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In \(2006,\) Boston Scientific sought approval for a new heart stent (a medical device used to open clogged arteries) called the Liberte. This stent was being proposed as an alternative to a stent called the Express that was already on the market. The following excerpt is from an article that appeared in The Wall Street Journal (August 14,2008 ): Boston Scientific wasn't required to prove that the Liberte was 'superior' than a previous treatment, the agency decided - only that it wasn't "inferior' to Express. Boston Scientific proposed - and the FDA okayed - a benchmark in which Liberte could be up to three percentage points worse than Express meaning that if \(6 \%\) of Express patients' arteries reclog, Boston Scientific would have to prove that Liberte's rate of reclogging was less than \(9 \%\). Anything more would be considered 'inferior.'... In the end, after nine months, the Atlas study found that 85 of the patients suffered reclogging. In comparison, historical data on 991 patients implanted with the Express stent show a \(7 \%\) rate. Boston \(S\) cientific then had to answer this question: Could the study have gotten such results if the Liberte were truly inferior to Express?" Assume a \(7 \%\) reclogging rate for the Express stent. Explain why it would be appropriate for Boston Scientific to carry out a hypothesis test using the following hypotheses: \(H_{0}: p=.10\) \(H_{a}: p<.10\) where \(p\) is the proportion of patients receiving Liberte stents that suffer reclogging. Be sure to address both the choice of the hypothesized value and the form of the alternative hypothesis in your explanation.

Step by step solution

01

Identify the problem

The problem is to determine if the new Liberte stent is inferior to the Express stent in terms of the reclogging rate. To do so, a hypothesis test is proposed with the null hypothesis \(H_{0}: p=0.10\) and alternative hypothesis \(H_{a}: p<0.10\), where \(p\) represents the proportion of patients suffering from reclogging.

02

Understand the proposed hypotheses

The null hypothesis suggests a reclogging rate for the Liberte stents at 10%, that is, 3% weaker than the Express stents rate of 7%, as suggested by the FDA approval conditions. The alternative hypothesis considers any reclogging rate less than 10% - if the Liberte reclogging rate is found to be less than 10%, it can be inferred it is not inferior to the Express stent.

03

Explain the reasoning behind hypotheses

The null hypothesis \(H_{0}: p=0.10\) is set to the maximum allowable reclogging rate for the Liberte stent for it to be considered not 'inferior' to the Express stent. This represents a scenario which assumes the worst case within the acceptable boundaries. The alternative hypothesis \(H_{a}: p<0.10\) is constructed to demonstrate that the reclogging rate is less than the postulated 10% level. In the context of the test, rejecting the null hypothesis in favor of the alternative would provide evidence that the Liberte stent is not inferior to the Express stent.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis represents a statement of no effect or no difference. It serves as a starting point and is usually denoted as \( H_{0} \). For the example provided, the null hypothesis is formulated as \( H_{0}: p = 0.10 \), which indicates the maximum allowable reclogging rate for the Liberte stent to still be considered not inferior to the existing Express stent.
This hypothesized rate of 10% comes from the condition that Liberte's performance should not exceed the Express stent's reclogging rate by more than 3%. This acts as our baseline assumption from which we can test the alternatives.

Alternative Hypothesis

The alternative hypothesis presents a statement that contradicts the null hypothesis, suggesting there is an effect or difference. In this context, the alternative hypothesis is \( H_{a}: p < 0.10 \).
Operations against the null hypothesis would entail showing that the Liberte stent's reclogging rate is, in fact, less than 10%. If the Liberte's rate is found to be below this threshold, it supports the claim that Liberte is not inferior to the Express stent. This alternative forms the basis for rejecting the null hypothesis during testing.

Inferiority Testing

Inferiority testing is employed when the goal is to determine whether a new treatment is not worse than an existing treatment by a specified margin. In the exercise, Boston Scientific sought to establish that the Liberte stent is not inferior to the Express stent.
It's important to understand that saying 'not inferior' does not imply it's superior, merely that it meets a minimum benchmark, which here is exemplified by the \(10\%\) reclogging threshold. By setting up the hypothesis as \( p = 0.10 \) versus \( p < 0.10 \), this methodology attempts to confirm the Liberte is within acceptable performance limits without needing to prove it's superior.

Statistical Significance

Statistical significance is an essential aspect of hypothesis testing. It refers to the probability that an observed effect or difference in data is not due to chance. When testing the Liberte stent, statistical significance would suggest that any observed difference in reclogging rates is unlikely due to random variations.
Achieving statistical significance allows Boston Scientific to reject the null hypothesis confidently. This implies the data supports the idea that the Liberte stent is not inferior to the Express, assuming other factors such as sample size and variability are properly accounted for. Importantly, significance helps to provide a strong basis for decision-making in medical device approval.