Baye’s Rule and Medical Screening Tests

Baye’s Rule Baye’s Rule is used in medicine and epidemiology to calculate the probability that an individual has a disease, given that they test positive on a screening test. Baye’s Rule is used in medicine and epidemiology to calculate the probability that an individual has a disease, given that they test positive on a screening test. Example: Down syndrome is a variable combination of congenital malformations caused by trisomy 21. It is the most commonly recognized genetic cause of mental retardation, with an estimated prevalence of 9.2 cases per 10,000 live births in the United States. Because of the morbidity associated with Down syndrome, screening and diagnostic testing for this condition are offered as optional components of prenatal care. Example: Down syndrome is a variable combination of congenital malformations caused by trisomy 21. It is the most commonly recognized genetic cause of mental retardation, with an estimated prevalence of 9.2 cases per 10,000 live births in the United States. Because of the morbidity associated with Down syndrome, screening and diagnostic testing for this condition are offered as optional components of prenatal care. Many studies have been conducted looking at the effectiveness of screening methods used to identify “likely” Down syndrome cases. Many studies have been conducted looking at the effectiveness of screening methods used to identify “likely” Down syndrome cases.

Study of “Triple Test” Effectiveness The results of a study looking at the effectiveness of the “triple-test” are presented below: The results of a study looking at the effectiveness of the “triple-test” are presented below: How well does the triple-test perform? How well does the triple-test perform?

The General Situation Patient presents with symptoms, and is suspected of having some disease. Patient either has the disease or does not have the disease. Patient presents with symptoms, and is suspected of having some disease. Patient either has the disease or does not have the disease. Physician performs a diagnostic test to assist in making a diagnosis. Physician performs a diagnostic test to assist in making a diagnosis. Test result is either positive (diseased) or negative (healthy). Test result is either positive (diseased) or negative (healthy).

The General Situation Test Result True Disease Status Diseased (+) Healthy (-) Diseased (+)Correct False Negative Healthy (-) False Positive Correct

Definitions False Positive: Healthy person incorrectly receives a positive (diseased) test result. False Positive: Healthy person incorrectly receives a positive (diseased) test result. False Negative: Diseased person incorrectly receives a negative (healthy) test result. False Negative: Diseased person incorrectly receives a negative (healthy) test result.

Goal Minimize chance (probability) of false positive and false negative test results. Minimize chance (probability) of false positive and false negative test results. Or, equivalently, maximize probability of correct results. Or, equivalently, maximize probability of correct results.

Accuracy of Tests in Development Sensitivity: probability that a person who truly has the disease correctly receives a positive test result. Sensitivity: probability that a person who truly has the disease correctly receives a positive test result. Specificity: probability that a person who is truly healthy correctly receives a negative test result. Specificity: probability that a person who is truly healthy correctly receives a negative test result.

Triple-Test Performance How does the triple-test perform using these measures? 87/118 = /118 = /4072 = /4072 =.0499

Test Result  What ? Now suppose you are have just been given the news the results of the “triple test” are positive for Down syndrome. Now suppose you are have just been given the news the results of the “triple test” are positive for Down syndrome. What do you want to know now? What do you want to know now? You probably would like to know what the probability that your unborn child actually has Down syndrome. You probably would like to know what the probability that your unborn child actually has Down syndrome.

Accuracy of Tests in Use Positive predictive value: probability that a person who has a positive test result really has the disease. Positive predictive value: probability that a person who has a positive test result really has the disease. Negative predictive value: probability that a person who has a negative test result really is healthy. Negative predictive value: probability that a person who has a negative test result really is healthy. To find these we use Baye’s Rule to “reverse the conditioning”. To find these we use Baye’s Rule to “reverse the conditioning”.

Case-Control Nature of Study Generally these studies are case-control in nature, meaning that the disease is not random! Generally these studies are case-control in nature, meaning that the disease is not random! We specifically choose individuals with the disease to perform the screening test on, therefore we cannot talk about or calculate P(D + ) or P(D - ) using our results. We specifically choose individuals with the disease to perform the screening test on, therefore we cannot talk about or calculate P(D + ) or P(D - ) using our results. To find these we use Baye’s Rule to “reverse the conditioning”. To find these we use Baye’s Rule to “reverse the conditioning”.

Baye’s Rule This requires prior knowledge about the probability of having the disease, P(D + ), and hence the probability of not having the disease, P(D - ). This probability is called the positive predictive value (PPV) of the triple-test.

Negative Predictive Value This also requires prior knowledge of the probabilities of having and not having the disease.

Example: PPV for Triple-Test Prior knowledge about Down’s Syndrome suggests P(D+) = Prior knowledge about Down’s Syndrome suggests P(D+) = or roughly 1 in 1,000 (i.e. P(D+) =.001). or roughly 1 in 1,000 (i.e. P(D+) =.001). We also now from our earlier work that… We also now from our earlier work that… P(T + |D + ) =.7373 P(T - |D + ) =.2627 P(T - |D - ) =.9501 P(T + |D - )=.0499

Comparing to Prior Probability In the absence of any test result the probability of having a child with Down’s Syndrome is P(D + ) = In the absence of any test result the probability of having a child with Down’s Syndrome is P(D + ) = Given a positive test result we have Given a positive test result we have P(D + |T + ) =.0134 P(D + |T + ) =.0134 Comparing these two probabilities in the form of a ratio we find.0134/ = Comparing these two probabilities in the form of a ratio we find.0134/ = When we look at in practical terms however there is still only a 1.34% chance that the unborn child has Down’s Syndrome. When we look at in practical terms however there is still only a 1.34% chance that the unborn child has Down’s Syndrome.

Example: NPV for Triple-Test Prior knowledge about Down’s Syndrome suggests P(D-) = Prior knowledge about Down’s Syndrome suggests P(D-) = or roughly 999 in 1,000 (i.e. P(D-) =.999). or roughly 999 in 1,000 (i.e. P(D-) =.999). We also now from our earlier work that… We also now from our earlier work that… P(T + |D + ) =.7373 P(T - |D + ) =.2627 P(T - |D - ) =.9501 P(T + |D - )=.0499

Comparing to Prior Probability In the absence of any test result the probability of NOT having a child with Down’s Syndrome is P(D - ) = In the absence of any test result the probability of NOT having a child with Down’s Syndrome is P(D - ) = Given a negative test result we have Given a negative test result we have P(D - |T - ) = P(D - |T - ) = Comparing these two probabilities in the form of a ratio we find.99975/ = Comparing these two probabilities in the form of a ratio we find.99975/ = When we look at in practical terms the probability only increases by when we have a negative test result. When we look at in practical terms the probability only increases by when we have a negative test result.

Questions to Think About What do you think of the “Triple-Test” for Down’s Syndrome? What do you think of the “Triple-Test” for Down’s Syndrome? Would you advocate having it administered to all women during their first trimester of pregnancy? Would you advocate having it administered to all women during their first trimester of pregnancy? Would you have it done if you were pregnant? Would you have it done if you were pregnant?