## Calculating Residual Risk

Residual risk is the risk remaining after a negative screening test. At the core of the calculation of residual risk resides Bayes’ Theorem. This theorem is used to calculate the probability of an outcome given the probability of events required for that outcome. The probability of the required events is compounded to calculate the combined probability of the outcome. In the case of reproductive risk, we calculate the probability that a child has a disorder given the probability that the parents are carriers. With appropriate genetic screening, these probabilities can be more precisely defined.

Given the Bayes’ Theorem, we can calculate the residual risk based on the combination of the carrier frequency of the disorder in the population, and the detection rate for the carrier allele. Here is the starting formula, let’s fill in the values:

To fill in the values we will need two values specific to the disorder or condition that we are evaluating.

**Carrier Frequency**– The proportion of individuals in the patient population who carry a single copy of the allele responsible for the disorder or condition. These alleles follow a recessive expression pattern where individuals with a single copy don’t show any symptoms of the disorder or condition.**Detection Rate**– This is the disorder-specific value that represents how well disorder or condition-causing variants can be detected. This value is based on the lab-specific test procedures and the publications that support them.

With these terms, we can fill in the formula for calculating the residual risk, which is the probability that an individual has the disorder allele but it is not detected:

We can fill in the terms as follows:

– the probability of having the disorder allele for the population at large; this is the carrier frequency**P(HasDisorder)***CF*.is a complement of the detection rate, which is**P(NotDetected|HasDisorder)***1-DetectionRate*.There are two parts of the not detected probability: there is the probability that an individual is a carrier, but the carrier allele was not detected, and there is the probability that the individual is not a carrier (and the disorder allele is not detected). The first term is the same as the numerator; this leaves just the second term, which is the probability that an individual does not have the disorder allele. This is the complement of the carrier frequency (1-cf) since we assume a 100% probability of not detecting the disorder allele when it is not there.**P(NotDetected)**

Putting this all together, we get the following formula:

This allows for the computation of the residual risk for each individual. Note that individuals from different populations may have different residual risks. Next, we will combine the risk values for each parent of a child to calculate the reproductive risk.

## Calculating Reproductive Risk

The reproductive risk is the likelihood that a couple may have a child with a genetic disorder or condition. With two negative test results for the presence of a carrier allele, the reproductive risk can be calculated as follows:

Here, we take the residual risk (the probability of having a carrier allele) and multiply it by the probability of passing it on. If a parent has a carrier allele, that parent has a 50% probability of passing on the allele to the child. If one partner has a positive test result, the residual risk becomes 1 since there is a 100% chance that the sample has an allele with the carrier allele. That partner still only has a 50% chance of passing on the allele, though, so the 0.5 probability of transmission term remains unchanged.

## Cystic Fibrosis

Here is a quick example to illustrate how reproductive risk is calculated. Given a detection rate of 99%, a caucasian carrier frequency of 0.04 (1/25), and a Southeast Asian carrier frequency of 0.025 (1/40). If a couple undergoes carrier screening, and both individuals have negative test results for carrier alleles, the reduction in residual risk can be calculated as follows:

Before Testing:

If you would like to find out more about how the Golden Helix suite of software can be used to automate the reporting of carrier variants or if you have questions related to carrier screening, don’t hesitate to contact us at support@goldenhelix.com. For more details related to the calculations here, you can check out this article: Bayesian Analysis and Risk Assessment in Genetic Counseling and Testing