Question

A new diagnostic test is developed for treating some disease is being tested among 10,000 hospital patients, among which 2000 have the disease. If the test came back positive in 1975 of the diseased subjects and came back negative in 7600 of the healthy patients,
a. Calculate SN and SP along with their approximate 95% CI’s

2. Calculate the probability of having the disease given a positive test

3. Calculate the probability of having the disease given a negative test

4. Calculate the PPV, NPV and their approximate 95% CI. Bonus marks for the exact CI’s.

5. If the true prevalence of disease is actually 5 per 100, calculate PPV and NPV. Comment

   on any differences observed and how you would feel about a positive or negative result.

Answer 1

Given:

	Disease	Healthy(No disease)	Total
Positive	1,975 (a)	400 (b)	2,375 (a+b)
Negative	25 (c)	7,600 (d)	7,625 (c+d)
Total	2,000 =n1	8,000 =n2	10,000 (N)

a.

SN =Sensitivity =a/n1 =1,975/2,000 =0.9875

SP =Specificity =d/n2 =7,600/8,000 =0.95

Confidence Intervals(CI's):

Standard Error for SN =SE(SN) = = =0.00248

Standard Error for SP =SE(SP) = = =0.00244

95% CI for SN =SN(Z*SE(SN)) =0.9875(1.96*0.00248) =(0.9826, 0.9924)

95% CI for SP =SP(Z*SE(SP)) =0.95(1.96*0.00244) =(0.9452, 0.9548)

b.

The probability of having the disease given a positive test =P(D/+ve) =PPV =a/(a+b) =1,975/2,375 =0.8316

The probability of having the disease given a negative test =P(D/-ve) =c/(c+d) =25/7,625 =0.0033

c.

PPV =Positive Predictive Value =The probability of having the disease given a positive test =a/(a+b)=1,975/2,375 =0.8316

NPV =Negative Predictive Value =The probability of not having the disease given a negative test =d/(d+c) =7,600/7,625 =0.9967

Confidence Intervals (CI's):

SE(PPV) = = =0.00768

SE(NPV) = = =0.00066

95% CI for PPV =0.8316 (1.96*00768) =(0.8165, 0.8467)

95% CI for NPV =0.9968 (1.96*0.00066) =(0.9955, 0.9981)

d.

If the true prevalence of disease is 5 per 100 =5% =0.05:

Prevalence rate =n1/N =n1/10,000 =0.05

So, n1 =10,000*0.05 =500 i.e., 500 out of 10,000 have the disease. Thus, 9,500 are healthy.

(Now, in order to calculate PPV and NPV, we need new values of a, b, c and d. in the above table - that is, in how many of these 500 diseased subjects, the test came positive and in how many of 9,500 healthy patients, the test came negative(or positive)).