Scaling, alignment and challengers

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1. Why “700 points” means nothing on its own

A points score is a linear function of a log-odds. Which log-odds, in which orientation, and through which map, is what gives the number its meaning. Two scorecards can both print “700” for a customer and disagree completely about that customer’s risk unless both raw scores were taken to the same scale by the same procedure. That is the whole subject of this vignette.

The scale is one statement

base_score, base_odds and pdo are not three independent knobs; they are a single sentence: at base_score points the odds are base_odds, and every pdo points they double. From that sentence follow two constants,

\[ \mathrm{factor} = \frac{\mathrm{pdo}}{\ln 2}, \qquad \mathrm{offset} = \mathrm{base\_score} - \mathrm{factor} \cdot \ln(\mathrm{base\_odds}), \]

and the map from log-odds to points, \(\mathrm{score} = \mathrm{offset} + \mathrm{factor} \cdot \ln(\mathrm{odds})\). The textbook example is 600 points at odds 50:1 with a PDO of 20 (Siddiqi, 2017); it is an example, not a standard.

factor <- 20 / log(2)
offset <- 600 - factor * log(50)
c(factor = factor, offset = offset)
#>   factor   offset 
#>  28.8539 487.1229

The orientation trap

The word odds changes meaning between the two literatures. Under objective = "risk" the event is the bad case, more points are safer, and base_odds is non-event:event (safe:event). Under objective = "propensity" the event is the good case, more points mean a higher chance of the event, and base_odds is event:non-event (event:safe). Feeding the same number 50 to both conventions is the most common sign error in scorecard work, and it is silent: nothing fails, the points simply mean something else.

scr_align() records odds_orientation on the object precisely so that this can never be implicit. Let us see the trap on a simulated, well-calibrated event logit.

set.seed(2026)
n   <- 4000
raw <- stats::rnorm(n, mean = stats::qlogis(0.12), sd = 1.2)   # event logit
y   <- stats::rbinom(n, 1, stats::plogis(raw))                 # calibrated outcome

With method = "direct" the model is trusted as calibrated: the log-odds is the raw logit, in the orientation the direction implies (S = -1 under higher_is_safer, +1 under higher_is_riskier, I = 0).

al_risk <- scr_align(raw, y, base_score = 600, base_odds = 50, pdo = 20,
                     direction = "higher_is_safer", method = "direct")
al_risk
#> <scr_align> 600 points at odds 50:1 (safe:event), PDO 20 | higher_is_safer
#>   factor = 28.853901 | offset = 487.122876
#>   calibration: direct (model assumed calibrated; I = 0.000, S = -1)
#>   score = 487.122876 + -28.853901 * raw

The same 50, passed under the propensity convention, states that at 600 points the event is fifty times more likely than not. The equivalent statement in that orientation is base_odds = 1/50.

al_prop_wrong <- scr_align(raw, y, base_score = 600, base_odds = 50, pdo = 20,
                           direction = "higher_is_riskier", method = "direct")
al_prop_right <- scr_align(raw, y, base_score = 600, base_odds = 1 / 50, pdo = 20,
                           direction = "higher_is_riskier", method = "direct")

# what does "600 points" mean under each object? Invert score = a + b * raw
prob_at <- function(al, score) predict(al, (score - al$a) / al$b, type = "prob")
data.table(
  object      = c("risk 50 (safe:event)", "propensity 50 (event:safe)", "propensity 1/50 (event:safe)"),
  orientation = c(al_risk$odds_orientation, al_prop_wrong$odds_orientation, al_prop_right$odds_orientation),
  p_event_at_600 = round(c(prob_at(al_risk, 600), prob_at(al_prop_wrong, 600), prob_at(al_prop_right, 600)), 4)
)
#>                          object orientation p_event_at_600
#>                          <char>      <char>          <num>
#> 1:         risk 50 (safe:event)  safe:event         0.0196
#> 2:   propensity 50 (event:safe)  event:safe         0.9804
#> 3: propensity 1/50 (event:safe)  event:safe         0.0196

Under the correct pair of statements the two scales are mirror images of each other around 600, and both return the same event probability for the same customer; under the wrong pair, “600 points” is a 2% customer on one card and a 98% customer on the other.

head(data.table(
  raw   = round(raw, 3),
  p_true = round(stats::plogis(raw), 4),
  score_risk = round(predict(al_risk, raw), 1),
  score_prop = round(predict(al_prop_right, raw), 1),
  p_risk = round(predict(al_risk, raw, type = "prob"), 4),
  p_prop = round(predict(al_prop_right, raw, type = "prob"), 4)
), 5)
#>       raw p_true score_risk score_prop p_risk p_prop
#>     <num>  <num>      <num>      <num>  <num>  <num>
#> 1: -1.368 0.2030      526.6      673.4 0.2030 0.2030
#> 2: -3.288 0.0360      582.0      618.0 0.0360 0.0360
#> 3: -1.825 0.1388      539.8      660.2 0.1388 0.1388
#> 4: -2.094 0.1097      547.5      652.5 0.1097 0.1097
#> 5: -2.792 0.0577      567.7      632.3 0.0577 0.0577
all.equal(predict(al_risk, raw) + predict(al_prop_right, raw), rep(1200, n))
#> [1] TRUE

Now the default, method = "regression". On a calibrated logit it should recover an intercept near 0 and a slope near -1 (the sign of the direction), and it does. The slope sits a little below 1 in magnitude because the regression runs on band means: averaging the logit inside a band is not the same as taking the logit of the band’s event rate, and the difference attenuates the slope slightly. That is a property of any banded calibration, not a defect of this one.

al_reg <- scr_align(raw, y, base_score = 600, base_odds = 50, pdo = 20,
                    direction = "higher_is_safer")
al_reg
#> <scr_align> 600 points at odds 50:1 (safe:event), PDO 20 | higher_is_safer
#>   factor = 28.853901 | offset = 487.122876
#>   calibration: ln(odds) = 0.082592 + -0.915197 * raw  (adj. R2 = 0.9787, 10 bands)
#>   score = 489.505966 + -26.406998 * raw

2. The two-step regression alignment

method = "regression" does not trust the raw score to be a log-odds. It measures how the raw score maps to the empirical log-odds and composes that map with the PDO map:

band the raw score by quantiles on the reference data (n_bands, default 10);
in every band, compute the empirical log-odds with Laplace smoothing, in the orientation the direction implies;
regress the band log-odds on the band mean of the raw score, weighted by band size: \(\ln(\mathrm{odds}) = I + S \cdot \mathrm{raw}\);
compose with the scale: \(\mathrm{score} = \mathrm{offset} + \mathrm{factor}\,(I + S \cdot \mathrm{raw}) = a + b \cdot \mathrm{raw}\).

The bands are kept on the object, so the regression can be audited row by row.

al_reg$calibration$bands
#>               band   n events    raw_mean    ln_odds ln_odds_fit
#> 1     [-Inf,-3.53] 400     10 -4.07429239  3.6160527 3.811370945
#> 2    (-3.53,-3.01] 400     14 -3.24729758  3.2829832 3.054507943
#> 3    (-3.01,-2.63] 400     28 -2.81324430  2.5703330 2.657263770
#> 4    (-2.63,-2.31] 400     37 -2.47321872  2.2714384 2.346073457
#> 5    (-2.31,-2.01] 400     48 -2.16009650  1.9834868 2.059505001
#> 6    (-2.01,-1.71] 400     53 -1.85637134  1.8710830 1.781536714
#> 7    (-1.71,-1.39] 400     61 -1.55681460  1.7084368 1.507383339
#> 8   (-1.39,-0.995] 400     86 -1.20975072  1.2908397 1.189751591
#> 9  (-0.995,-0.429] 400    125 -0.72959054  0.7862819 0.750310530
#> 10   (-0.429, Inf] 400    222  0.08708293 -0.2203385 0.002893594
c(intercept = al_reg$calibration$intercept, slope = al_reg$calibration$slope,
  adj_r2 = al_reg$calibration$r2)
#>   intercept       slope      adj_r2 
#>  0.08259161 -0.91519679  0.97874564

How to read the three numbers:

intercept \(I\): the log-odds at raw = 0, in the scale’s orientation. A value away from 0 means the raw score is shifted relative to the population it is being aligned on (a prior shift, a reweighted sample, a model fitted on a different base rate).
slope \(S\): how many log-odds units the empirical odds move per unit of raw score. Its sign must agree with the direction (-1 expected under higher_is_safer, +1 under higher_is_riskier); scr_align() warns when it does not. Its magnitude measures over- or under-confidence: \(|S| < 1\) means the raw logit spreads more than the data support, \(|S| > 1\) that it is too timid.
adj. R2: how well a straight line explains the band log-odds. A low value means the raw score is not linear in log-odds, and a linear alignment will be systematically wrong in the tails; that is a signal to inspect the bands, not to trust the map.

Why not always use "direct"? Because raw scores are rarely calibrated on the population they are deployed to. Shrink and shift the simulated logit and compare the two methods:

raw_bad <- 0.6 * raw - 1            # over-timid and shifted
al_bad_direct <- scr_align(raw_bad, y, method = "direct")
al_bad_reg    <- scr_align(raw_bad, y)
al_bad_reg$calibration[c("intercept", "slope", "r2")]
#> $intercept
#> [1] -1.442736
#> 
#> $slope
#> [1] -1.525328
#> 
#> $r2
#> [1] 0.9787456
data.table(
  method = c("direct", "regression"),
  mean_predicted = c(mean(predict(al_bad_direct, raw_bad, type = "prob")),
                     mean(predict(al_bad_reg,    raw_bad, type = "prob"))),
  observed = mean(y)
)
#>        method mean_predicted observed
#>        <char>          <num>    <num>
#> 1:     direct      0.1174596    0.171
#> 2: regression      0.1714736    0.171

The true map is \(\ln(\mathrm{odds}_{\mathrm{safe:event}}) = -(1 + \mathrm{raw\_bad}) / 0.6\), so intercept and slope should both be near \(-1.67\). The regression lands at about \(-1.4\) and \(-1.5\), attenuated by the banding as above but in the right place, and its mean predicted probability matches the observed event rate; the direct map reports whatever the raw score claims, here a third too low. This is why scr_scorecard() always aligns and defaults to "regression".

The regression aligns the score to the event rate of the sample it is fitted on, a point-in-time rate. It is not a calibration to a long-run default rate with rating grades and margins of conservatism; that step is covered in PD calibration and rating grades.

3. Two targets, two conventions, one scale

The bundled scr_demo table carries two binary targets: default, a risk target, and churn, which we shall treat as a propensity target (the campaign wants to reach customers who will churn). Each is selected with the other target dropped, so that neither becomes a candidate for the other.

cfg_risk <- scr_config(objective = "risk", verbose = FALSE, nthread = 1, use_glmnet = TRUE,
                       use_ranger = FALSE, use_lightgbm = FALSE, xgb_rounds = 60, n_boot = 20)
cfg_prop <- scr_config(objective = "propensity", verbose = FALSE, nthread = 1, use_glmnet = TRUE,
                       use_ranger = FALSE, use_lightgbm = FALSE, xgb_rounds = 60, n_boot = 20)

res_risk <- scr_select(scr_demo, "default", config = cfg_risk, drop = c("id", "churn"),
                       date_col = "ref_date")
res_prop <- scr_select(scr_demo, "churn", config = cfg_prop, drop = c("id", "default"),
                       date_col = "ref_date")

sc_risk <- scr_scorecard(res_risk)
sc_prop <- scr_scorecard(res_prop)

Both scorecards declare the same scale, 600/50/20, but the two alignment objects differ in direction and in odds orientation, and therefore in the sign of the map from logit to points.

sc_risk$alignment
#> <scr_align> 600 points at odds 50:1 (safe:event), PDO 20 | higher_is_safer
#>   factor = 28.853901 | offset = 487.122876
#>   calibration: ln(odds) = 0.141187 + -0.912143 * raw  (adj. R2 = 0.9668, 10 bands)
#>   score = 491.196658 + -26.318891 * raw
sc_prop$alignment
#> <scr_align> 600 points at odds 50:1 (event:safe), PDO 20 | higher_is_riskier
#>   factor = 28.853901 | offset = 487.122876
#>   calibration: ln(odds) = 0.004026 + 1.008929 * raw  (adj. R2 = 0.9904, 10 bands)
#>   score = 487.239034 + 29.111532 * raw
c(risk = sc_risk$odds_orientation, propensity = sc_prop$odds_orientation)
#>         risk   propensity 
#> "safe:event" "event:safe"

scr_score_metrics() reports the AUC in the direction of the scale: above 0.5 whenever the score ranks correctly in its own convention. The direction column is carried on the table so the reader knows which way the score was read.

scr_score_metrics(sc_risk)[, .(sample, direction, auc, auc_lo, auc_hi, ks)]
#>     sample       direction       auc    auc_lo    auc_hi        ks
#>     <char>          <char>     <num>     <num>     <num>     <num>
#> 1:   train higher_is_safer 0.7856428 0.7662170 0.8069608 0.4411466
#> 2: holdout higher_is_safer 0.7394060 0.7066284 0.7770357 0.3889033
scr_score_metrics(sc_prop)[, .(sample, direction, auc, auc_lo, auc_hi, ks)]
#>     sample         direction       auc    auc_lo    auc_hi        ks
#>     <char>            <char>     <num>     <num>     <num>     <num>
#> 1:   train higher_is_riskier 0.7404593 0.7218347 0.7547629 0.3715330
#> 2: holdout higher_is_riskier 0.7285271 0.7107000 0.7557903 0.3276755

Compare the mean score by outcome on hold-out. Under risk, events score lower; under propensity, events score higher. The same points scale, read the other way round.

rbind(
  sc_risk$samples$holdout[, .(scorecard = "default (risk)", n = .N, mean_score = round(mean(score), 1)), by = y],
  sc_prop$samples$holdout[, .(scorecard = "churn (propensity)", n = .N, mean_score = round(mean(score), 1)), by = y]
)[order(scorecard, y)]
#>        y          scorecard     n mean_score
#>    <int>             <char> <int>      <num>
#> 1:     0 churn (propensity)   976      449.0
#> 2:     1 churn (propensity)   424      472.3
#> 3:     0     default (risk)  1197      554.9
#> 4:     1     default (risk)   203      529.6

Reading a risk score as if it were a propensity score is the same error in reverse: scr_metrics() with the wrong higher_is_event returns one minus the AUC.

s <- sc_risk$samples$holdout
c(read_as_safer  = scr_metrics(s$score, s$y, higher_is_event = FALSE, ci = FALSE)$auc,
  read_as_riskier = scr_metrics(s$score, s$y, higher_is_event = TRUE,  ci = FALSE)$auc)
#>   read_as_safer read_as_riskier 
#>        0.739406        0.260594

4. From logit to points

With score = a + b * logit and logit = alpha + sum(beta_j * woe_ij), the score splits into a constant and one term per variable:

\[ \mathrm{base} = a + b\,\alpha, \qquad \mathrm{points}_{ij} = b\,\beta_j\,\mathrm{WOE}_{ij}. \]

Under the default points_style = "base_plus_deviation" the constant is kept apart as base_points and each bin carries its deviation, so a bin with a WOE of 0 (the average risk) scores 0 points. "distributed" spreads the constant over the variables instead (Siddiqi, 2017), and no bin has a zero reference. The exact points are kept in points_raw; the published points are rounded.

al <- sc_risk$alignment
c(a = al$a, b = al$b, alpha = unname(sc_risk$coef["(Intercept)"]),
  base_exact = sc_risk$base_points_raw, base_points = sc_risk$base_points)
#>           a           b       alpha  base_exact base_points 
#>  491.196658  -26.318891   -1.789721  538.300127  538.000000
sc_risk$points[variable == "vl_score_01", .(bin, woe, coef, points_raw = round(points_raw, 2), points)]
#>                      bin         woe     coef points_raw points
#>                   <char>       <num>    <num>      <num>  <num>
#> 1:      (-Inf;33.360000] -2.06253559 1.128507      61.26     61
#> 2: (33.360000;38.150000] -0.73104946 1.128507      21.71     22
#> 3: (38.150000;44.240000] -0.65810792 1.128507      19.55     20
#> 4: (44.240000;48.060000] -0.52261206 1.128507      15.52     16
#> 5: (48.060000;63.940000]  0.03978629 1.128507      -1.18     -1
#> 6: (63.940000;72.610000]  0.70394095 1.128507     -20.91    -21
#> 7:      (72.610000;+Inf]  0.99617148 1.128507     -29.59    -30

A risky bin has a positive WOE, the coefficient is positive and b is negative under higher_is_safer, so the riskiest bin of vl_score_01 loses 30 points and the safest gains 61. scr_apply() returns both scores: score is exact, and score_points is base_points plus the rounded points of each bin, which is what a points table on paper gives. The two differ by the rounding of one constant and twelve bin points.

sc_new <- scr_apply(sc_risk, head(scr_demo, 5), what = "points")
var_pts <- sc_new[, paste0(sc_risk$features, "_points"), with = FALSE]
sc_new[, .(score = round(score, 2), score_points,
           check = sc_risk$base_points + rowSums(var_pts), diff = round(score - score_points, 2))]
#>     score score_points check  diff
#>     <num>        <num> <num> <num>
#> 1: 546.53          546   546  0.53
#> 2: 562.33          562   562  0.33
#> 3: 560.52          559   559  1.52
#> 4: 507.06          507   507  0.06
#> 5: 536.76          536   536  0.76

5. Do the odds double every PDO?

The scale is a claim about odds: at 600 points 50:1, and twice the odds every 20 points. scr_score_gains() gives the observed odds (non-events per event) in each band of the score, the training deciles applied frozen to the hold-out, and the claim can be checked against them.

g <- scr_score_gains(sc_risk)
g[, scale_odds := exp((mean_score - al$offset) / al$factor)]
g[sample == "holdout", .(band, n, events, mean_score = round(mean_score, 1),
                         odds = round(odds, 2), scale_odds = round(scale_odds, 2))]
#>           band     n events mean_score  odds scale_odds
#>         <char> <int>  <int>      <num> <num>      <num>
#>  1: [-Inf,510]   127     45      495.2  1.81       1.32
#>  2:  (510,524]   128     35      517.0  2.63       2.81
#>  3:  (524,533]   139     37      528.7  2.73       4.22
#>  4:  (533,542]   149     26      537.7  4.66       5.77
#>  5:  (542,550]   151     17      545.9  7.69       7.68
#>  6:  (550,558]   148     18      554.0  7.05      10.15
#>  7:  (558,567]   149     11      562.4 12.04      13.59
#>  8:  (567,577]   128      5      571.0 22.45      18.32
#>  9:  (577,590]   124      4      583.0 26.78      27.74
#> 10: (590, Inf]   157      5      608.0 27.73      66.05
g[, .(pdo_implied = log(2) / coef(stats::lm(log_odds ~ mean_score, weights = n))[[2]]), by = sample]
#>     sample pdo_implied
#>     <char>       <num>
#> 1:   train    20.00000
#> 2: holdout    24.03442

On train the implied PDO is 20 by construction: the same deciles fed the alignment regression. On hold-out the odds double every 24 points or so, and the safest band is well short of the scale (odds near 28 against about 66 on the scale). The ranking holds, but the out-of-time population is less separated than the development one, and the top of the scale promises more than it delivers. This is the table to show before a cut-off is quoted in odds, and the one to recompute when monitoring.

6. A tree challenger on the same scale

A gradient-boosted model can be fitted on the same WOE columns as the scorecard and aligned to the same scale by the same regression. That makes its output comparable point for point with the champion; it does not make it a scorecard. The object says so explicitly: no points, no reason codes, supports_scorecard = FALSE.

sc_x <- scr_scorecard(res_risk, challenger = "xgboost")

sc_x$challenger$supports_scorecard
#> [1] FALSE
sc_x$challenger$points
#> [1] "NOT_APPLICABLE_ENGINE"
sc_x$challenger$alignment
#> <scr_align> 600 points at odds 50:1 (safe:event), PDO 20 | higher_is_safer
#>   factor = 28.853901 | offset = 487.122876
#>   calibration: ln(odds) = -1.398883 + -1.883869 * raw  (adj. R2 = 0.9710, 10 bands)
#>   score = 446.759642 + -54.356970 * raw
rbind(sc_x$challenger$metrics[, .(model = "xgboost challenger", sample, auc, auc_lo, auc_hi, ks)],
      scr_score_metrics(sc_x)[sample == "holdout", .(model = "scorecard", sample, auc, auc_lo, auc_hi, ks)])
#>                 model  sample       auc    auc_lo    auc_hi        ks
#>                <char>  <char>     <num>     <num>     <num>     <num>
#> 1: xgboost challenger holdout 0.7340725 0.7021223 0.7720179 0.3574453
#> 2:          scorecard holdout 0.7394060 0.7066284 0.7770357 0.3889033

The challenger’s alignment has its own intercept and slope: two engines, two raw scores, one declared scale. Here the slope is well above 1 in magnitude, which says the boosted logit is compressed (a few dozen rounds at a low learning rate leave it under-confident) and the alignment stretches it back to the empirical odds; the scorecard logit, by contrast, needed a slope close to 1. Its AUC interval overlaps the scorecard’s, which is the usual finding on a WOE feature set.

The swap set

Aggregate AUCs hide where two models disagree. The swap-set table holds the approval rate fixed and asks which customers one model would approve and the other would not:

sc_x$challenger$swapset
#>    approval_rate n_swap_in n_swap_out event_rate_swap_in event_rate_swap_out
#>            <num>     <int>      <int>              <num>               <num>
#> 1:           0.5        96         96          0.1145833           0.1250000
#> 2:           0.7        75         75          0.1866667           0.1866667
#> 3:           0.9        36         36          0.4166667           0.3611111
#>    event_rate_champion event_rate_challenger
#>                  <num>                 <num>
#> 1:          0.06000000            0.05857143
#> 2:          0.08367347            0.08367347
#> 3:          0.12142857            0.12301587

How to read one row, say approval_rate = 0.5:

both models approve the safest half of hold-out by their own score;
n_swap_in customers are approved by the challenger but not by the champion; n_swap_out the reverse. At a fixed rate the two counts are equal;
event_rate_swap_in versus event_rate_swap_out is the decision: the challenger is worth its complexity only if the customers it brings in are safer than the ones it sends out, by a margin that survives the sample size of the swap;
event_rate_champion and event_rate_challenger are the event rates of the two approved books.

Because both scores sit on the same scale, the swap set can also be read in points: a customer swapped in at the 70% rate crossed the challenger’s cut-off and not the champion’s, and the two cut-offs are directly comparable numbers.

7. The portfolio view

With several targets in flight, scr_compare() gives one row per target (funnel, best hold-out model of the selection stage, warning signs) and scr_core() the variables approved on more than one target.

runs <- list(default = res_risk, churn = res_prop)
scr_compare(runs)[, .(target, rows, event_rate, approved, best_model, auc, ks, relaxation)]
#>     target  rows event_rate approved best_model    auc     ks
#>     <char> <int>      <num>    <int>     <char>  <num>  <num>
#> 1: default  4200     0.1425       12    xgboost 0.7375 0.3695
#> 2:   churn  4200     0.2846        9     glmnet 0.7262 0.3288
#>                relaxation
#>                    <char>
#> 1:                   none
#> 2: min_votes reduced to 1
scr_core(runs, min_targets = 2)
#>        feature n_targets mean_rank        targets
#>         <char>     <int>     <num>         <char>
#> 1: vl_score_02         2       2.0 churn, default
#> 2: vl_score_04         2       3.5 churn, default
#> 3:   ds_region         2       6.0 churn, default
#> 4: vl_score_06         2       6.0 churn, default
#> 5: vl_score_05         2       6.0 churn, default
#> 6: vl_score_07         2       7.5 churn, default
#> 7:  vl_hist_04         2      10.0 churn, default

A variable approved on both targets is not an artefact of one target definition. relaxation records whether the consensus loosened its vote threshold to reach the minimum number of variables, as it did for churn. The AUC here is that of the selection-stage models; the scorecards’ own metrics are in scr_score_metrics().

8. Rescaling an existing scorecard

A committee may ask for a different convention, say 700 points at 30:1 with a PDO of 25. The model does not change: calling scr_scorecard() with the new scale refits the same regression on the same data, so the logit and the calibration regression come out identical and only the PDO map moves.

sc_700 <- scr_scorecard(res_risk, base_score = 700, base_odds = 30, pdo = 25)

sc_700$alignment
#> <scr_align> 700 points at odds 30:1 (safe:event), PDO 25 | higher_is_safer
#>   factor = 36.067376 | offset = 577.327735
#>   calibration: ln(odds) = 0.141187 + -0.912143 * raw  (adj. R2 = 0.9668, 10 bands)
#>   score = 582.419962 + -32.898614 * raw
# the calibration is the same; only factor and offset change
c(intercept_600 = sc_risk$alignment$calibration$intercept, intercept_700 = sc_700$alignment$calibration$intercept,
  slope_600 = sc_risk$alignment$calibration$slope, slope_700 = sc_700$alignment$calibration$slope)
#> intercept_600 intercept_700     slope_600     slope_700 
#>     0.1411865     0.1411865    -0.9121433    -0.9121433

Ranking metrics are invariant to a monotone transformation, so AUC, KS and Gini are identical to the last digit, while every point value moves.

rbind(scr_score_metrics(sc_risk)[, .(scale = "600/50/20", sample, auc, ks, gini)],
      scr_score_metrics(sc_700)[, .(scale = "700/30/25", sample, auc, ks, gini)])
#>        scale  sample       auc        ks      gini
#>       <char>  <char>     <num>     <num>     <num>
#> 1: 600/50/20   train 0.7856428 0.4411466 0.5712856
#> 2: 600/50/20 holdout 0.7394060 0.3889033 0.4788120
#> 3: 700/30/25   train 0.7856428 0.4411466 0.5712856
#> 4: 700/30/25 holdout 0.7394060 0.3889033 0.4788120
merge(sc_risk$points[, .(variable, bin, points_600 = points)],
      sc_700$points[, .(variable, bin, points_700 = points)],
      by = c("variable", "bin"))[1:6]
#> Key: <variable, bin>
#>      variable    bin points_600 points_700
#>        <char> <char>      <num>      <num>
#> 1:    ds_band      A        -11        -14
#> 2:    ds_band      B          0         -1
#> 3:    ds_band      C          2          3
#> 4:    ds_band      D         13         16
#> 5: ds_channel    APP         -5         -6
#> 6: ds_channel  STORE          9         11

The two scores are the same log-odds seen through two maps, so one can be recovered from the other exactly:

a600 <- sc_risk$alignment; a700 <- sc_700$alignment
ln_odds <- (sc_risk$samples$holdout$score - a600$offset) / a600$factor
all.equal(a700$offset + a700$factor * ln_odds, sc_700$samples$holdout$score)
#> [1] TRUE

That identity is what “comparable” means: two scorecards are on the same scale when the same log-odds gives the same points, and a scorecard is rescaled, not remodelled, when only factor and offset change.

9. A checklist for the model committee

Everything the committee needs to reproduce, compare and later monitor the scale is on the model card. Record it verbatim.

mc <- sc_risk$model_card
keys <- c("target", "objective", "direction", "odds_orientation",
          "base_score", "base_odds", "pdo", "factor", "offset",
          "align_method", "align_intercept", "align_slope", "align_r2",
          "n_train", "event_rate_train", "split_method", "split_cutoff",
          "challenger", "challenger_supports_scorecard", "auc_holdout", "ks_holdout")
data.table(field = keys, value = vapply(mc[keys], function(v) format(v, digits = 6), character(1)))
#>                             field           value
#>                            <char>          <char>
#>  1:                        target         default
#>  2:                     objective            risk
#>  3:                     direction higher_is_safer
#>  4:              odds_orientation      safe:event
#>  5:                    base_score             600
#>  6:                     base_odds              50
#>  7:                           pdo              20
#>  8:                        factor         28.8539
#>  9:                        offset         487.123
#> 10:                  align_method      regression
#> 11:               align_intercept        0.141187
#> 12:                   align_slope       -0.912143
#> 13:                      align_r2        0.966774
#> 14:                       n_train            2800
#> 15:              event_rate_train          0.1425
#> 16:                  split_method     out-of-time
#> 17:                  split_cutoff      2026-05-01
#> 18:                    challenger            none
#> 19: challenger_supports_scorecard              NA
#> 20:                   auc_holdout        0.739406
#> 21:                    ks_holdout        0.388903
#>                             field           value
#>                            <char>          <char>

What to write down, and why:

The scale as one sentence: base_score, base_odds, pdo, and the derived factor and offset. Anyone with these five numbers can turn a log-odds into points and back.
The orientation: objective, direction and odds_orientation. Without them base_odds = 50 is ambiguous; with them it is not.
The alignment method and its coefficients: align_method, align_intercept, align_slope, align_r2, and the number of bands. A "regression" alignment on one population is not the same map as a "regression" alignment on another; the coefficients are what make the two comparable, or show that they are not.
The population the alignment was fitted on: n_train, event_rate_train, the split method and cut-off. An alignment absorbs the base rate of that population; if the deployment population differs, the intercept will drift first. Calibration to a long-run default rate is a separate step, described in PD calibration and rating grades.
The challenger’s status: engine, supports_scorecard = FALSE, and the swap-set table at the approval rates the business uses.
The invariants to check on every rescale: AUC/KS/Gini unchanged, calibration intercept and slope unchanged, only factor and offset moved.

Two scorecards that share items 1 to 3 are comparable; two that share only item 1 are not, however similar their numbers look.

Reference

Siddiqi, N. (2017). Intelligent Credit Scoring: Building and Implementing Better Credit Risk Scorecards, 2nd edition. Wiley.

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