91Ó°ÊÓ

Genetically altered mice. Genetic influences on cancer can be studied by manipulating the genetic makeup of mice. One of the processes that turn genes on or off (so to speak) in particular locations is called "DNA methylation." Do low levels of this process help cause tumors? Compare mice altered to have low levels with normal mice. Of 33 mice with lowered levels of DNA methylation, 23 developed tumors. None of the control group of 18 normal mice developed tumors in the same time period. \({ }^{18}\) (a) Explain why we cannot safely use either the large-sample confidence interval or the test for comparing the proportions of normal and altered mice that develop tumors. (b) The plus four method adds two observations, a success and a failure, to each sample. What are the sample sizes and the numbers of mice with tumors after you do this? (c) Give a 99\% confidence interval for the difference in the proportions of the two populations that develop tumors.

Step by step solution

01

Understand the Problem Context

We are comparing the proportion of tumors in two groups of mice, one with low levels of DNA methylation and the other a control group. We want to determine if this difference is statistically significant and calculate the confidence interval for the difference.

02

Evaluate Large-Sample Methods Appropriateness

Large-sample methods require a certain number of successes and failures in each group to be valid. Typically, both are greater than 5. In this case, we have 23 successes (tumors) and 10 failures in the altered group, and 0 successes with 18 failures in the control group, which doesn't satisfy this condition for the control group. Hence, large-sample methods cannot be safely used.

03

Apply Plus Four Method

The plus four method involves adding two additional observations (one success and one failure) to each sample. For the altered group, this transforms the sample size from 33 to 35, with 24 successes. For the control group, the sample size becomes 20, with 1 success and 19 failures.

04

Calculate Proportions with Adjustments

Calculate the proportion of tumors in each group with the plus four adjustments:- Altered group: \( p_1 = \frac{24}{35} \)- Control group: \( p_2 = \frac{1}{20} \)

05

Calculate the Standard Error for the Difference in Proportions

Use the formula for the standard error of the difference in proportions:\[ SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \] where \( n_1 = 35 \) and \( n_2 = 20 \).

06

Determine the 99% Confidence Interval

The 99% confidence interval is given by:\[ (p_1 - p_2) \pm z \times SE \] where \( z \) is the z-score corresponding to 99% confidence (approximately 2.576). Substitute \( p_1 \), \( p_2 \), \( SE \), and \( z \) to compute the interval.

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Key Concepts

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

DNA Methylation and Cancer

DNA methylation is a biological process that modifies the DNA to regulate gene activity. In simple terms, it acts like a switch that can turn genes on or off. Scientists are particularly interested in DNA methylation because of its potential role in cancer formation.
In our study with genetically altered mice, the aim was to understand if lower DNA methylation links to an increased tumor rate. By comparing two groups of mice—one with altered methylation levels and one control group with normal levels—we seek insights into the genetic influence on cancer.
• Lower methylation as a potential cause for tumors emphasizes the need for genetic investigation.
• Understanding these influences could lead to better cancer prevention and treatment strategies.

Confidence Intervals

A confidence interval provides a range of values that we believe contain the true difference in proportions between two groups. It offers an assurance level or 'confidence' that this range captures the actual figure we're estimating.
For our mouse study, a 99% confidence interval means we can be 99% sure that the interval calculated contains the true difference in tumor development proportions between the altered and control mice population.
The confidence interval considers the variability in data and provides a more robust picture than a simple point estimate. Therefore, it's an essential statistical tool to quantify the uncertainty in our sample data and make meaningful inferences about the larger populations of interest.

Plus Four Method

The plus four method is a statistical technique often used when the sample size or number of successes or failures is small. It involves adding one success and one failure to each group, adjusting the sample size when calculating the confidence intervals or significance tests. This adjustment helps to stabilize the variability inherent in smaller samples.
In the mouse study, after applying the plus four method, the altered group sample size became 35 with 24 instances of tumors, while the control group’s size became 20 with 1 tumor case. These adjustments help create a more reliable estimation of proportions, facilitating the generation of accurate confidence intervals.
• Enhances reliability of statistical inference in small samples
• Facilitates comparison between groups when assumptions of larger samples are not met

Statistical Significance

Statistical significance helps determine if the observed differences in sample data likely reflect true differences in the overall population. It answers the question: Is the observed effect due to chance, or is it a true difference?
In the context of our study, statistical significance assesses whether the difference in tumor proportions between altered and control mice is enough to infer that low levels of DNA methylation could likely cause cancer. If a result is statistically significant, it's unlikely to occur by random variation alone.
• Helps draw conclusions about population differences
• Strengthens the evidence for or against a hypothesized effect, such as a genetic cause for cancer
Deciding statistical significance typically involves comparing a p-value with a chosen significance level, commonly decision points like 0.05, 0.01, or 0.001, depending on how strongly you wish to control for Type I errors.