Question

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 93 percent reliable, this means that the test will yield an accurate positive result in 93% of the cases where the disease is actually present. Gestational diabetes affects 4 percent of the population in our patient’s age group, and that our test has a false positive rate of 7 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

1. If 100,000 people take the blood test, how many people would you expect to test positive and actually have gestational diabetes?

	Have gestational diabetes	Do not have gestational diabetes	Total
Test positive	0.96 X 0.93 = 0.8928	0.04 X 0.07 = 0.0028	0.8956
Test negative	0.96 - 0.8928=0.0672	0.04 - 0.0028 = 0.0372	0.1016
Total	1 - 0.04 = 0.96	0.04	1

Number of people Test positive AND Have gestational diabetes = 100,000 X 0.8928 = 89,280

1. What is the probability of having the disease given that you test positive?

P(Have gestational diabetes/ Test positive) = P(Have gestational diabetes AND Test positive)/ P(Test positive)

= 0.8928/0.8956

= 0.9969

1. If 100,000 people take the blood test, how many people would you expect to test negative despite actually having gestational diabetes?

Number of people Test negative AND Have gestational diabetes = 100,000 X 0.0672 = 6,720

1. What is the probability of having the disease given that you tested negative?

P(Have gestational diabetes/ Test negative) = P(Have gestational diabetes AND Test negative)/ P(Test negative)

= 0.0672/0.1016

= 0.6614

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

	Have gestational diabetes	Do not have gestational diabetes	Total
Test positive	a= 0.8928 (True Positive)	b= 0.0028 (False Positive)	0.8956
Test negative	c=0.0672 (False Negative)	d= 0.0372 (True Negative)	0.1016
Total	0.96	0.04	1

Sensitivity = a/ (a+c)

              =0.8928/(0.8928 + 0.0672)

            = 0.8918/0.96

          = 0.93

Since sensitivity, i.e., the ability to correctly identify the proportion of patients with the disease = 93% is high, we can conclude that the test can be trusted.