value of latent_ilink needs to contain the linear predictors transformed to the original response space, with the following structure:

• If is.null(family$cats) (after taking latent_y_unqs into account): an $S x N$ matrix.
• If !is.null(family$cats) (after taking latent_y_unqs into account): an $S x N x C_cat$ array. In that case, latent_ilink needs to return probabilities (for the response categories given in family$cats, after taking latent_y_unqs into account).

The function supplied to argument latent_ll_oscale needs to have the prototype

latent_ll_oscale(ilpreds, y_oscale, wobs = rep(1, length(y_oscale)), cl_ref,
            wdraws_ref = rep(1, length(cl_ref)))

where:

• ilpreds accepts the return value from latent_ilink.
• y_oscale accepts a vector of length $N$ containing response values on the original response scale.
• wobs accepts a numeric vector of length $N$ containing observation weights.
• cl_ref accepts the same input as argument cl_ref of latent_ilink.
• wdraws_ref accepts the same input as argument wdraws_ref of latent_ilink.

The return value of latent_ll_oscale needs to be an $S x N$

matrix containing the response-scale (not latent-scale) log-likelihood values for the $N$ observations from its inputs.

The function supplied to argument latent_ppd_oscale needs to have the prototype

latent_ppd_oscale(ilpreds_resamp, wobs, cl_ref,
             wdraws_ref = rep(1, length(cl_ref)), idxs_prjdraws)

where:

• ilpreds_resamp accepts the return value from latent_ilink, but possibly with resampled (clustered) draws (see argument nresample_clusters of proj_predict()).

• wobs accepts a numeric vector of length $N$ containing observation weights.

• cl_ref accepts the same input as argument cl_ref of latent_ilink.

• wdraws_ref accepts the same input as argument wdraws_ref of latent_ilink.

• idxs_prjdraws accepts a numeric vector of length dim(ilpreds_resamp)[1]

  containing the resampled indices of the projected draws (i.e., these indices are values from the set c("{1, ...,\n", "dim(ilpreds)[1]}") where ilpreds denotes the return value of latent_ilink).

The return value of latent_ppd_oscale needs to be a $dim(ilpreds_resamp)[1] x N$

matrix containing the response-scale (not latent-scale) draws from the posterior(-projection) predictive distributions for the $N$ observations from its inputs.

If the bodies of these three functions involve parameter draws from the reference model which have not been projected (e.g., for latent_ilink, the thresholds in an ordinal model), cl_agg() is provided as a helper function for aggregating these reference model draws in the same way as the draws have been aggregated for the first argument of these functions (e.g., lpreds in case of latent_ilink).

In fact, the weights passed to argument wdraws_ref are nonconstant only in case of cv_varsel() with cv_method = "LOO" and validate_search = TRUE. In that case, the weights passed to this argument are the PSIS-LOO CV weights for one observation. Note that although argument wdraws_ref has the suffix _ref, wdraws_ref does not necessarily obtain weights for the initial

reference model's posterior draws: In case of cv_varsel() with cv_method = "kfold", these weights may refer to one of the $K$ reference model refits (but in that case, they are constant anyway).

If family$cats is not NULL (after taking latent_y_unqs into account), then the response vector resulting from extract_model_data (see init_refmodel()) is coerced to a factor (using as.factor()) at multiple places throughout this package. Inside of init_refmodel(), the levels of this factor have to be identical to family$cats (after applying extend_family() inside of init_refmodel()). Everywhere else, these levels have to be a subset of <refmodel>$family$cats (where <refmodel> is an object resulting from init_refmodel()).