Suppose that the probability of a disease is 0.00369 in a population of unvaccin

Question

Suppose that the probability of a disease is 0.00369 in a population of unvaccinated subjects and that the probability of the disease is 0.001 in a population of vaccinated subjects. (a) What are the odds of disease without vaccine relative to the odds of disease with vaccine? (b) How many people out of 100,000 would get the disease if they were not treated? (c) How many people out of 100,000 would get the disease if they were vaccinated? (d) What proportion of people out of 100,000 who would have gotten the disease would be spared from it if all 100,000 were vaccinated? (This is called the protection rate.) (e) Follow the steps in parts (a)-(d) to derive the odds ratio and the protection rate if the unvaccinated probability of disease is 0.48052 and the vaccinated probability is 0.2. (The point is that the odds ratio is the same in the two situations, but the total benefit of vaccination also depends on the probabilities.)

Explanation / Answer

The probability of disease in unvaccinated subjects is 0.00369, and that of vaccinated subjects is 0.001

(a)The odds of disease in unvaccinated subjects is 369:99631 and that of vaccinated subjects is 100:99900 or 1:999.

(b) The number of individuals who will get disease in unvaccinated subjects is (0.00369*100000)=369 out of 100000

(c) The number of individuals who will get disease in vaccinated subjects is (0.001*100000)=100 out of 100000

(d) The number of people who would have spared from the disease if they were vaccinated is

(369-100)/100000=269/100000

(e)

Now the probability of disease in unvaccinated subjects is 0.48052, and that of vaccinated subjects is 0.2

(ea)The odds of disease in unvaccinated subjects is 48052:51948 and that of vaccinated subjects is 20000:80000 or 1:999.

(eb) The number of individuals who will get disease in unvaccinated subjects is (0.48052*100000)=48052 out of 100000

(ec) The number of individuals who will get disease in vaccinated subjects is (0.2*100000)=20000 out of 100000

(ed) The number of people who would have spared from the disease if they were vaccinated is

(48052-20000)/100000=28052/100000

Calculating Odds Ratios

We will calculate odds ratios (OR) using following formula
Where

a = Number of exposed cases(Vaccinated, Diseased)
b = Number of exposed non-cases(Vaccinated, NotDiseased)
c = Number of unexposed cases(NotVaccinated, Diseased)
d = Number of unexposed non-cases(NotVaccinated,NotDiseased)

Odds Ratio=(a/c)/(b/d)0=ad/bc

For Ist part a=100, b=99900, c=369,d=99631
Odds Ratio=(ad/bc)= (100*99631)/(99900*369)=0.270
For Second Case a=20000,b=800000,c= 48052,d=51984
Odds Ratio=(ad/bc)=(20000*51984)/(80000*48052)=0.270

The odds ratio is same yet total benefit of Ist vaccine is greater than second.

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