Risk score

Risk score are designed to represent an underlying probability of an adverse event denoted ${\displaystyle \lbrace Y=1\rbrace }$ given a vector of ${\displaystyle P}$ explaining variables ${\displaystyle \mathbf {X} }$ containing measurements of the relevant risk factors. In order to establish the connection between the risk factors and the probability we estimate a set of weights ${\displaystyle \beta }$ is estimated using a generalized linear model:

${\displaystyle {\begin{aligned}\operatorname {E} (\mathbf {Y} |\mathbf {X} )=\mathbf {P} (\mathbf {Y} =1|\mathbf {X} )=g^{-1}(\mathbf {X} \beta )\end{aligned}}}$

Where ${\displaystyle g^{-1}:\mathbb {R} \rightarrow [0,1]}$ is a real-valued, monotonically increasing function that maps the values of the linear predictor ${\displaystyle \mathbf {X} \beta }$ to the interval ${\displaystyle [0,1]}$. GLM methods typically uses the logit or probit as the link function.

While it's possible to estimate ${\displaystyle \mathbf {P} (\mathbf {Y} =1|\mathbf {X} )}$ using other statistical or machine learning methods, the requirements of simplicity and easy interpretation (and monotonicity per risk factor) make most of these methods difficult to use for scoring in this context:

• With more sophisticated methods it becomes difficult to attribute simple weights for each risk factor and to provide a simple formula for the calculation of the score. A notable exception are tree-based methods like CART, that can provide a simple set of decision rules and calculations, but cannot ensure the monotonicity of the scale across the different risk factors.
• The fact that we are estimating underlying risk across the population, and therefore cannot tag people in advance on an ordinal scale (we can't know in advance if a person belongs to a "high risk" group, we only see observed incidences) classification methods are only relevant if we want to classify people into 2 groups or 2 possible actions.

When using GLM, the set of estimated weights ${\displaystyle \beta }$ can be used to assign different values (or "points") to different values of the risk factors in ${\displaystyle \mathbf {X} }$ (continuous or nominal as indicators). The score can then be expressed as a weighted sum:

${\displaystyle {\begin{aligned}{\text{Score}}=\mathbf {X} \beta =\sum _{j=1}^{P}\mathbf {X} _{j}\beta _{j}\end{aligned}}}$

• Some scoring methods will translate the score into probabilities by using ${\displaystyle g^{-1}}$ (e.g. SAPS II score[2] that gives an explicit function to calculate mortality from the score[3]) or a look-up table (e.g. ABCD² score[4][5] or the ISM7 (NI) Scorecard[6]). This practice makes the process of obtaining the score more complicated computationally but has the advantage of translating an arbitrary number to a more familiar scale of 0 to 1.
• The columns of ${\displaystyle \mathbf {X} }$ can represent complex transformations of the risk factors (including multiple interactions) and not just the risk factors themselves.
• The values of ${\displaystyle \beta }$ are sometimes scaled or rounded to allow working with integers instead of very small fractions (making the calculation simpler). While scaling has no impact ability of the score to estimate risk, rounding has the potential of disrupting the "optimality" of the GLM estimation.

Let ${\displaystyle \mathbf {A} =\lbrace \mathbf {a} _{1},...,\mathbf {a} _{m}\rbrace }$ denote a set of ${\displaystyle m\geq 2}$ "escalating" actions available for the decision maker (e.g. for credit risk decisions: ${\displaystyle \mathbf {a} _{1}}$ = "approve automatically", ${\displaystyle \mathbf {a} _{2}}$ = "require more documentation and check manually", ${\displaystyle \mathbf {a} _{3}}$ = "decline automatically"). In order to define a decision rule, we want to define a map between different values of the score and the possible decisions in ${\displaystyle \mathbf {A} }$. Let ${\displaystyle \tau =\lbrace \tau _{1},...\tau _{m-1}\rbrace }$ be a partition of ${\displaystyle \mathbb {R} }$ into ${\displaystyle m}$ consecutive, non-overlapping intervals, such that ${\displaystyle \tau _{1}<\tau _{2}<\ldots <\tau _{m-1}}$.

The map is defined as follows:

${\displaystyle {\begin{aligned}{\text{If Score}}\in [\tau _{j-1},\tau _{j})\rightarrow {\text{Take action }}\mathbf {a} _{j}\end{aligned}}}$

• The values of ${\displaystyle \tau }$ are set based on expert opinion, the type and prevalence of the measured risk, consequences of miss-classification, etc. For example, a risk of 9 out of 10 will usually be considered as "high risk", but a risk of 7 out of 10 can be considered either "high risk" or "medium risk" depending on context.
• The definition of the intervals is on right open-ended intervals but can be equivalently defined using left open ended intervals ${\displaystyle (\tau _{j-1},\tau _{j}]}$.
• For scoring methods that are already translated the score into probabilities we either define the partition ${\displaystyle \tau }$ directly on the interval ${\displaystyle [0,1]}$ or translate the decision criteria into ${\displaystyle [g^{-1}(\tau _{j-1}),g^{-1}(\tau _{j}))}$, and the monotonicity of ${\displaystyle g}$ ensures a 1-to-1 translation.

• Framingham Risk Score
• QRISK
• TIMI
• Rockall score
• ABCD² score
• CHA2DS2–VASc score
• SAPS II

(see more examples on the category page Category:Medical scoring system)

The primary use of scores in the financial sector is for Credit scorecards, or credit scores:

• In many countries (such as the US) credit score are calculated by commercial entities and therefore the exact method is not public knowledge (for example the Bankruptcy risk score, FICO score and others). Credit scores in Australia and UK are often calculated by using logistic regression to estimate probability of default, and are therefore a type of risk score.
• Other financial industries, such as the insurance industry also use scoring methods, but the exact implementation remains a trade secret, except for some rare cases[6]

• COMPAS score for recidivism, as reverse-engineered by ProPublica[7] using logistic regression and Cox's proportional hazard model.