Summarise age-to-age factor statistics

Compute group-wise summary statistics for age-to-age factors from an object of class "ATA". This function serves two purposes:

1. Diagnostics: provides descriptive statistics (mean, median, wt, cv) that help the user assess the stability and consistency of observed ata factors across cohorts.

2. Estimation: fits a no-intercept weighted least squares model per ata link to produce the WLS-estimated factor (f), its standard error (f_se), relative standard error (rse), and Mack sigma (sigma). These are used downstream by find_ata_maturity() and fit_ata().

Usage

# S3 method for class 'ATA'
summary(object, alpha = 1, digits = 3, ...)

Arguments

object

An object of class "ATA", typically produced by build_ata().

alpha

Numeric scalar controlling the variance structure in the WLS fit. Default is 1.

digits

Number of decimal places to round numeric columns. Default is 3. Pass NULL to skip rounding.

...

Additional arguments passed to the internal WLS estimation.

Value

A data.table with class "ATASummary" containing one row per ata link:

ata_from, ata_to, ata_link

Link identifiers.

mean

Arithmetic mean of observed ata factors.

median

Median of observed ata factors.

wt

Volume-weighted mean: \(\sum C_{i,k+1} / \sum C_{i,k}\), independent of alpha.

cv

Coefficient of variation of observed ata factors (\(SD / mean\)). Used by find_ata_maturity() to assess stability.

f

WLS-estimated factor. Equals wt when alpha = 2 and no zero value_from rows are present.

f_se

Standard error of the WLS-estimated factor.

rse

Relative standard error of the WLS-estimated factor (\(f\_se / f\)).

sigma

Mack sigma (residual standard deviation from the WLS fit). Used in Mack variance estimation.

n_obs

Total number of observations for the link.

n_valid

Number of finite ata values.

n_inf

Number of infinite ata values.

n_nan

Number of NaN ata values.

valid_ratio

Proportion of finite ata values (\(n\_valid / n\_obs\)).

Relationship between wt and f

Both wt and f are weighted averages of the observed ata factors, but they differ in how weights are assigned and which observations are included:

wt

Volume-weighted mean: \(wt = \sum C_{i,k+1} / \sum C_{i,k}\). Computed from all rows where value_from and value_to are finite, including rows where either value is zero. Independent of alpha.

f

WLS-estimated factor. Only rows where value_from > 0 are used, since value_from = 0 causes numerical issues in the WLS weights (\(w = value\_from^{\alpha}\)). When alpha = 2, f and wt are numerically equivalent (assuming no zero value_from rows). When alpha \ne 2, they diverge.

Therefore wt and f can differ for two reasons:

1. Zero exclusion: rows with value_from = 0 are included in wt but excluded from f. This typically affects early development periods where some cohorts have not yet accumulated any claims.

2. Alpha effect: when alpha \ne 2, the WLS weights differ from the volume weights used in wt, leading to different estimates. Comparing wt and f can help diagnose whether the choice of alpha materially affects the estimated factor.

Weights

When the input "ATA" object contains a weight column (added by build_ata() when weight_var is supplied), that column is automatically used as the WLS weight in place of value_from. This is useful when value_var = "clr", where value_from carries no exposure information and an external exposure variable such as crp should be used instead.

Coefficient of variation (cv)

The coefficient of variation is defined as: $$cv = \frac{SD(f_k)}{\bar{f}_k}$$ where \(f_k\) are the individual observed ata values for link \(k\) and \(\bar{f}_k\) is their arithmetic mean. The cv reflects the relative spread of observed factors across cohorts, regardless of the exposure scale. It is used by find_ata_maturity() as one of the criteria for determining the maturity point.

Relative standard error (rse)

The relative standard error is defined as: $$rse = \frac{SE(\hat{f}_k)}{\hat{f}_k}$$ where \(SE(\hat{f}_k)\) is the standard error of the WLS-estimated factor. Unlike cv, which treats all cohorts equally, rse gives more weight to cohorts with larger exposures (via the WLS weights). A small rse indicates that the WLS estimate is precise, which tends to occur when: (1) there are many cohorts, (2) exposures are large, and (3) the observed ata values are consistent across cohorts.

See also

build_ata(), find_ata_maturity(), fit_ata()