Question

2

2. Suppose next that we have even less knowledge of our patient, and we are only given the accuracy of the blood test and prevalence of the disease in our population. We are told that the blood test is 9# percent reliable, this means that the test will yield an accurate positive result in 9#% of the cases where the disease is actually present. Gestational diabetes affects #+1 percent of the population in our patient’s age group, and that our test has a false positive rate of #+4 percent. Use your knowledge of Bayes’ Theorem and Conditional Probabilities to compute the following quantities based on the information given only in part 2:

1. If 100,000 people take the blood test, how many people would you expect to test positive and actually have gestational diabetes?
2. What is the probability of having the disease given that you test positive?
3. If 100,000 people take the blood test, how many people would you expect to test negative despite actually having gestational diabetes?
4. What is the probability of having the disease given that you tested negative?

Comment on what you observe in the above computations. How does the prevalence of the disease affect whether the test can be trusted?

Answer 1

Let # be 8% and TP = Test Positive and TN = Test Negative D = Disease and ND = No Disease

So,

P(TP|D) = 0.98

P(TN|D) = 1-P(TP|D)

= 0.02

P(D) = 0.09

P(TP|ND) = 0.12

P(TN|ND) = 1-P(TN|ND)

= 0.88

A.

n = 100000

No. of diseased people = n*P(D)

= 9000

No. of people who will test positive and actually have disease = 9000*0.98

= 8820

B.

P(D|TP) = P(TP|D)*P(D)/[P(TP|D)*P(D) + P(TP|ND)*P(ND)]

= 0.98*0.09/[0.98*0.09 + 0.12*0.91]

= 0.0882/[0.0882+0.1092]

= 0.4468

C.

No. of people testing -ve (when they do have the disease) = 9000-8820

= 180

D.

P(D|TN) = P(TN|D)*P(D)/[P(TN|D)*P(D) + P(TN|ND)*P(ND)]

= 0.02*0.09/[0.02*0.09 + 0.88*0.91]

= 0.0018/[0.0018+0.8008]

= 0.00224

Diagnostic accuracy of a particular test increases as the disease prevalence decreases.

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