Add coefficient restrictions to a refinement workflow

Fix selected risk-factor levels at user-supplied relativities before the refined pricing GLM is fitted. This can be appropriate when sampling variation produces an implausible local effect, when an actuarial assumption is supported by additional information, or when a documented tariff constraint must be applied consistently.

Usage

add_restriction(
  model,
  restrictions,
  allow_new_levels = TRUE,
  allow_new_risk_factors = FALSE,
  replaces = NULL
)

Arguments

model: Object of class rating_refinement, created with prepare_refinement(). A fitted GLM, including a model returned by refit(), is not accepted directly; retain and modify the corresponding refinement specification instead.
restrictions: Data frame with exactly two columns. The first column must have the same name as the risk factor to restrict and contains the levels to adjust. This can also be the output_variable from an earlier add_relativities() step. The second column contains the replacement relativities. Levels that are not supplied are fixed at their current effective relativities.
allow_new_levels: Logical. If TRUE (default), restrictions may contain levels that were not observed in the model data. Their supplied relativities are treated as explicit tariff assumptions rather than model estimates. If FALSE, an unknown level results in an error.
allow_new_risk_factors: Logical. If FALSE (default), the first column of restrictions must identify a variable included in the fitted GLM or a tariff factor created by an earlier refinement step. Set this to TRUE to add an external variable that is present in the refinement data but absent from both the model and preceding refinement steps. All observed levels must then have supplied relativities, which are treated as fixed tariff assumptions.
replaces: NULL (default) or a character string naming an existing standalone model term that the new fixed risk factor replaces. During refit(), this term is removed before the restricted relativity column is added. Supplying replaces also provides the explicit opt-in required for a new risk factor; allow_new_risk_factors = TRUE is then unnecessary. Existing terms used in transformations or interactions cannot be replaced through this argument.

Value

A rating_refinement object containing the stored restriction specification. The pricing GLM is not fitted again until refit() is called.

Details

add_restriction() stores a restriction step on a rating_refinement object. It does not alter the fitted GLM immediately. The restriction is evaluated in the recorded step order and applied when refit() is called. Retain the refinement object when reviewing or revising the specification.

The restrictions data frame identifies the risk factor to restrict by its first column. This may be a variable from the original GLM or a tariff factor created by an earlier refinement step. The second column contains the relativities used for those levels in the refined model.

Actuarial interpretation

The restriction table may contain all levels of the model variable, or only the levels that need a manual adjustment. If only a subset is supplied, the missing levels are automatically filled with their current effective relativities at that point in the refinement workflow. These may be the original fitted GLM relativities or values produced by preceding refinement steps. This makes it possible to change one level explicitly while fixing all other levels at their current values.

Levels that were not observed when the GLM was fitted can also be supplied. Such a level has no coefficient estimate from the model data. Its relativity is therefore an explicit tariff assumption, for example based on expert judgement, external experience or a planned extension of the tariff. Existing levels that are not supplied remain fixed at their fitted relativities.

With allow_new_levels = TRUE, which is the default, these new tariff levels are retained in the refinement metadata and subsequently shown by rating_table(). An informational message identifies every newly added level, its supplied relativity and the fact that it was not observed in the model data. Set allow_new_levels = FALSE when the restriction table should be checked strictly against the levels observed by the fitted model, for example to detect spelling errors in level names.

When a newly supplied level is written as a numeric interval, the function checks it against the existing interval levels. A warning is issued if the new interval overlaps the current classification. The level is still retained because overlapping labels may occasionally be intentional, but they do not form an unambiguous tariff partition. To replace an interval classification, first add the complete new classification as a separate column in the refinement data, use that column as the first column of restrictions, and identify the old model term with replaces.

A variable that is present in the refinement data but was not included in the fitted GLM can be added with allow_new_risk_factors = TRUE. In that case all observed levels must have a supplied relativity. The new factor is applied as a fixed tariff factor during refit(); its effects are not estimated from the model data. This can be appropriate when an external classification or expert assumption must be incorporated, such as a hail zone derived from geographic information.

allow_new_risk_factors does not create the portfolio variable itself. The refinement data must already contain a column assigning every observation to a level. This is required to apply the supplied relativities to individual records.

Replacing an existing model variable

A new fixed tariff factor can either supplement the fitted GLM or replace an existing model variable. Supply replaces when the new factor represents an alternative tariff classification for an effect already present in the model. During refit(), the named existing term is removed and the supplied fixed relativities are inserted in its place. With replaces = NULL, the new factor is added alongside the existing model terms, which preserves the previous behaviour.

Supplying replaces is itself an explicit request to add the new risk factor, so allow_new_risk_factors = TRUE does not also need to be supplied. The replacement relationship is retained in the ordered refinement metadata and is shown by print(), summary() and audit_refinement(). This makes clear that the new factor substitutes for an earlier model effect rather than adding further multiplicative differentiation.

replaces is intentionally limited to a standalone main-effect term in the current refinement formula. A variable used in an interaction or transformed expression cannot be removed unambiguously through this argument. Such model structures should be revised explicitly before the refinement is prepared. This argument is therefore not a general-purpose facility for deleting model terms.

Updating an existing restriction

A later call to add_restriction() for the same risk factor and the same restricted model variable updates the restriction already stored in the refinement. Relativities supplied in the later call replace the previously stored values for those levels. Restrictions for levels that are not supplied again are retained.

The existing and new values are first combined and the resulting restriction table is then validated as one specification. This is useful when an actuarial assumption is revised during model refinement: only the affected levels need to be supplied again, while the remaining tariff assumptions stay unchanged. The restriction step keeps its original position in the workflow, so subsequent steps such as add_relativities() use the revised restricted coefficients.

The second column must retain the same name when an existing restriction is updated, because that name identifies the restricted model variable used by refit(). A message reports levels whose previously supplied relativity is changed.

Restricting a factor created by add_relativities()

An output_variable introduced by an earlier add_relativities() step is already part of the ordered refinement specification. It is therefore not treated as a new external risk factor and does not require allow_new_risk_factors = TRUE. add_restriction() identifies the preceding relativity step from its stored metadata and replaces the corresponding derived tariff effect during refit().

When only one level of such a refined variable is supplied, that level receives the new relativity and every other level is fixed at the relativity produced by add_relativities(). Mathematically, the resulting restriction therefore covers all current levels. Only the explicitly supplied level changes. This is useful when actuarial review supports a local adjustment but the remaining expert split should not be re-estimated.

Refinement order remains material. A restriction added after add_relativities() operates on the derived split relativities. A restriction added before add_relativities() instead changes the coefficient basis from which the split is derived.

Author

Martin Haringa

Examples

portfolio <- data.frame(
  claims = c(1, 2, 1, 3, 2, 4),
  exposure = rep(1, 6),
  postal_area = factor(c("A", "B", "C", "A", "B", "C"))
)

model <- glm(
  claims ~ postal_area + offset(log(exposure)),
  family = poisson(),
  data = portfolio
)

restrictions <- data.frame(
  postal_area = c("C", "D"),
  relativity = c(1.10, 1.20)
)

refined <- prepare_refinement(model, data = portfolio) |>
  add_restriction(restrictions)
#> Added new level `D` to risk factor `postal_area` with relativity 1.2. This level was not observed in the model data.

# Postal area D was not observed in the portfolio. Its relativity is an
# explicit tariff assumption and becomes available after refitting.
refined_model <- refit(refined)
rating_table(refined_model, exposure = FALSE)
#>   risk_factor       level est_refined_model
#> 1 (Intercept) (Intercept)          2.096774
#> 2  relativity           D          1.200000
#> 3  relativity           C          1.100000
#> 4  relativity           A          1.000000
#> 5  relativity           B          1.000000

# A factor absent from the fitted GLM can replace an existing model term.
# The portfolio must already assign every observation to a hail zone.
portfolio$hail_zone <- factor(c("low", "high", "low", "high", "low", "high"))
hail_restrictions <- data.frame(
  hail_zone = c("low", "high"),
  hail_relativity = c(1.00, 1.20)
)

prepare_refinement(model, data = portfolio) |>
  add_restriction(
    hail_restrictions,
    replaces = "postal_area"
  )
#> <rating_refinement>
#> Base model: Poisson GLM (log link)
#> Steps: 1
#>   1. Restriction: hail_zone -> hail_relativity (2 levels) [expert-specified new risk factor] [replaces postal_area]
# During refit(), hail_zone replaces postal_area rather than supplementing it.

# Without `replaces`, a new fixed factor supplements the existing terms.
# A later actuarial review changes only the relativity for the low hail zone.
# The high-zone relativity remains 1.20 and the existing step is updated.
revised_hail_restrictions <- data.frame(
  hail_zone = "low",
  hail_relativity = 1.10
)

hail_refinement <- prepare_refinement(model, data = portfolio) |>
  add_restriction(
    hail_restrictions,
    allow_new_risk_factors = TRUE
  ) |>
  add_restriction(revised_hail_restrictions)
#> Updated existing restriction for `hail_zone = "low"`: 1 -> 1.1

refit(hail_refinement)
#> Refined generalized linear model
#> 
#> Original formula:
#>   claims ~ postal_area + offset(log(exposure))
#> 
#> Refitted formula:
#>   claims ~ postal_area + offset(log(hail_relativity) + log(exposure))
#> 
#> Family: poisson (link: log)
#> Intercept-only refit: no
#> Refinement steps:
#>   1. Restriction: hail_zone -> hail_relativity (2 levels) [expert-specified new risk factor]
#> 
#> 
#> Call:  glm(formula = claims ~ postal_area + offset(log(hail_relativity) + 
#>     log(exposure)), family = poisson(link = "log"), data = refined_data)
#> 
#> Coefficients:
#>  (Intercept)  postal_areaB  postal_areaC  
#>    5.534e-01    -6.563e-11     2.231e-01  
#> 
#> Degrees of Freedom: 5 Total (i.e. Null);  3 Residual
#> Null Deviance:	    2.714 
#> Residual Deviance: 2.563 	AIC: 24.05