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SVD-Based Prior for Age or Age-Sex Profiles

Source: R/bage_prior-constructors.R
SVD.Rd

Use components from a Singular Value Decomposition (SVD) to model a main effect or interaction involving age.

Usage

SVD(ssvd, n_comp = NULL, indep = TRUE)

Arguments

ssvd

Object of class "bage_ssvd" holding a scaled SVD. See below for scaled SVDs of databases currently available in bage.

n_comp

Number of components from scaled SVD to use in modelling. The default is half the number of components of ssvd.

indep

Whether to use separate or combined SVDs in terms involving sex or gender. Default is TRUE. See below for details.

Value

An object of class "bage_prior_svd".

Details

A SVD() prior assumes that the age, age-sex, or age-gender profiles for the quantity being modelled looks like they were drawn at random from an external demographic database. For instance, the prior obtained via

SVD(HMD)

assumes that age or age-sex profiles look like they were drawn from the Human Mortality Database.

If SVD() is used with an interaction involving variables other than age and sex/gender, separate profiles are constructed within each combination of other variables.

bage chooses the appropriate age-specific or age-sex-specific SVD values internally. The choice depends on the model term that the SVD() prior is applied to, and on the age labels used in data argument to mod_pois(), mod_binom() or mod_norm(). bage makes its choice when set_prior() is called.

Joint or independent SVDs

Two possible ways of extracting patterns from age-sex-specific data are

  1. carry out separate SVDs on separate datasets for each sex/gender; or

  2. carry out a single SVD on dataset that has separate entries for each sex/gender.

Option 1 is more flexible. Option 2 is more robust to sampling or measurement errors. Option 1 is obtained by setting the joint argument to FALSE. Option 1 is obtained by setting the indep argument to TRUE. The default is TRUE.

Mathematical details

Case 1: Term involving age and no other variables

When SVD() is used with an age main effect,

$$\pmb{\beta} = \pmb{F} \pmb{\alpha} + \pmb{g},$$

where

  • \(\pmb{\beta}\) is a main effect or interaction involving age;

  • \(J\) is the number of elements of \(\pmb{\beta}\);

  • \(n\) is the number of components from the SVD;

  • \(\pmb{F}\) is a known matrix with dimension \(J \times n\); and

  • \(\pmb{g}\) is a known vector with \(J\) elements.

\(\pmb{F}\) and \(\pmb{g}\) are constructed from a large database of age-specific demographic estimates by performing an SVD and standardizing.

The elements of \(\pmb{\alpha}\) have prior $$\alpha_k \sim \text{N}(0, 1), \quad k = 1, \cdots, K.$$

Case 2: Term involving age and non-sex, non-gender variable(s)

When SVD() is used with an interaction that involves age but that does not involve sex or gender,

$$\pmb{\beta}_u = \pmb{F} \pmb{\alpha}_u + \pmb{g},$$

where

  • \(\pmb{\beta}_u\) is a subvector of \(\pmb{\beta}\) holding values for the \(u\)th combination of the non-age variables;

  • \(V\) is the number of elements of \(\pmb{\beta}_u\);

  • \(n\) is the number of components from the SVD;

  • \(\pmb{F}\) is a known matrix with dimension \(V \times n\); and

  • \(\pmb{g}\) is a known vector with \(V\) elements.

Case 3: Term involving age, sex/gender, and no other variables

When SVD() is used with an interaction that involves age and sex or gender, there are two sub-cases, depending on the value of indep.

When indep is TRUE,

$$\pmb{\beta}_{s} = \pmb{F}_s \pmb{\alpha}_{s} + \pmb{g}_s,$$

and when indep is FALSE,

$$\pmb{\beta} = \pmb{F} \pmb{\alpha} + \pmb{g},$$

where

  • \(\pmb{\beta}\) is an interaction involving age and sex/gender;

  • \(\pmb{\beta}_{s}\) is a subvector of \(\pmb{\beta}\), holding values for sex/gender \(s\);

  • \(J\) is the number of elements in \(\pmb{\beta}\);

  • \(S\) is the number of sexes/genders;

  • \(n\) is the number of components from the SVD;

  • \(\pmb{F}_s\) is a known \((J/S) \times n\) matrix, specific to sex/gender \(s\);

  • \(\pmb{g}_s\) is a known vector with \(J/S\) elements, specific to sex/gender \(s\);

  • \(\pmb{F}\) is a known \(J \times n\) matrix, with values for all sexes/genders; and

  • \(\pmb{g}\) is a known vector with \(J\) elements, with values for all sexes/genders.

The elements of \(\pmb{\alpha}_s\) and \(\pmb{\alpha}\) have prior $$\alpha_k \sim \text{N}(0, 1).$$

Case 4: Term involving age, sex/gender, and other variable(s)

When SVD() is used with an interaction that involves age, sex or gender, and other variables, there are two sub-cases, depending on the value of indep.

When indep is TRUE,

$$\pmb{\beta}_{u,s} = \pmb{F}_s \pmb{\alpha}_{u,s} + \pmb{g}_s,$$

and when indep is FALSE,

$$\pmb{\beta}_u = \pmb{F} \pmb{\alpha}_u + \pmb{g},$$

where

  • \(\pmb{\beta}\) is an interaction involving sex/gender;

  • \(\pmb{\beta}_{u,s}\) is a subvector of \(\pmb{\beta}\), holding values for sex/gender \(s\) for the \(u\)th combination of the other variables;

  • \(V\) is the number of elements in \(\pmb{\beta}_u\);

  • \(S\) is the number of sexes/genders;

  • \(n\) is the number of components from the SVD;

  • \(\pmb{F}_s\) is a known \((V/S) \times n\) matrix, specific to sex/gender \(s\);

  • \(\pmb{g}_s\) is a known vector with \(V/S\) elements, specific to sex/gender \(s\);

  • \(\pmb{F}\) is a known \(V \times n\) matrix, with values for all sexes/genders; and

  • \(\pmb{g}\) is a known vector with \(V\) elements, with values for all sexes/genders.

Scaled SVDs of demographic databases in bage

References

  • For details of the construction of scaled SVDS see the vignette here.

See also

Examples

SVD(HMD) 
#>   SVD(HMD) 
#>       ssvd: HMD
#>     n_comp: 3
SVD(HMD, n_comp = 3)
#>   SVD(HMD) 
#>       ssvd: HMD
#>     n_comp: 3