Math Problem Statement

Consider a permutation test of the null hypothesis that there is no effect of the mutagen, i.e. that the mean number of microsatellite nuclei is the same in the two groups. We will use the difference in means (dose of 80 mg/ml - control) as our test statistic. So, a positive value for the test statistic indicates that the 80 mg/ml dose had a larger number of microsatellite nuclei. The observed difference (calculated from the data above) is 8.2 There are a total of 126 permutations of 9 things into a group of 4 and group of 5. The 126 values of the test statistic (difference in means as dose - no), sorted from smallest to largest, are: -9.35 -8.9 -8 -8 -8 -8 -8 -7.1 -7.1 -7.1 -7.1 -6.65 -6.65 -6.65 -6.65 -5.75 -5.75 -5.75 -5.75 -5.75 -5.75 -5.3 -4.4 -4.4 -4.4 -4.4 -3.95 -3.95 -3.95 -3.95 -3.05 -3.05 -3.05 -3.05 -3.05 -3.05 -3.05 -3.05 -3.05 -3.05 -2.15 -2.15 -2.15 -2.15 -2.15 -2.15 -1.7 -1.7 -1.7 -1.7 -1.7 -1.7 -1.25 -0.8 -0.8 -0.8 -0.8 -0.35 -0.35 -0.35 -0.35 0.1 0.1 0.1 0.1 1 1 1 1 1 1 1 1 1 1 1 1.9 1.9 1.9 1.9 1.9 1.9 2.35 2.35 2.35 2.35 2.35 2.35 3.25 3.25 3.25 3.25 3.7 3.7 3.7 3.7 4.6 4.6 4.6 4.6 4.6 4.6 5.05 5.05 5.05 5.05 5.05 5.05 5.95 5.95 5.95 5.95 5.95 5.95 5.95 5.95 5.95 5.95 6.85 6.85 6.85 6.85 7.3 7.3 7.3 8.2 a) If the null hypothesis (no difference) is true, what is the probability of 8.2 or a more extreme positive value? b) What is the two-sided p-value for the test of the null hypothesis of no difference?

Solution

Let's break this down step by step to address each question.

(a) Probability of 8.28.2 or a more extreme positive value

In permutation tests, the p-value is calculated as the proportion of the permutation test statistics that are as extreme or more extreme than the observed test statistic. The observed test statistic in this case is 8.28.2.

To calculate the probability of getting 8.28.2 or a more extreme positive value, we need to count how many of the 126 permutations have a test statistic of 8.28.2 or greater.

Looking at the sorted list of test statistics:

  • The values equal to or greater than 8.28.2 are only 8.28.2 itself.

Since there is only one permutation with a value of 8.28.2, the probability is:

Probability=Number of values8.2Total number of permutations=1126\text{Probability} = \frac{\text{Number of values} \geq 8.2}{\text{Total number of permutations}} = \frac{1}{126}

So, the probability of observing 8.28.2 or a more extreme positive value under the null hypothesis is 11260.00794\frac{1}{126} \approx 0.00794.


(b) Two-sided p-value

A two-sided p-value considers both the extreme positive and extreme negative values. To calculate this, we need to find how many of the 126 permutations are as extreme as or more extreme than 8.28.2 (on the positive side) and its negative counterpart 8.2-8.2 (on the negative side).

From the sorted list, we can count:

  • There is one value equal to 8.28.2.
  • On the negative side, the smallest value is 9.35-9.35, and the negative values more extreme than 8.2-8.2 are 9.35,8.9,8,8,8,8,8-9.35, -8.9, -8, -8, -8, -8, -8. There are 7 values that are 8\leq -8.

Thus, there are 1+7=81 + 7 = 8 values that are as extreme as or more extreme than 8.28.2 or 8.2-8.2.

The two-sided p-value is:

Two-sided p-value=Number of values 8.2 or 8.2Total number of permutations=81260.0635\text{Two-sided p-value} = \frac{\text{Number of values } \leq -8.2 \text{ or } \geq 8.2}{\text{Total number of permutations}} = \frac{8}{126} \approx 0.0635


Summary:

  • (a) The probability of observing 8.28.2 or a more extreme positive value is 11260.00794\frac{1}{126} \approx 0.00794.
  • (b) The two-sided p-value is 81260.0635\frac{8}{126} \approx 0.0635.

Would you like further details or clarifications on any steps?


Here are 5 related questions you might find useful:

  1. How is a permutation test different from a t-test?
  2. What are the advantages of using a permutation test?
  3. How would the results change if we used a different test statistic (e.g., median difference)?
  4. How does the size of the sample affect the results of a permutation test?
  5. Can permutation tests be used for more than two groups?

Tip: Permutation tests are a powerful non-parametric method because they don’t rely on normal distribution assumptions!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Permutation Test
Hypothesis Testing
P-Value Calculation

Formulas

P-value = (Number of values >= observed test statistic) / Total number of permutations
Two-sided P-value = (Number of extreme values) / Total number of permutations

Theorems

Permutation Test Theorem
Non-parametric Hypothesis Testing

Suitable Grade Level

Undergraduate (Statistics/Biostatistics Level)