A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and 9.0 infections per 1000 in the placebo group.
I ran a computer simulation to predict possible outcomes of the trial if the null hypothesis is true—that is, if vaccinated and unvaccinated people are equally likely to contract HIV. I ran 1000 virtual trials of 20,000 people (10,000 per group) assuming that the vaccine is ineffective. Outcomes are expressed as excess infections in the placebo group. Here are the results of the 1000 virtual trials displayed as a histogram.
1. Roughly estimate the (two-sided) p-value associated with the trial’s outcome from the histogram.
2. From the simulation, I learn that the statistic ‘excess infections in the placebo group’ follows a normal distribution with a mean of 0 and a standard deviation (standard error) of 12.3. Use this information to calculate the two-sided p-value more precisely than in (11). Round to the nearest thousandth.
4. The effect size in this hypothetical trial (2.7 fewer infections per 1000 vaccinated) is slightly smaller than the effect size found in the real 2009 HIV vaccine trial (2.8 fewer infections per 1000 vaccinated); so why is the p-value smaller?
A) The hypothetical trial has less variability.
B) The sample size of the hypothetical trial is larger.
C) The real trial did not follow a normal distribution.
D) The real trial used an intention to treat analysis.
Can some one help us to answer this question
To answer these questions, we need to understand the concept of p-value and its relationship with the trial's outcome. The p-value helps determine whether the observed results are statistically significant or due to chance.
1. Estimating the p-value from the histogram:
To estimate the p-value from the histogram, we need to assess the proportion of trials where the outcome (excess infections in the placebo group) is as extreme as, or more extreme than, the observed outcome (9.0 infections per 1000 in the placebo group). This is a two-sided test as any extreme outcome, higher or lower, is of interest.
By examining the histogram, we can count the number of trials where the excess infections in the placebo group is at least as extreme as 9.0. Dividing this count by the total number of trials (1000) will give us an estimate of the p-value.
2. Calculating a more precise p-value using the mean and standard deviation:
Given that the excess infections in the placebo group follows a normal distribution with a mean of 0 and a standard deviation (standard error) of 12.3, we can calculate the z-score for the observed outcome (9.0 infections per 1000 in the placebo group).
The z-score is calculated as (observed outcome - mean) / standard deviation. In this case, it would be (9.0 - 0) / 12.3.
Once we have the z-score, we can refer to a standard normal distribution table or use statistical software to find the proportion of values that are as extreme or more extreme than the observed outcome. This proportion gives us the precise two-sided p-value.
3. Comparing the effect size and p-value in the hypothetical and real trials:
The effect size refers to the magnitude of the difference between the vaccine and placebo groups. In the hypothetical trial, the effect size was 2.7 fewer infections per 1000 vaccinated, while in the real 2009 HIV vaccine trial, it was 2.8 fewer infections per 1000 vaccinated.
The p-value is not directly related to the effect size. A smaller p-value indicates stronger evidence against the null hypothesis (no difference between vaccinated and unvaccinated groups). In this case, the p-value for the hypothetical trial is smaller, suggesting stronger evidence.
The options provided are:
A) The hypothetical trial has less variability.
B) The sample size of the hypothetical trial is larger.
C) The real trial did not follow a normal distribution.
D) The real trial used an intention to treat analysis.
Among these options, the correct answer is B) The sample size of the hypothetical trial is larger. A larger sample size generally increases the statistical power of a study, making it easier to detect smaller effect sizes. The hypothetical trial's larger sample size likely contributed to generating a smaller p-value despite a slightly smaller effect size.
This explanation breaks down the steps involved in estimating the p-value and highlights the key factor influencing the smaller p-value in the hypothetical trial.