Question

The growth factor pleiotrophin is associated with cancer progression in humans. In an attempt to monitor the growth of tumors, doctors measured serum pleiotrophin levels in patients with pancreatic cancer and in a control group of patients. They found that only 2 of 28 control patients had serum levels more than two standard deviations above the control group mean, whereas 20 of 41 cancer patients had serum levels this high. \(^{27}\) Use Fisher's exact test to determine whether a discrepancy this large \((2\) of 28 versus 20 of 41\()\) is likely to happen by chance. Use a directional alternative and let \(\alpha=0.05\)

Short answer

Using Fisher's exact test, we determine the p-value. If it's less than 0.05, there is a significant difference in pleiotrophin levels between control and cancer patients; otherwise, the discrepancy could be due to chance.

Step-by-step solution

Set up the contingency table

For Fisher's exact test, you first need to organize the data into a 2x2 contingency table. The rows should represent the two groups (control patients and cancer patients), while the columns should represent whether the serum levels were more than two standard deviations above the mean or not.The table looks like this:| Group | > 2 SD above mean | ≤ 2 SD above mean | Total ||------------------|-------------------|--------------------|-------|| Control Patients | 2 | 26 | 28 || Cancer Patients | 20 | 21 | 41 || Total | 22 | 47 | 69 |

Calculate the Fisher's exact test

Fisher's exact test calculates the probability of obtaining a table that is at least as extreme as the one observed, assuming that there is no association between the groups and the characteristics. To perform the calculation, you use the hypergeometric distribution which considers all the ways that the observed sum could occur given the row and column totals.Given our observed data, the probability (p-value) of obtaining a result as extreme or more extreme by random chance is calculated using the formula for the hypergeometric probability distribution.

Determine the p-value

You can use statistical software or a calculator with Fisher's exact test capability to find the p-value. Input the frequencies from your contingency table, select the appropriate alternative hypothesis (directional in this case), and calculate the p-value.

Compare the p-value to the significance level

Compare the calculated p-value to the predetermined significance level \( \alpha = 0.05 \). If the p-value is less than \( \alpha \), you reject the null hypothesis and conclude that there is a statistically significant difference in serum pleiotrophin levels between the control and cancer patient groups.

Draw a conclusion

Based on the comparison between the p-value and the value of \( \alpha \), if the p-value is less than 0.05, you conclude that there is a statistically significant difference and that the discrepancy in pleiotrophin levels is unlikely to happen by chance. If the p-value is greater than 0.05, you fail to reject the null hypothesis and cannot conclude that there is a statistically significant difference.

Key concepts

Contingency Table

A contingency table is a useful tool for summarizing the relationship between two categorical variables. It displays the frequency distributions of the variables in a matrix format, allowing researchers to visualize the interaction between variables.

For example, in the given exercise, doctors are comparing serum pleiotrophin levels in two distinct groups: those with pancreatic cancer and a control group without it. Here's a simplified version of how we might set up the contingency table based on the provided data:

• One axis (rows) represents the two patient groups.
• The other axis (columns) shows whether serum levels were more than two standard deviations above the mean.
• Each cell then records the count of observations that fall at the intersection of each row and column category.

This systematic arrangement is fundamental for employing Fisher's exact test, as it helps in the organization and comprehension of the observed data when evaluating associated probabilities.

Hypergeometric Distribution

The hypergeometric distribution is central to understanding Fisher's exact test. This distribution models the probability of a given number of successes in draws from a finite population without replacement. It's a discrete probability distribution that has three parameters: the population size, the number of successes in the population, and the sample size.

In the context of our contingency table, the hypergeometric distribution can provide the probability of observing the exact configuration of our table (or one more extreme), assuming that the variables are independent. When calculating the p-value for Fisher's exact test, the hypergeometric distribution takes into account all possible combinations of how the observed totals could occur given the fixed margins (row and column totals).

P-value

The p-value is a crucial concept in statistical hypothesis testing. It represents the probability of obtaining results at least as extreme as those observed during the study, assuming the null hypothesis is true. In other words, it tells us how likely it is that we would see the data we have (or something more extreme) just by random chance.

For Fisher's exact test, the p-value is calculated with the aid of the hypergeometric distribution. A low p-value suggests that the data observed is unusual under the null hypothesis. If this value is below a predetermined threshold (known as the alpha level, commonly set at 0.05 for many tests), there is evidence to reject the null hypothesis, thus indicating the presence of a statistically significant association between the variables under study.

Statistical Significance

Statistical significance plays a pivotal role in hypothesis testing. It is a determination by an analyst that the results in the data are not explainable by chance alone. A result is said to be statistically significant if it is unlikely to occur by random variation, and this is often described in terms of a p-value.

A common threshold for the significance level is 0.05, which corresponds to a 5% risk of concluding that a difference exists when there is no actual difference. In our example, if the p-value calculated using Fisher's exact test is less than 0.05, we would declare the difference in serum pleiotrophin levels between the control and cancer patient groups to be statistically significant. This would imply that the variation in levels is likely attributed to the presence of cancer rather than occurring by chance.