Question

The number of cases of Lyme disease, a tick-borne illness, more than doubled between 2001 and 2015 in North America. A vaccine for Lyme disease was approved by the U.S. Food and Drug Administration in 1998, but it was then removed from the market just 3 years later (Nigrovic and Thompson 2007). Did it work? A clinical trial of the efficacy of this vaccine compared 5469 people who received the vaccine to 5467 people who received a placebo (Steere et al. 1998). Of these, 15 people with the vaccine had developed Lyme disease after 2 years, and 65 people in the placebo control group had developed the disease. In the same study, side effects were also tracked. The most severe possible side effect was joint pain. Of the vaccinated group, 71 people had joint pain, while in the control group 6 people had joint pain.

a. Calculate the relative risk of getting Lyme disease for patients who received the vaccine compared to those who did not.
b. Based on that relative risk calculation, by what amount does the vaccine reduce the rate of getting Lyme disease?
c. Do these data provide evidence that the vaccine was effective in reducing the rate of contracting Lyme disease? Carry out an appropriate hypothesis test.
d. What is the relative risk of joint pain for vaccinated patients compared to the controls?
e. Is there evidence that the probability of joint pain is different between vaccinated and control groups? Do a hypothesis test.

Answer 1

The vaccine reduced the risk of contracting Lyme disease by 86%, which supports its effectiveness. The relative risk of joint pain for vaccinated individuals was significantly higher than that of the control group.

Step-by-step explanation:

a. To calculate the relative risk of getting Lyme disease after vaccination compared to not being vaccinated, we use the formula: Relative risk = (incidence in the exposed group) / (incidence in the control group).

In this trial, the incidence in the vaccinated group is 15 cases out of 5469, and the incidence in the placebo group is 65 cases out of 5467. Thus, the relative risk is (15/5469) / (65/5467), which equals approximately 0.14. This suggests that vaccinated individuals had a 14% risk of developing Lyme disease compared to the control group.

b. To find out by what amount the vaccine reduces the rate of contracting Lyme disease, we calculate 1 - relative risk. This equals 1 - 0.14 which is 0.86 or 86%. Therefore, the vaccine reduces the risk of contracting Lyme disease by 86%.

c. For evidence of vaccine efficacy in reducing Lyme disease, a chi-square test can be used to test the difference in proportions: null hypothesis (H0) is that there is no difference, and alternative hypothesis (Ha) is that there is a difference. With the given data, the chi-square statistic yields a value that is significantly different from the expected if H0 were true, indicating that the vaccine was effective (p-value < 0.05).

d. The relative risk of joint pain for vaccinated patients compared to controls is calculated in the same way. Here, the relative risk is (71/5469) / (6/5467), equating to about 19.67. This implies that vaccinated individuals had a much higher risk of developing joint pain compared to the control group.

e. To determine if there is a significant difference in the probability of joint pain between the two groups, a chi-square test can be conducted, using the same hypothesis framework as for Lyme disease. The results would indicate a statistically significant higher risk of joint pain in the vaccinated group (p-value < 0.05).