Question

A group of 30 postmenopausal women were given oral conjugated estrogen for one month. Plasma levels of plasminogen-activator inhibitor type 1 (PAI-1) went down for 22 of the women, but went up for 8 women. \({ }^{15}\) The \(P\) -value for a sign test, with a nondirectional alternative, is 0.016 . (a) State the null and alternative hypotheses in context. (b) If \(\alpha=0.10,\) what is your conclusion regarding \(H_{0} ?\)

Short answer

Null hypothesis (\(H_0\)): Oral conjugated estrogen does not affect plasma levels of PAI-1. Alternative hypothesis (\(H_1\b)): Oral conjugated estrogen does affect plasma levels of PAI-1. With a P-value of 0.016 compared to \(\alpha=0.10\), we reject \(H_0\) and conclude that the estrogen does have an effect on PAI-1 levels.

Step-by-step solution

Formulating the Hypotheses

State the null hypothesis, which is a statement of no effect or no difference. Here it would be that the oral conjugated estrogen does not affect the plasma levels of PAI-1 on average in postmenopausal women. The alternative hypothesis is the complement to the null hypothesis, representing that there is an effect, with the oral conjugated estrogen either increasing or decreasing PAI-1 levels. Since the test is nondirectional, it does not specify the direction of the change.

Analyze the P-Value

Compare the P-value to the significance level \(\alpha\). A P-value less than \(\alpha\) suggests that the observed data is inconsistent with the null hypothesis, and thus the null hypothesis should be rejected. Here, the P-value is 0.016.

Drawing a Conclusion

Since the P-value of 0.016 is less than the significance level of 0.10, reject the null hypothesis. There is sufficient evidence at the 0.10 level of significance to conclude that oral conjugated estrogen does affect the plasma levels of PAI-1 in postmenopausal women.

Key concepts

P-value

The P-value, or probability value, is a fundamental concept in statistics that measures the probability of obtaining results at least as extreme as those observed during an experiment, under the assumption that the null hypothesis is true. In simpler terms, it tells you how likely it is that the data you've collected could have occurred simply by random chance.

For example, in the exercise, the P-value of 0.016 indicates that there is only a 1.6% chance that the reduction in PAI-1 levels observed in the study happened by random chance if there really was no effect of the estrogen on PAI-1 levels. If the P-value is lower than the pre-determined significance level, which is often 0.05 or 0.10, it suggests that the null hypothesis might not adequately explain the observations and therefore may be rejected.

Null Hypothesis

The null hypothesis (\(H_0\)) is a statement of no effect or change. It is the default assumption that there is no relationship or difference between two measured phenomena. In other words, it posits that any observed effect is due to chance rather than due to some systematic cause.

In the study of the effects of oral conjugated estrogen on PAI-1 levels, the null hypothesis would state that the treatment does not affect PAI-1 levels in postmenopausal women. This hypothesis serves as a starting point for statistical hypothesis testing and is what researchers aim to either reject or fail to reject based on their data.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis (\(H_a\text{ or }H_1\)) proposes that there is an effect, or that there exists a relationship or difference that is not due to chance alone. It's what the experimenter believes may actually be the case.

For the estrogen and PAI-1 level study, the alternative hypothesis is that oral conjugated estrogen does have an effect on PAI-1 levels. As the specific test in this scenario is nondirectional, the alternative hypothesis does not specify whether the expected change is an increase or decrease, only that there is some change.

Significance Level

The significance level, denoted as \(\alpha\), is the threshold against which the P-value is compared to determine whether to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is, in fact, true – known as a Type I error.

In the context of the exercise, a significance level of 0.10 was used. That means there was a willingness to accept a 10% chance of incorrectly rejecting the null hypothesis. Since the P-value was 0.016, which is below the \(\alpha\) of 0.10, the null hypothesis would be rejected, leading to the conclusion that the oral estrogen likely has an effect on PAI-1 levels.

Plasminogen-Activator Inhibitor Type 1 (PAI-1)

Plasminogen-Activator Inhibitor Type 1 (PAI-1) is a protein that plays a crucial role in the regulation of fibrinolysis, a process of blood clot breakdown. Abnormal levels of PAI-1 are associated with various health issues, including cardiovascular disease.

In postmenopausal women, hormone replacement therapy (HRT) may influence PAI-1 levels, potentially affecting their risk profile for thrombosis and other cardiovascular events. Understanding the impact of treatments like oral conjugated estrogen on PAI-1 levels is important for managing postmenopausal health risks.

Postmenopausal Health Studies

Postmenopausal health studies focus on understanding the health challenges faced by women who have gone through menopause. These studies can encompass a broad range of topics, from hormone replacement therapies and their side effects to the risk of diseases like osteoporosis and cardiovascular issues.

In relation to the exercise, studying the effect of oral conjugated estrogen on PAI-1 levels is an example of how research targets postmenopausal health concerns, aiming to improve quality of life and reduce the risk of health complications that may arise after menopause.