Area under the curve and risk assessment

This month, Greg was finally released from prison. In fact, many people in his county were released from prison this month, some of them under parole supervision. Unfortunately, some of the released individuals will commit at least one crime in the coming several years. Many people relapse into criminal behaviour because the factors that initially facilitated their crimes often persist. Additionally, it can be quite challenging to reintegrate into society after prison.

Although three years later some former prisoners will reoffend and end up in prison again, the majority will desist from crime. Policymakers and practitioners would like to know in advance who among released prisoners is likely to reoffend, so they can keep a closer eye on them and provide additional support if necessary, such as access to substance abuse treatment programmes or vocational training.

But how can they do it, and how can they do it effectively?

Risk assessment

The main way to predict potential future reoffending is through the use of some form of risk assessment, such as standardised risk assessment instruments. These instruments evaluate various risk factors and protective factors that comprise an individual’s characteristics and circumstances.

Many different instruments exist, and they have slightly different abilities to predict potential future reoffending. The ability of a risk assessment instrument to predict future outcomes is called predictive validity, and one of the measures of predictive validity is the AUC ROC curve.

AUC has several technical interpretations, but our goal is to create a simple, actionable intuition behind it.

Area Under the Curve (AUC) explanatory primer

You can watch the video accompanying this text on YouTube ->

Imagine that we implemented a risk assessment instrument and measured the reoffending risk of Greg, using a risk score that ranges from 0 to 100. He scored somewhere around 30. Then we measured the risk in 10 more individuals, who each received different scores, ranging from 5 to 90.

We let these people go about their lives and checked on them after 3 years.

Different AUC corresponds to different degree of separation

As you can see, in the first three scenarios, some of the released prisoners reoffended, and all offenders are among the top scorers. This perfect separation between reoffenders and desisters corresponds to an AUC of 1. It means that our hypothetical risk assessment tool can perfectly distinguish one group of individuals from the other. The number of people who reoffended and their scores don’t matter; as long as they are clearly grouped, the AUC is 1.

If our risk assessment instrument is no better at predicting than a random coin flip, then the AUC will be close to 0.5. Again, different arrangements are possible, but since all of them are almost evenly mixed, the AUC is 0.5. However, in practice, we usually don’t see AUC values of 1 or 0.5. Most risk assessment instruments have AUC values around 0.7–0.8. These values can correspond to different arrangements, with some overlap in scores between reoffenders and desisters.

The larger AUC, the better?

In general, yes. However, AUC, although an important metric, only presents half of the picture. To decide who we consider at high risk of reoffending, we need to choose a cutoff risk score. Individuals scoring at or above the chosen score would be considered high risk, while those scoring lower would be considered low risk.

In the hypothetical example on the right, if we set the cutoff score at around 60, we would correctly identify some reoffenders and desisters but also misidentify a few. We would get 7 correct and 4 wrong. If we set a lower cutoff score, we would get 6 correct and 5 wrong. Overall, in this hypothetical case, we would be more likely to be correct using the higher threshold. However, in real-life applications, the choice of the threshold depends on the intended application of risk assessment results and other societal factors external to risk assessment.

Prioritisation

Overall, such properties of the risk assessment instrument as sensitivity and specificity depend on the choice of the cutoff. Practitioners and policymakers should pay closer attention to them and not focus too much on AUC, as most commonly used risk assessment tools have comparable AUC values.

If we want to prioritise the prevention of crime and provide intensive supervision to most potential reoffenders, we might consider a lower cutoff. Reoffenders would have a higher weight in our decision-making. On the other hand, if we want to prioritise less intrusive supervision, such as in a cohort of individuals who committed less dangerous crimes, and facilitate independence and reintegration, we might choose a higher threshold. This way, desisters will have a higher weight in our decision-making.

Advanced explanation. Harrel’s c-index

There is another widely used metric, equivalent to AUC, which can provide a complementary interpretation of the concept. The metric is called the c-index, and it was proposed by Frank Harrell.

Think of the c-index as a way to check how well a prediction model gets the ranking right. A pair of individuals selected from a sample is considered concordant if the individual with the higher predicted risk actually experiences the event (e.g., recidivism) sooner than the one with the lower predicted risk. You can see the animation of checking the pairs below.

So, the c-index measures the proportion of these “concordant pairs” among all pairs of individuals. If the c-index is 1.0, it means all pairs are perfectly ranked according to their predicted risks. If it is 0.5, it is akin to a random arrangement, where the model is no better than chance at predicting who will experience the event earlier.

In simpler terms, the c-index helps us see how often the model gets the order of risks right, with higher values indicating better predictive performance.