‘Along’ and ‘by’ dimensions

Each β^(m), m > 0, can be a main effect, involving a single dimension, or an interaction, involving two dimensions. Some priors, when applied to an interaction, treat one dimension, referred to as the ‘along’ dimension, differently from the remaining dimensions, referred to as ‘by’ dimensions. A random walk prior (Section @ref(sec:pr-rw)), for instance, consists of an independent random walk along the ‘along’ dimension, within each combination of the ‘by’ dimensions.

We use v = 1, ⋯, V_m to denote position within the ‘along’ dimension, and u = 1, ⋯, V_m to denote position within a classification formed by the ‘by’ dimensions. When there are no sum-to-zero constraints (see below), U_m = ∏_kd_k where d_k is the number of elements in the kth ‘by’ variable. When there are sum-to-zero constraints, U_m = ∏_k(d_k − 1).

If a prior involves an ‘along’ dimension but the user does not specify one, the procedure for choosing a dimension is as follows:

• if the term involves time, use the time dimension;
• otherwise, if the term involves age, use the age dimension;
• otherwise, raise an error asking the user to explicitly specify a dimension.

Sum-to-zero constraints

Priors that involve ‘along’ dimensions can optionally include sum-to-zero constraints. If these constrains are applied, then within each element v of the ‘along’ dimension, the sum of the β_j^(m) across each ‘by’ dimension is zero. For instance, if β^(m) is an interaction between time, region, and sex, with time as the ‘along’ variable, then within each combination of time and region, the values for females and males sum to zero, and within each combination of time and sex, the values for regions sum to zero.

Except in the case of dynamic SVD-based priors (eg Sections @ref(sec:pr-svd-rw)), the sum-to-zero constraints are implemented internally by drawing values within an unrestricted lower-dimensional space, and then transforming to the restricted higher-dimensional space. For instance, a random walk prior for a time-region interaction with R regions consists of R − 1 unrestricted random walks along time, which are converted into R random walks that sum to zero across region. Matrices for transforming between the unrestricted and restricted spaces are constructed using the QR decomposition, as described in Section 1.8.1 of Wood (2017). With dynamic SVD-based priors, we draw values for the SVD coefficients with no constraints, convert these to unconstrained values for β^(m), and then subtract means.

Algorithm for assigning default priors

• If β^(m) has one or two elements, assign β^(m) a fixed-normal prior (Section @ref(sec:pr-fnorm));
• otherwise, if β^(m) involves time, assign β^(m) a random walk prior (Section @ref(sec:pr-rw)) along the time dimension;
• otherwise, if β^(m) involves age, assign β^(m) a random walk prior (Section @ref(sec:pr-rw)) along the age dimension;
• otherwise, assign β^(m) a normal prior (Section @ref(sec:pr-norm))

The intercept term β⁽⁰⁾ can only be given a fixed-normal prior (Section @ref(sec:pr-fnorm)) or a Known prior (Section @ref(sec:pr-known)).

Model

Exchangeable normal

Contribution to posterior density

Forecasting

Code

N(s = 1)

• s is A_τ^(m). Defaults to 1.

Model

Exchangeable normal, with fixed standard deviation

Contribution to posterior density

Forecasting

Code

NFix(sd = 1)

• sd is A_τ^(m). Defaults to 1.

Model

Random walk

A₀^(m) can be 0, implying that β_u, 1^(m) is fixed at 0.

When U_m > 1, sum-to-zero constraints (Section @ref(sec:sum-to-zero)) can be applied.

Contribution to posterior density

Forecasting

If the prior includes sum-to-zero constraints, means are subtracted from the forecasted values within each combination of ‘along’ and ‘by’ variables.

Code

RW(s = 1, along = NULL, zero_sum = TRUE)

• s is A_τ^(m). Defaults to 1.
• sd is A₀^(m). Defaults to 1.
• along used to identify ‘along’ and ‘by’ dimensions.
• if zero_sum is TRUE, sum-to-zero constraints are applied.

Model

Second-order random walk

A₀^(m) can be 0, implying that β_u, 1^(m) is fixed at 0.

When U_m > 1, sum-to-zero constraints (Section @ref(sec:sum-to-zero)) can be applied.

Contribution to posterior density

Forecasting

If the prior includes sum-to-zero constraints, means are subtracted from the forecasted values within each combination of ‘along’ and ‘by’ variables.

Code

RW2(s = 1, sd = 1, sd_slope = 1, along = NULL, zero_sum = TRUE)

• s is A_τ^(m)
• sd is A₀^(m)
• sd_slope is A_η^(m)
• along used to identify ‘along’ and ‘by’ dimensions
• if zero_sum is TRUE, sum-to-zero constraints are applied

Model

Random walk with seasonal effect

A₀^(m) can be 0, implying that α_u, 1^(m) is fixed at 0.

A_ω^(m)2 can be set to zero, implying that seasonal effects are fixed over time.

When U_m > 1, sum-to-zero constraints (Section @ref(sec:sum-to-zero)) can be applied.

Contribution to posterior density

Forecasting

Code

RW_Seas(n_seas, s = 1, sd = 1, s_seas = 0, sd_seas = 1, along = NULL, zero_sum = FALSE)

• n_seas is S_m
• s is A_τ^(m)
• sd is A₀^(m)
• s_seas is A_ω^(m)
• sd_seas is A_λ^(m)
• along used to identify ‘along’ and ‘by’ dimensions
• if zero_sum is TRUE, sum-to-zero constraints are applied

Model

Second-order random work, with seasonal effect

A₀^(m) can be 0, implying that α_u, 1^(m) is fixed at 0.

A_ω^(m)2 can be set to zero, implying that seasonal effects are fixed over time.

When U_m > 1, sum-to-zero constraints (Section @ref(sec:sum-to-zero)) can be applied.

Contribution to posterior density

Forecasting

Code

RW2_Seas(n_seas, s = 1, sd = 1, sd_slope = 1, s_seas = 0, sd_seas = 1, along = NULL, zero_sum = FALSE)

• n_seas is S_m
• s is A_τ^(m)
• sd is A₀^(m)
• sd_slope is A_η^(m)
• s_seas is A_ω^(m)
• sd_seas is A_λ^(m)
• along used to identify ‘along’ and ‘by’ dimensions
• if zero_sum is TRUE, sum-to-zero constraints are applied

Model

Internally, TMB derives values for β_u, v^(m), v = 1, ⋯, K_m, and for ω_m, that imply a stationary distribution, and that give every term β_u, v^(m) the same marginal variance. We denote this marginal variance τ_m², and assign it a prior Each of the ϕ_k^(m) has prior

Contribution to posterior density

where p(β_u, 1^(m), ⋯, β_u, Vm^(m) ∣ ϕ₁^(m), ⋯, ϕ_Km^(m), τ_m) is calculated internally by TMB.

Forecasting

Code

AR(n_coef = 2,
   s = 1,
   shape1 = 5,
   shape2 = 5,
   along = NULL,
   zero_sum = FALSE)

• n_coef is K_m
• s is A_τ^(m)
• shape1 is S₁^(m)
• shape2 is S₂^(m)
• along is used to indentify the ‘along’ and ‘by’ dimensions

Model

This is adapted from the specification used for AR1 densities in TMB. It implies that the marginal variance of all β_u, v^(m) is τ_m². We require that −1 < a_0m < a_1m < 1.

Contribution to posterior density

Forecasting

Code

AR1(s = 1,
    shape1 = 5,
    shape2 = 5,
    min = 0.8,
    max = 0.98,
    along = NULL,
    zero_sum = TRUE)

• s is A_τ^(m)
• shape1 is S₁^(m)
• shape2 is S₂^(m)
• min is a_0m
• max is a_1m
• along is used to identify ‘along’ and ‘by’ dimensions

The defaults for min and max are based on the defaults for function ets() in R package forecast (Hyndman and Khandakar 2008).

Model

Note that $\sum_{v=1}^{V_m} \alpha_{u,v}^{(m)} = 0$.

Contribution to posterior density

Forecasting

Code

Lin(s = 1, mean_slop = 0, sd_slope = 1, along = NULL, zero_sum = FALSE)

• s is A_τ^(m)
• mean_slope is B_η^(m)
• sd_slope is A_η^(m)
• along is used to indentify ‘along’ and ‘by’ dimensions

Model

Note that $\sum_{v=1}^{V_m} \alpha_{u,v}^{(m)} = 0$.

Internally, TMB derives values for ϵ_u, v^(m), v = 1, ⋯, K_m, and for ω_m, that provide the ϵ_u, v^(m) with a stationary distribution in which each term has the same marginal variance. We denote this marginal variance τ_m², and assign it a prior Each of the individual ϕ_k^(m) has prior

Contribution to posterior density

where p(ϵ_u, 1^(m), ⋯, ϵ_u, Vm^(m) ∣ ϕ₁^(m), ⋯, ϕ_Km^(m), τ_m) is calculated internally by TMB.

Forecasting

Code

Lin_AR(n_coef = 2,
       s = 1,
       shape1 = 5,
       shape2 = 5,
       mean_slope = 0,
       sd_slope = 1,
       along = NULL,
       zero_coef = FALSE)

• n_coef is K_m
• s is A_τ^(m)
• shape1 is S₁^(m)
• shape2 is S₂^(m)
• mean_slope is B_η^(m)
• sd_slope is A_η^(m)
• along is used to indentify ‘along’ and ‘by’ variables

Model

Penalised spline (P-spline)

where β_u^(m) is the subvector of β^(m) composed of elements from the uth combination of the ‘by’ variables, B^(m) is a V_m × K_m matrix of B-splines, and α_u^(m) has a second-order random walk prior (Section @ref(sec:pr-rw2)).

B^(m) = (b₁^(m)(v), ⋯, b_Km^(m)(v)), with v = (1, ⋯, V_m)^⊤. The B-splines are centered, so that 1^⊤b_k^(m)(v) = 0, k = 1, ⋯, K_m.

Contribution to posterior density

Forecasting

Terms with a P-Spline prior cannot be forecasted.

Code

Sp(n = NULL, s = 1)

• n is K_m. Defaults to max (0.7J_m, 4).
• s is the A_τ^(m) from the second-order random walk prior. Defaults to 1.
• along is used to identify ‘along’ and ‘by’ variables

Model

Age but no sex or gender

Let β_u be the age effect for the uth combination of the ‘by’ variables. With an SVD prior, where F^(m) is a V_m × K_m matrix, and g^(m) is a vector with V_m elements, both derived from a singular value decomposition (SVD) of an external dataset of age-specific values for all sexes/genders combined. The construction of F^(m) and g^(m) is described in Appendix @ref(app:svd). The centering and scaling used in the construction allow use of the simple prior

Joint model of age and sex/gender

In the joint model, vector β_u represents the interaction between age and sex/gender for the uth combination of the ‘by’ variables. Matrix F^(m) and vector g^(m) are calculated from data that separate sexes/genders. The model is otherwise unchanged.

Independent models for each sex/gender

In the independent model, vector β_s, u represents age effects for sex/gender s and the uth combination of the ‘by’ variables, and we have Matrix F_s^(m) and vector g_s^(m) are calculated from data that separate sexes/genders. The prior is

Contribution to posterior density

for the age-only and joint models, and for the independent model

Forecasting

Terms with an SVD prior cannot be forecasted.

Code

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

where - ssvd is an object containing F and g - n_comp is the number of components to be used (which defaults to ceiling(n/2), where n is the number of components in ssvd - indep determines whether and independent or joint model will be used if the term being modelled contains a sex or gender variable.

Model

The SVD_RW() prior is identical to the SVD() prior except that the coefficients evolve over time, following independent random walks. For instance, in the combined-sex/gender and joint models with K_m SVD components,

Contribution to posterior density

In the combined-sex/gender and joint models,

and in the independent model,

Forecasting

Code

SVD_RW(ssvd,
       n_comp = NULL,
       indep = TRUE,
       s = 1,
       sd = 1,
       zero_sum = FALSE)

where - ssvd is an object containing F and g - n_comp is K_m - indep determines whether and independent or joint model will be used if the term being modelled contains a sex or gender variable. - s is A_τ^(m) - sd is A₀^(m)

Model

Elements of β^(m) are treated as known with certainty.

Contribution to posterior density

Known priors make no contribution to the posterior density.

Forecasting

Main effects with a known prior cannot be forecasted.

Code

Known(values)

• values is a vector containing the β_j^(m).

Standard prior

Shrinkage prior

Regularized horseshoe prior (Piironen and Vehtari 2017)

where p₀ is an initial guess at the number of ζ_p that are non-zero, and σ̂ is obtained as follows:

• Poisson. Using maximum likelihood, fit the GLM and set
• Binomial. Using maximum likelihood, fit the GLM and set
• Normal. Using maximum likelihood, fit the linear model and set σ̂ = ξ̂.

The quantities used for Poisson and binomial likelihoods are derived from normal approximations to GLMs (Piironen and Vehtari 2017; Gelman et al. 2014, sec. 16.2).

Random Rounding to Base 3

Random rounding to base 3 (RR3) is a confidentialization method used by some statistical agencies. It is applied to counts data. Each count x is rounded randomly as follows:

• If x mod  3 = 0, then x is left unchanged;
• if x mod  3 = 1 then x is changed to x − 1 with probability 2/3, and is changed to x + 2 with probability 1/3; and
• if x mod  3 = 2 then x is changed to x − 2 with probability 1/3, and is changed to x + 1 with probability 2/3.

RR3 data models can be used with Poisson or binomial likelihoods. Let y_i denote the observed value for the outcome, and y_i^* the true value. The likelihood with a RR3 data model is then

Step 0: Select ‘inner’ and ‘outer’ variables

Select variables to be used in inner model. By default, these are the age, sex, and time variables in the model. All remaining variables are ‘outer’ variables.

Step 1: Fit inner model

Aggregate the data using the classification formed by the inner variables. (See Section @ref(sec:filter-ag) on aggregation procedures.) Remove all terms not involving ‘inner’ variables, other than the intercept term, from the model. Set dispersion to 0. Fit the resulting model.

Step 2: Fit outer model

Let μ̂_iⁱⁿ be point estimates for the linear predictor μ_i obtained from the inner model.

Poisson model

Aggregate the data using the classification formed by the outer variables. Remove all terms involving the ‘inner’ variables, plus the intercept, from the model. Set dispersion to 0. Set exposure to w_i^out = μ̂ⁱⁿw_i. Fit the model.

Binomial model

Fit the original model, but set dispersion to 0, and for all terms from the ‘inner’ model, use Known priors using point estimates from the inner model.

Normal model

Aggregate the data using the classification formed by the outer variables. Remove all terms involving the ‘inner’ variables, plus the intercept, from the model. Set the outcome variable to to $y_i^{} = y_i - ^{}. Fit the model.

Step 4: Concatenate estimates

Concatenate posterior distributions for the inner terms from the inner model to posterior distributions for the outer terms from the outer model.

Step 5: Calculate dispersion

If the original model includes a dispersion term, then estimate dispersion. Let μ̂_i^comb be point estimates for the linear predictor obtained from the concatenated estimates.

Poisson model

Use the original disaggregated data, or, if the original data contains more then 10,000 rows, select 10,000 rows at random from the original data. Remove all terms from the original model except for the intercept. Set exposure to w_i^out = μ̂^combw_i.

Binomial model

Fit the the original model, but with all terms except the intercept having Known priors, where the values are obtained from point estimates from the concatenated estimates.

Normal model

Use the original disaggregated data, or, if the original data contains more then 10,000 rows, select 10,000 rows at random from the original data. Remove all terms from the original model except for the intercept. Set the outcome to y_i^out = y_i − μ̂^comb.

Poisson likelihood

Binomial likelihood

Normal likelihood

Data models for outcomes

If the overall model includes a data model for the outcome, then a further set of draws is made, deriving values for the observed outcomes, given values for the true outcomes.