91Ó°ÊÓ

In a 2009 study reported in the New England Journal of Medicine, Boyer et al. randomly assigned children aged 6 months to 18 years who had nonlethal scorpion stings to receive an experimental antivenom or a placebo. "Good" results were no symptoms after four hours and no detectable plasma venom. $$ \begin{array}{|lccc|} \hline & \text { Antivenom } & \text { Placebo } & \text { Total } \\ \hline \text { No Improvement } & 1 & 6 & 7 \\ \hline \text { Improvement } & 7 & 1 & 8 \\ \hline \text { Total } & 8 & 7 & 15 \\ \hline \end{array} $$ The alternative hypothesis is that the antivenom leads to improvement. The p-value for a one-tailed Fisher's Exact Test with these data is \(0.009\). a. Suppose the study had turned out differently, as in the following table. $$ \begin{array}{|lcc|} \hline & \text { Antivenom } & \text { Placebo } \\ \hline \text { Bad } & 0 & 7 \\ \hline \text { Good } & 8 & 0 \\ \hline \end{array} $$ Would Fisher's Exact Test have led to a p-value larger or smaller than 0.009? Explain. b. Suppose the study had turned out differently, as in the following table. $$ \begin{array}{|lcc|} \hline & \text { Antivenom } & \text { Placebo } \\ \hline \text { Bad } & 2 & 5 \\ \hline \text { Good } & 6 & 2 \\ \hline \end{array} $$ Would Fisher's Exact Test have led to a p-value larger or smaller than 0.009? Explain. c. Try the two tests, and report the p-values. Were you right? Search for a Fisher's Exact Test calculator on the Internet, and use it.

Step by step solution

01

Predict the p-value in the perfect study

Given that the effectiveness of antivenom is much more prominent in this scenario, it is expected that the p-value is going to be smaller than 0.009.

02

Predict the p-value in the less perfect study

In this case, the effect of the antivenom is less definitive. Thus, the p-value is expected to be larger than 0.009 as the clear efficacy of the antivenom is lessened.

03

Compute the p-values

To check the predictions from Steps 1 and 2, use a Fisher's Exact Test calculator. Input the values from the contingency tables given in parts a and b, and let the calculator do the computation for the p-value. Repeat this for both tables.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fisher's Exact Test

Fisher's Exact Test is a statistical method used to determine if there are nonrandom associations between two categorical variables. It is particularly useful when dealing with small sample sizes, as it provides an exact p-value rather than an approximation. In the context of our study, it allows us to assess whether the observed distribution of 'Good' and 'Bad' results across the antivenom and placebo groups is due to chance or is statistically significant.

It is based on the principle of exact probabilities and uses a hypergeometric distribution to calculate the likelihood of obtaining the observed data, or data that is more extreme, assuming that there is no association between the variables (i.e., under the null hypothesis). This test is ideal for 2x2 contingency tables like those presented in the exercise, making it a perfect choice for small samples in medical research like the antivenom study.

p-value

The p-value is a critical concept in hypothesis testing and represents the probability of finding the observed data, or something more extreme, given that the null hypothesis is true. A lower p-value indicates stronger evidence against the null hypothesis, suggesting that the alternative hypothesis may be true.

• If the p-value is below a predefined threshold (commonly 0.05), we reject the null hypothesis.
• A p-value of 0.009, as found in the initial study, suggests strong evidence against the null hypothesis.

In our study example, we are interested in whether the antivenom is effective. The p-value helps us understand the likelihood that our observed pattern of effects - the differences in improvement between the antivenom and placebo groups - happened by mere chance.

Contingency Tables

Contingency tables are a fundamental tool for summarizing data in categorical format. They help organize data into a matrix format, allowing easy visualization of the relationships between two qualitative variables. They form the basis for tests like Fisher's Exact Test.

For instance, let's consider the study on scorpion antivenom. Here, a contingency table displays cases with 'Good' or 'Bad' outcomes for each treatment group (Antivenom vs. Placebo). It essentially provides a snapshot of how subjects respond differently to the two treatments.

• Columns represent the different treatments or categories (e.g., Antivenom, Placebo).
• Rows represent different outcomes (e.g., Bad, Good).
• The total counts in cells allow for calculating probabilities and performing statistical tests.

These tables are vital for conducting statistical analyses as they offer a structured way to examine relationships in data.

Alternative Hypothesis

The alternative hypothesis is an essential aspect of inferential statistics, representing the statement that contradicts the null hypothesis. It posits that there is a significant effect or a relationship between variables.

In the context of the scorpion antivenom study, the alternative hypothesis is that the antivenom leads to a better improvement rate compared to the placebo. It's what the researchers hope to show evidence for through statistical testing.

• It suggests that the positive or negative effect observed is unlikely to be due to random chance alone.
• If the p-value is low, it challenges the null hypothesis and suggests support for the alternative hypothesis.
• For the improvement seen in the antivenom group, a significant p-value would indicate that these results are not likely due to random variations.

Therefore, accepting the alternative hypothesis implies that the intervention (antivenom) has a real impact compared to the control (placebo). This is the ultimate goal of such clinical trials.