91Ó°ÊÓ

A study group of 576 working women \(30-49\) years of age who took phenacetin- containing analgesics and a control group of 533 comparably aged women without such intake were identified in 1968 and followed for mortality and morbidity outcomes. One hypothesis to be tested was that phenacetin intake may influence renal (kidney) function and hence have an effect on specific indices of renal morbidity and mortality. The mortality status of these women was determined from 1968 to \(1987 .\) The researchers found that 16 of the women in the study group and 1 of the women in the control group died, where at least one cause of death was considered renal [13]. Test for differences in renal mortality between the two groups in either direction, and report a two-tailed \(p\) -value. The cohort was also followed for total mortality. The researchers found that 74 women in the study group died, compared with 27 in the control group.

Step by step solution

01

Define the Hypotheses

First, define the null and alternative hypotheses for the test. We want to see if there is a significant difference in renal mortality between the two groups. The null hypothesis ( H_0 ) states that there is no difference in the incidence of renal mortality between women who took phenacetin-containing analgesics and those who did not. The alternative hypothesis ( H_1 ) states that there is a difference.

02

Calculate Mortality Rates

Calculate the mortality rates in both the study and control groups. For the study group, the renal mortality rate is \( \frac{16}{576} \approx 0.0278 \)\, and for the control group, it is \( \frac{1}{533} \approx 0.00188 \)\.

03

Use a Chi-Squared Test

To compare the rates between the two groups, use a chi-squared test. The observed frequencies are: 16 renal deaths in the study group and 1 in the control group; the expected frequencies are calculated based given the assumption of no difference between the groups.

04

Calculate Chi-Squared Statistic

The formula for the chi-squared statistic is \( \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \), where O_i is the observed frequency and E_i is the expected frequency. Plug in your values: observed = [16, 1], expected based on total mortality proportions.

05

Determine Significance Level

Determine the significance level, which is typically 0.05 for a two-tailed test. Using a chi-squared distribution table, find the critical value for 1 degree of freedom.

06

Calculate P-value

Using the chi-squared statistic calculated in Step 4, find the p-value using a chi-squared distribution. Compare this p-value with the significance level to determine whether to reject or accept the null hypothesis.

07

Interpret Results

If the p-value is less than the significance level, reject the null hypothesis. This would indicate a significant difference in renal mortality between the two groups. Otherwise, there is no sufficient evidence to claim a significant difference.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing in statistics involves making an educated guess about a population parameter. In this exercise, we are interested in checking if there is a significant difference in renal mortality between women who took phenacetin-containing analgesics and those who did not. We start by defining hypotheses:

• Null Hypothesis ( H_0 ): There is no difference in renal mortality rates between the two groups.
• Alternative Hypothesis ( H_1 ): There is a difference in renal mortality rates between the two groups.

By setting up these hypotheses, we can use statistical tests to determine if any observed differences are due to chance or if they are statistically significant. This process allows researchers to infer conclusions backed by statistical evidence.

Chi-Squared Test

The Chi-Squared Test is a statistical method used to compare observed frequencies with expected frequencies under a specific hypothesis. It is particularly useful for categorical data to assess how likely it is that any observed difference between the sets arose by chance.

In this scenario, we calculate the renal mortality rates for both the study and control groups. Then, using these rates, we conduct a chi-squared test:

• The study group observed 16 renal deaths out of 576 women.
• The control group observed 1 renal death out of 533 women.

We calculate the expected frequencies assuming no difference between the groups. The chi-squared statistic is then calculated using:\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \]Where: - \(O_i\) is the observed frequency. - \(E_i\) is the expected frequency.
This statistical tool helps in assessing the association between phenacetin intake and renal mortality.

Mortality Rates

Mortality rates are a key measure in epidemiology, indicating the frequency of death in a specific population, expressed as per unit of population within a certain time frame. In our study, we focus on renal mortality rates among two groups of women.

To find these rates:

• For the study group: The renal mortality rate is calculated as \( \frac{16}{576} \approx 0.0278 \).
• For the control group: The renal mortality rate is \( \frac{1}{533} \approx 0.00188 \).

These rates are essential as they help determine whether the phenacetin intake might be influencing the life expectancy due to renal complications. Comparing these rates can indicate the broader impact of the drug on health.

Statistical Significance

Statistical significance is a fundamental concept that helps us understand whether the findings from our data are likely due to chance or if they might reflect a true effect. In the context of this exercise, we're determining if the differences in renal mortality between the study and control groups are statistically significant.

To make this determination, we compare the p-value obtained from the chi-squared test with a pre-defined significance level, typically 0.05 for a two-tailed test. If the p-value is less than 0.05, the result is considered statistically significant, and we reject the null hypothesis. This implies that the difference in mortality rates is unlikely due to random variation alone and may be attributed to the phenacetin intake.
Understanding statistical significance helps researchers make informed decisions and draw meaningful conclusions from their data.