September 16, 1999
FACT SHEET ON ROTAVIRUS VACCINE
Addendum: A Note on Probability and Risk-Benefit Analysis
Should the 3 cases of intussusception in the vaccine trial group of about 10,000, occurring within a week of vaccination, raise warning flags, even though "not significantly different" from the one case in the control group of 4,633, which occurred sometime within an unspecified time interval of up to one year? (See MMWR 1999 (July 16)48:577-581, "Intussusception among recipients of rotavirus vaccine-United States 1998-1999," for statistics quoted on this fact sheet.)
How often would one observe an incidence of 3 cases in the post-vaccine week in a sample of 10,000, by chance alone, when the actual risk is only 1/100,000 infant- weeks, as estimated from the CDC's figures for "normal" incidence within an "infant-week"?
The probability P that a sample of n individuals contains r with condition A (intussusception) and (n-r) with condition B (no intussusception), when I is the probability of A and (1 - I) the probability of B, is given by:
n!/r!(n-r)! Ir (1-I)(n-r) (assuming a binomial distribution function)
The probability of no cases of intussusception in 10,000 subjects within one week of the vaccine is found thus:
n = 10,000, r = 0, I = 1/100,000
P0 = 1 x 1 x (0.99999 raised to the 10,000th power) or about 90%.
In other words, 90% of trials of this size will miss a complication occurring with this probability altogether, and it will only be discovered in post-licensure surveillance. The probability of one or more cases being seen is about 10%.
The probability of 3 cases being seen in the trial, by chance alone, is 0.00015 or 0.015%. The standard for rejecting the null hypothesis (no difference between test group and normal population) is P < 0.05 or 5%.
Is a risk of 0.0003 important? A risk this high would mean 150 additional cases of intussusception resulting from 1.5 million doses of vaccine (the majority requiring surgery) if each baby required 3 doses. If the risk is 0.00006 per dose, the level seen in post-licensure trials, there would be 90 excess cases. If the vaccine is given to the entire population, say 3 doses x around 3.6 million children < 1 year old, that would be between 216 and 1080 cases. Is it worth it, to prevent 20 to 40 deaths from rotavirus (assuming the vaccine is 100% effective) and 55,000 hospitalizations to treat severe diarrhea with intravenous rehydration? At $240 per series of three shots, the vaccine cost is $22 - $43 million per life saved and $16,000 per hospitalization saved - before you add the costs of treating the bowel obstructions (some of which themselves could be fatal).
Even more to the point, does the benefit justify overriding patients' rights to decline the vaccine? Does any hypothetical benefit to society justify the abrogation of patients' rights to decline a medical treatment?
Note: The Poisson distribution, rather than the binomial, is appropriate for events occurring randomly over time. The binomial distrubion approximates the Poisson, and the approximation gets better and better as n increases and the probability decreases. The approximation is quite excellent already with a probability as high as 0.05 and n as low as 100. The results will be indistinguishable for practical purposes with a probability of 0.00001 and n >> 1,000.