Risk score (or risk scoring) is the name given to a general practice in applied statistics, bio-statistics, econometrics and other related disciplines, of creating an easily calculated number (the score) that reflects the level of risk in the presence of some risk factors (e.g. risk of mortality or disease in the presence of symptoms or genetic profile, risk financial loss considering credit and financial history, etc.).
Risk scores are designed to be: Simple to calculate: In many cases all you need to calculate a score is a pen and a piece of paper (although some scores use rely on more sophisticated or less transparent calculations that require a computer program). Easily interpreted: The result of the calculation is a single number, and higher score usually means higher risk. Furthermore, many scoring methods enforce some form of monotonicity along the measured risk factors to allow a straight forward interpretation of the score (e.g. risk of mortality only increases with age, risk of payment default only increase with the amount of total debt the customer has, etc.). Actionable: Scores are designed around a set of possible actions that should be taken as a result of the calculated score. Effective score-based policies can be designed and executed by setting thresholds on the value of the score and associating them with escalating actions.
Formal definition
A typical scoring method is composed of 3 components: A set of consistent rules (or weights) that assign a numerical value ("points") to each risk factor that reflect our estimation of underlying risk. A formula (typically a simple sum of all accumulated points) that calculates the score. A set of thresholds that helps to translate the calculated score into a level of risk, or an equivalent formula or set of rules to translate the calculated score back into probabilities (leaving the nominal evaluation of severity to the practitioner).
Items 1 & 2 can be achieved by using some form of regression, that will provide both the risk estimation and the formula to calculate the score. Item 3 requires setting an arbitrary set of thresholds and will usually involve expert opinion.
Estimating risk with GLM
Risk score are designed to represent an underlying probability of an adverse event denoted given a vector of explaining variables 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 is estimated using a generalized linear model:
Where is a real-valued, monotonically increasing function that maps the values of the linear predictor to the interval . GLM methods typically uses the logit or probit as the link function.
Estimating risk with other methods
While it's possible to estimate 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.
Constructing the score
When using GLM, the set of estimated weights can be used to assign different values (or "points") to different values of the risk factors in (continuous or nominal as indicators). The score can then be expressed as a weighted sum: