Question

 

Sensitivity and Specificity

We are interested in looking at the connection between a test and a disease to investigate the ability of the test to distinguish between sick and healthy.

We look at a sample of 50,000 people who have been tested for a particular disease. Of these, 100 have the disease. Of the 100 who have the disease, 95 are receiving positive test results. Of those who are healthy, there are 48902 people who get negative test results.

Set up a table showing the number of sick / healthy with positive / negative tests.

What is the probability in this sample to have the disease?

What is the sensitivity and specificity of the test and what does this mean in words?

4. What is the predictive value of positive test? What does this mean in words and what does this mean for the practical value of the test? Also, find out what the predictive value of the negative test is.Also record positive predictive value and negative predictive value using Bayes rule.

5. If we instead test vulnerable risk groups, the probability of the disease increases to 5%. The sensitivity and specificity of the test are the same as you found in the assignment. 6.3. What happens to predictive positive value if we look at 50,000 people at risk of the disease? Set up a new table and rain out PPV.

We are looking at another type of test, the probability of having the disease in vulnerable countries is 10%. The probability that the test is positive when infected with the disease is 0.999 and the probability that the test is negative when not infected is 0.99.

What is the sensitivity and specificity of the test?

7. What will be positive predictive value? Use Baye's rule.

Worldwide, the likelihood of having the disease is 1%. If the test's sensitivity and specificity are the same, what will be the positive predictive value?

Do you see a connection between the prevalence of the disease and the positive predictive value?

Answer 1

4.

	Disease	Non-Disease	Total
Positive	95	998	1093
Negative	5	48902	48907
Total	100	49900	50,000

Predictive value of positive test = Number of having Disease with positive results / Number with positive results

= 95 / 1093 = 0.0869 = 8.69%

Predictive value of positive test means the chance that a person with a positive test truly has the disease.

A higher predictive value of positive test shows that the test is more accurate in prediction of the disease.

Predictive value of negative test = Number of having no Disease with negative results / Number with negative results

= 48902 / 48907 = 0.9999 = 99.99 %

Using Bayes Rule,

positive predictive value = Given positive, probability of having disease = P(D | P) = P(D P) / P(P) =

= [95 / 50000 ] / [1093 / 50000] =  95 / 1093 = 0.0869

and negative predictive value = Given negative results, probability of having no disease = P(~D | N) = P(~D N) / P(N) = [48902 / 50000 ] / [48907 / 50000] = 48902 / 48907 = 0.9999

5.

New table is

	Disease	Non-Disease	Total
Positive	2375	950	3325
Negative	125	46550	46675
Total	2500	47500	50000

The probability of the disease increases to 5%.

So, Total Disease = 5% of 50,000 = 2500

Total non- Disease = 50000 - 2500 = 47500

From 6.3,

sensitivity = 95 / 100 = 0.95 = 95%

and specificity = 48902 / 49900 = 0.98 = 98%

Number of Positive and Disease = sensitivity * Total Disease = 0.95 * 2500 = 2375

Number of Negative and Disease = 2500 - 2375 = 125

Number of Negative and Non-Disease = specificity * Total Non-Disease = 0.98 * 47500 = 46550

Number of Positive and Non-Disease = 47500 - 46550 = 950

Predictive value of positive test = Number of having Disease with positive results / Number with positive results

= 2375 / 3325 = 0.7143 = 71.43%

7.

From earlier part,

sensitivity = 0.999 = 99.9%

and specificity = 0.99 = 99%

The likelihood of having the disease is 1% .

Assuming the total population size as 100, number of preople having disease = 1% of 100 = 1,

Number of people having no disease = 100 - 1 = 99

Number of people having disease and have positive result = sensitivity * Total Disease = 0.999 * 1 = 0.999

Number of Negative and Non-Disease = specificity * Total Non-Disease = 0.99 * 99 = 98.01

Number of people having no disease and have positive result = 99 - 98.01 = 0.99

Total number of having positive results = Number of people having disease and have positive result + Number of people having no disease and have positive result = 0.999 + 0.99 = 1.989

Using Bayes Rule,

positive predictive value = Given positive, probability of having disease = P(D | P) = P(D P) / P(P) =

= [0.999 / 100 ] / [1.989 / 100] =  0.999 / 1.989 = 0.5022624 = 50.22 %

So, high prevalence of the disease gives high positive predictive value. Thus, there is positive relation between the prevalence of the disease and the positive predictive value.