From Grok:
Question:
in a drug development trial, a subject developed an adverse event of skin infections while on active drug which is known to suppress the immune system. The drug is dosed monthly, and the adverse event developed 10 days after the dose. The skin infections resolved after 20 days. The drug was not given for a month and no new skin infections developed. If the subject was rechallenged with the same dose of drug, and developed skin infections again, what is the statistical test to determine is he probability that the drug is associated with skin infections. More generally, if the drug was dosed on alternate months, how many times would development of skin infections over the next year of treatment bring the probability that the drug is associated with developing skin infections reach a statistical significance of < 0.05?
Answers:
To determine the association between the drug and the development of skin infections, we need to use a statistical test suited for repeated measures or matched pairs data since the same subject is involved in both conditions (with and without the drug). Here are the steps and considerations:
Statistical Test for Immediate Rechallenge
To determine the association between the drug and the development of skin infections, we need to use a statistical test suited for repeated measures or matched pairs data since the same subject is involved in both conditions (with and without the drug). Here are the steps and considerations:
Statistical Test for Immediate Rechallenge
McNemar's Test: This test is appropriate if you are comparing the occurrence of skin infections before and after the rechallenge in the same subject. You would set up a 2x2 contingency table:
Cell A: Number of times skin infection occurs both times (not applicable here since no infections occurred without drug).
Cell B: Number of times infection occurs only with drug (1).
Cell C: Number of times infection occurs only without drug (0).
Cell D: Number of times no infection occurs with or without drug (1 if no infection during the off-month).
With this setup, since you have only one subject and one event:
Cell B = 1 (infection with drug)
Cell D = 1 (no infection without drug)
McNemar's test statistic would be calculated as:
χ2=(B−C)2B+C=(1−0)21=1
With one degree of freedom, this would give a p-value of exactly 0.5 (not significant at 0.05 level), but this is with very limited data.
For Multiple Doses Over a Year
If the drug is dosed on alternate months, and we want to see how many times skin infections must occur to achieve statistical significance:
Binomial Test: Here, each month's outcome (infection or no infection) can be considered a Bernoulli trial where:
p = probability of infection with drug (unknown, but we're testing if it's significantly greater than without drug)
q = probability of no infection without drug (assumed to be 1 if no infection occurs in off months)
Let's say we observe infections in 'x' out of 'n' months where the drug is given:
Null Hypothesis (H0): The drug does not increase the risk of infection (p = q).
Alternative Hypothesis (Ha): The drug increases the risk of infection (p > q).
Since we have a baseline where no infections occur without the drug, we're essentially testing if the proportion of infections with the drug is significantly greater than zero.
With 6 months of potential dosing in a year, if we observe infections in:
3 or more out of these 6 months, using a binomial test (where success is an infection), the p-value would be below 0.05, considering the null hypothesis that the probability of infection is 0 without the drug. This calculation could be done in R or Python like this:
R
pbinom(2, 6, 0, lower.tail = FALSE) # This would yield a p-value less than 0.05 for 3 or more infections
Exact number: With no infections in off months, observing 3 infections out of 6 doses gives a p-value of about 0.03125, which is just under 0.05.
Therefore, if skin infections occur in 3 or more of the 6 months the drug is administered over the next year, the probability that the drug is associated with developing skin infections would reach statistical significance at p < 0.05. Remember, this is under the assumption of zero infections during the months without the drug. If any infections occur during off months, you would need to adjust the null hypothesis probability accordingly.
I apologize that the formatting is lame, as I cannot get Grok to format properly., and I am not going to retype it. If you have Grok, copy the question and presumably get the full answer.
