Maturity of the event by score band

Follows the units of each score band over time and estimates, at each horizon, the cumulative share that has had the event, allowing for units whose follow-up ends before the horizon (censoring). It shows how fast each band matures, where the curve flattens (the performance window) and how the discrimination of the score changes with the horizon.

Usage

scr_maturity(x, ...)

# S3 method for class 'data.frame'
scr_maturity(
  x,
  score = "score",
  time = "time",
  event = "event",
  horizons,
  objective = "risk",
  direction = NULL,
  n_bands = 5L,
  cuts = NULL,
  level = 0.95,
  weight = NULL,
  max_cells = 1e+05,
  ...
)

Arguments

x: A data.frame with one row per unit.
...: Passed on to the methods; an unknown argument is an error.
score, time, event: Column names of the score, of the time to the event or to the end of the follow-up, and of the 0/1 event indicator.
horizons: Times at which the incidence is read, in the unit of time (positive numbers).
objective: "risk" (the event is the bad case) or "propensity" (the event is the good case).
direction: "higher_is_safer" or "higher_is_riskier"; NULL derives it from objective.
n_bands: Equal-share bands of the score (tie-safe, the event-richest first) when cuts is not given.
cuts: Optional ascending cuts of the score, or an object from scr_bands() or scr_tiers(), whose cuts, numbers and labels are then used, with its objective and direction.
level: Confidence level of the intervals.
weight: Optional column of non-negative case weights.
max_cells: Largest number of distinct score values kept exactly.

Value

An object of class c("scr_maturity", "list"):

table: One row per band and horizon, event-richest band first, then all units (band = NA, label = "all"): band, label, n, horizon, at_risk, events, censored, incidence, se, lo, hi and pct_of_final.
discrimination: One row per horizon: horizon, n (units with a complete window), events, auc and gini.
cuts, codes, labels: The ascending cuts, and the band number and label of every interval in ascending score order.
horizons, level, objective, direction, score, time, event, n, n_rows, n_dropped, weighted, call: The settings: n is the volume used (the sum of the weights), n_rows the rows used and n_dropped those left out.

Data

One row per unit (a loan, a customer), with the score at the origin, time and event:

time: A non-negative number of periods (days, months) from the origin to the event, or to the end of the follow-up.
event: 1 when the event was observed at time, 0 when the follow-up ended there without it (censored).

From a monthly panel, take per unit the months from its opening date to its first month in default (event = 1), or to its last observed month when it never defaulted (event = 0); the example does it for scr_demo_panel.

Rows with a missing or infinite score, a missing time or event, or a zero weight are left out (n_dropped).

Incidence

Per band and for all units, the Kaplan-Meier estimate of the survival is \(S(t) = \prod_{t_j \le t} (1 - d_j / n_j)\), with \(d_j\) the events at the distinct time \(t_j\) and \(n_j\) the units still followed just before it (units censored at \(t_j\) are at risk at \(t_j\)). incidence is \(1 - S(h)\) at the horizon \(h\), and se its standard error by Greenwood's formula, \(S(h) \sqrt{\sum_{t_j \le h} d_j / (n_j (n_j - d_j))}\). The interval lo, hi is the complementary log-log interval of \(S(h)\), \(S^{\exp(\pm z \sigma)}\) with \(\sigma = \sqrt{\sum d_j / (n_j (n_j - d_j))} / |\ln S|\), which stays inside [0, 1]. It is undefined when no event has happened by the horizon (lo = 0, hi = NA) and when every unit has had the event (NA).

Under weights, \(d_j\) and \(n_j\) are weighted sums and each term of Greenwood's sum is computed on the Kish effective size of the risk set, \(d_j / ((n_j - d_j)\, n^{eff}_j)\) with \(n^{eff}_j = n_j^2 / \sum w^2\); equal weights give the unweighted result.

Per band and horizon the table also counts events (events up to the horizon), censored (units censored before it) and at_risk (the other units: still followed at the horizon without the event), which add up to n. pct_of_final is the incidence over the incidence at the largest horizon: the share of the final events already seen.

Past the last follow-up time of a band nothing more is observed: the curve is carried flat, and a horizon beyond it repeats the last estimate with at_risk = 0. Read such a row as "no information", not as a plateau of the event rate.

Discrimination

At each horizon, the units with a complete window are those with the event by the horizon and those followed for at least the horizon without it; units censored earlier are left out. auc and gini are those of the score for "event by the horizon" among them, from the counts per score value. The complete window ignores the censored units, so it is unbiased only when censoring does not depend on the score.

References

Greenwood, M. (1926). The natural duration of cancer. Reports on Public Health and Medical Subjects, 33, 1-26. HMSO.

Kalbfleisch, J. D. and Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data, 2nd edition. Wiley. doi:10.1002/9781118032985

Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457-481. doi:10.1080/01621459.1958.10501452

Examples

# time and event from a monthly panel: months from the opening date to
# the first default (90 days past due), or to the last observed month
p <- scr_demo_panel[order(scr_demo_panel$id, scr_demo_panel$ref_date), ]
month <- 12 * as.integer(format(p$ref_date, "%Y")) + as.integer(format(p$ref_date, "%m"))
first <- !duplicated(p$id)
bad <- which(p$dpd >= 90)
bad <- bad[!duplicated(p$id[bad])]
u <- data.frame(id = p$id[first], score = p$score[first], open = month[first])
u$event <- as.integer(u$id %in% p$id[bad])
u$end <- as.vector(tapply(month, p$id, max)[as.character(u$id)])
u$end[u$event == 1] <- month[bad][match(u$id[u$event == 1], p$id[bad])]
u$time <- u$end - u$open

mt <- scr_maturity(u, horizons = c(6, 12, 24, 35), n_bands = 4)
mt
#> <scr_maturity> event "event" over "time" | objective risk | higher_is_safer
#>   600 units | 4 bands (equal shares) | horizons 6, 12, 24, 35 | 95% complementary log-log intervals
#> 
#> Cumulative incidence (Kaplan-Meier), event-richest band first
#>   band score                            n       t=6      t=12      t=24      t=35
#>      1 [-Inf, 556.5)                  149     12.1%     24.2%     47.7%     58.4%
#>      2 [556.5, 603.5)                 150      7.3%     12.0%     27.3%     34.7%
#>      3 [603.5, 643.5)                 150      3.3%      4.0%      8.7%     14.7%
#>      4 [643.5, Inf)                   151    0.662%      3.3%      5.3%     11.3%
#>        all                            600      5.8%     10.8%     22.2%     29.7%
#>        share of the final (all)                 20%       37%       75%      100%
#> 
#> Discrimination at each horizon (units with a complete window)
#>     horizon         n    events      AUC     Gini
#>           6       600        35   0.7492   0.4984
#>          12       600        65   0.7601   0.5202
#>          24       600       133   0.7965   0.5930
#>          35       600       178   0.7692   0.5384
mt$discrimination
#>    horizon     n events       auc      gini
#>      <num> <num>  <num>     <num>     <num>
#> 1:       6   600     35 0.7492035 0.4984071
#> 2:      12   600     65 0.7601006 0.5202013
#> 3:      24   600    133 0.7965014 0.5930028
#> 4:      35   600    178 0.7692103 0.5384206