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Chapter 11: Problem 77

True or false: Relative risk Interchanging the rows in a \(2 \times 2\) contingency table has no effect on the value of the relative risk.

Short Answer

Expert verified
False, interchanging rows affects the relative risk value.

Step by step solution

01

Understanding Relative Risk

Relative risk is a measure used to compare the risk of a particular event occurring in one group compared to another. In a 2x2 contingency table, it's often used to compare the risk between two populations, typically exposed vs. unexposed groups. The formula for relative risk (RR) is \( RR = \frac{a/(a+b)}{c/(c+d)} \), where \( a, b, c, \) and \( d \) are the frequencies in the 2x2 table.
02

Defining the 2x2 Contingency Table

A 2x2 contingency table has four fields: \( a \) (exposed and event occurs), \( b \) (exposed and event does not occur), \( c \) (not exposed and event occurs), and \( d \) (not exposed and event does not occur). The rows and columns can be interchanged or rearranged.
03

Effect of Interchanging Rows

Switching the rows in a contingency table would switch the numerator and denominator in the relative risk formula. The new table arrangement will make \( RR = \frac{c/(c+d)}{a/(a+b)} \). This is the reciprocal of the original relative risk, indicating that interchanging the rows changes the value of the relative risk.

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Key Concepts

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

Contingency Table
A contingency table is a helpful tool in statistics used to depict and analyze the relationship between two categorical variables. Specifically, a 2x2 contingency table offers a simple yet powerful way to display data for two groups, each having two possible outcomes. In our context, the rows and columns of the table categorize data into exposed vs. not exposed groups and whether an event occurs or not. This layout helps in calculating and comparing probabilities within each group.

If we have "exposed" and "not exposed" as row categories, and "event occurs" and "event does not occur" as column labels, we can clearly see the data distribution. Such tables help in quick visualization, which in turn aids in calculating statistical measures like relative risk, allowing us to easily interpret our data set and draw conclusions.
Risk Comparison
Risk comparison is vital in understanding how one group's risk of experiencing an event fares against another group. Through the use of relative risk, you can compare the likelihood of an event between two groups, often categorized as 'exposed' and 'not exposed'.

A relative risk value of 1 suggests equal risk in both groups. Below 1 indicates a reduced risk in the exposed group compared to the not exposed group, whereas above 1 suggests an increased risk. This way, calculating relative risk using the formula \( RR = \frac{a/(a+b)}{c/(c+d)} \) allows for a direct and meaningful comparison of risks between groups in context. Be mindful that changing how data is organized in the contingency table affects this risk ratio, as swapping rows changes the operation within the formula, thus altering the calculated value.
Statistical Measures
Statistical measures such as relative risk provide insights into the relationships within data. These measures are essential in creating interpretations from contingency tables, turning raw numbers into informative statistics.

Relative risk quantifies the likelihood of an event happening in one group compared to another. It is vital for evaluating efficacy in contexts like medicine or public health. A calculated relative risk helps in interpreting the effectiveness of interventions or exposures by demonstrating differences in outcome probabilities.

Understanding changes in statistical measures when variable orientation changes is crucial. If you interchange rows or columns in a 2x2 contingency table, it impacts the statistical calculation. Therefore, organizing your data correctly is as critical as using statistical formulas in making informed interpretations.

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Most popular questions from this chapter

Seat belt helps? The table refers to passengers in autos and light trucks involved in accidents in the state of Maine in a recent year. a. Use the difference of proportions to describe the strength of association. Interpret. b. Use the relative risk to describe the strength of association. Interpret. \begin{tabular}{lcc} \hline Maine Accident Data & \\ \hline & \multicolumn{2}{c} { Injury } \\ \cline { 2 - 3 } Seat Belt & Yes & No \\ \hline No & 3865 & 27,037 \\ Yes & 2409 & 35,383 \\ \hline \end{tabular} Source: Dr. Cristanna Cook, Medical Care Development, Augusta, Maine.

Normal and chi-squared with \(d f=1\) When \(d f=1,\) the P-value from the chi- squared test of independence is the same as the P-value for the two-sided test comparing two proportions with the \(z\) test statistic. This is because of a direct connection between the standard normal distribution and the chi-squared distribution with \(d f=1:\) Squaring a \(z\) -score yields a chi-squared value with \(d f=1\) having chisquared right-tail probability equal to the two-tail normal probability for the \(z\) -score. a. Illustrate this with \(z=1.96\), the \(z\) -score with a two-tail probability of \(0.05 .\) Using the chi-squared table or software, show that the square of 1.96 is the chi-squared score for \(d f=1\) with a P-value of 0.05 b. Show the connection between the normal and chi-squared values with P-value \(=0.01\).

Williams College admission Data from 2013 posted on the Williams College website shows that of all 3,195 males applying, \(18.2 \%\) were admitted, and of all 3,658 females applying, \(16.9 \%\) were admitted. Let \(\mathrm{X}\) denote gender of applicant and \(\mathrm{Y}\) denote whether admitted. a. Which conditional distributions do these percentages refer to, those of \(Y\) at given categories of \(X,\) or those of \(X\) at given categories of \(Y ?\) Set up a table showing the two conditional distributions. b. Are \(X\) and \(Y\) independent or dependent? Explain.

In an experiment on chlorophyll inheritance in corn, for 1,103 seedlings of self-fertilized heterozygous green plants, 854 seedlings were green and 249 were yellow. Theory predicts that \(75 \%\) of the seedlings would be green. a. Specify a null hypothesis for testing the theory. b. Find the value of the chi-squared goodness-of-fit statistic and report its \(d f\). c. Report the P-value and interpret.

Vioxx In September \(2004,\) the pharmaceutical company Merck withdrew its blockbuster drug rofecoxib (a painkiller, better known under its brand name, Vioxx) from the worldwide market amid concerns about its safety. By that time, millions of people had used the drug. In a 2000 study comparing rofecoxib to a control group (naproxen), it was mentioned that "Myocardial infarctions were less common in the naproxen group than in the rofecoxib group ( 0.1 percent vs. 0.4 percent; 95 percent confidence interval for the difference, 0.1 to 0.6 percent; relative risk, 0.2; 95 percent confidence interval, 0.1 to 0.7\()\) ". (Source: Bombadier et al., New England Journal of Medicine, vol. \(343,2000,\) pp. \(1520-8 .)\) a. Find and interpret the difference of proportions between the naproxen and rofecoxib groups. b. Interpret the stated relative risk. c. In this study, myocardial infarctions were how much more likely to occur in the rofecoxib group than in the naproxen group?
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