Question

Scientists are working to train dogs to smell cancer, including early stage cancer that might not be detected with other means. In previous studies, dogs have been able to distinguish the smell of bladder cancer, lung cancer, and breast cancer. Now, it appears that a dog in Japan has been trained to smell bowel cancer. \({ }^{15}\) Researchers collected breath and stool samples from patients with bowel cancer as well as from healthy people. The dog was given five samples in each test, one from a patient with cancer and four from healthy volunteers. The dog correctly selected the cancer sample in 33 out of 36 breath tests and in 37 out of 38 stool tests. (a) The cases in this study are the individual tests. What are the variables? (b) Make a two-way table displaying the results of the study. Include the totals. (c) What proportion of the breath samples did the dog get correct? What proportion of the stool samples did the dog get correct? (d) Of all the tests the dog got correct, what proportion were stool tests?

Short answer

The variables of this study include the type of the sample (breath or stool), the health status of the sample (cancerous or healthy), and the dog's ability to correctly identify the sample. The proportion of correct breath and stool sample detections were approximately 91.7% and 97.4% respectively. Among all correct detections, approximately 52.86% were stool tests.

Step-by-step solution

Identifying the Variables

The variables of this study can be classified as follows: \n- Independent variable: Type of the sample (breath or stool) and health status of the sample (cancerous or healthy). \n- Dependent variable: The dog's ability to correctly identify the sample.

Create a Two-Way Table

A two-way table can be formulated as follows:\n\n\t| \t| Breath \t| Stool\t| Total\t|\n\t|--------\t|--------\t|-------\t|-------\t|\n\t| Cancer \t| 33 \t| 37 \t| 70 \t|\n\t| Healthy\t| 3 \t| 1 \t| 4 \t|\n\t| Total \t| 36 \t| 38 \t| 74 \t|\n\nHere we consider 4 categories: Breath Cancer, Breath Healthy, Stool Cancer, Stool Healthy. Their respective counts have been filled in the table.

Proportion of Correct Samples

Using the data from the table above, we calculate the proportion of correct detections:\n- For breath samples: this is \(\frac{33}{36} = 0.917\) or 91.7%.\n- For stool samples: this is \(\frac{37}{38} = 0.974\) or 97.4%.

Proportion of Correct Stool Tests

Out of all the tests the dog got correct (70 in total based on the table), the proportion that were stool tests is \(\frac{37(correct stool tests)}{70(all correct tests)} = 0.52857\) or approximately 52.86%.

Key concepts

Dependent and Independent Variables

Understanding the relationship between dependent and independent variables is foundational in statistics and research. In experiments, independent variables are those that can be manipulated or controlled. They are the cause to the effect, which is measured by the dependent variables.

In our exercise, the type of sample (breath or stool) and the health status (cancerous or healthy) are independent variables. These variables are chosen by the researchers and remain fixed during the test cases to see how they affect the outcome. The dependent variable, in this case, is the dog's ability to identify the correct sample. It is dependent because the outcome varies based on the independent variables in the study.

Two-Way Table

A two-way table, also known as a contingency table, organizes data into categories that are defined by two variables. It allows for a clear visualization of the relationship between the variables. In the context of our exercise, a two-way table is used to display the results of the dog's ability to identify cancer in both breath and stool samples.

The table is constructed with one variable along the top row and another along the first column. The intersecting cells then show the number of occurrences for each combination of variables. This layout makes it easy to calculate totals and to perform further proportion calculations. A well-constructed two-way table can provide immediate insights and is a fundamental tool in statistical analysis.

Proportion Calculation

Proportion calculations are a significant aspect of statistical analysis. They help in understanding the relative size of one category compared to the whole. In our exercise, the proportion of correct samples is calculated by dividing the number of correct detections by the total number of trials.

The proportion effectively communicates the success rate of the dog's ability to detect cancer. It is typically presented as a percentage to make it more interpretable. For example, calculating the proportion of correct stool samples by using the data provided in the two-way table gives us valuable information about the dog’s detection accuracy specific to stool samples compared to breath samples. Proportion calculations are important in assessing the effectiveness of tests, treatments, and other experimental outcomes in a straightforward way.