Question

Women diagnosed with breast cancer whose tumors have not spread may be faced with a decision between two surgical treatments - mastectomy (removal of the breast) or lumpectomy (only the tumor is removed). In a long-term study of the effectiveness of these two treatments, 701 women with breast cancer were randomly assigned to one of two treatment groups. One group received mastectomies and the other group received lumpectomies and radiation. Both groups were followed for 20 years after surgery. It was reported that there was no statistically significant difference in the proportion surviving for 20 years for the two treatments (Associated Press, October 17,2002 ). What hypotheses do you think the researchers tested in order to reach the given conclusion? Did the researchers reject or fail to reject the null hypothesis?

Short answer

The researchers tested the Null Hypothesis that states no significant difference in the 20-year survival rate for patients who received mastectomies or lumpectomies and radiation. Since it was reported that there was no statistically significant difference in the proportions surviving for 20 years for the two treatments, this suggests that the researchers failed to reject the Null Hypothesis.

Step-by-step solution

Understanding the Null Hypothesis

The Null Hypothesis \(H_0\) for this study states that there is no significant difference in the survival proportion between the two groups. This means the proportion of women who survive for 20 years after being treated with mastectomy is the same as the proportion of women who survive for 20 years after being treated with lumpectomy and radiation.

Understanding the Alternative Hypothesis

The Alternative Hypothesis \(H_1\) for this study states that there is a significant difference in the survival proportion between the two groups. This means the proportion of women who survive for 20 years after being treated with mastectomy is not the same as the proportion of women who survive for 20 years after being treated with lumpectomy and radiation.

Interpreting the Conclusion of the researchers

The researchers did not find a statistically significant difference hence, 'Failed to reject the Null Hypothesis'. This means according to the data collected and analyzed, there isn't enough evidence to suggest that one treatment is significantly better than the other in terms of the 20-year survival rate.

Key concepts

Null Hypothesis

In hypothesis testing, the null hypothesis, denoted as \( H_0 \), forms the cornerstone of scientific inquiry. It suggests that there is no effect or no difference in the context of the variables being tested. In the case of the study on breast cancer treatments, the null hypothesis proposes that there is no statistically significant difference in the 20-year survival rates between women who underwent mastectomies and those who received lumpectomies with radiation.

For a study like this, the null hypothesis acts as a default assumption that researchers aim to test. It's comparable to a presumption of innocence in a court case; the null hypothesis is innocent until proven guilty by the data. If the evidence (data) does not strongly contradict this hypothesis, we continue to assume it's true.

Alternative Hypothesis

The alternative hypothesis, represented as \( H_1 \), is the statement that contradicts the null hypothesis. It is what researchers hope to find evidence for, essentially positing that a genuine effect or difference exists. In this breast cancer treatment study, the alternative hypothesis would assert that there is a significant difference in the 20-year survival rates of breast cancer patients between the two treatments: mastectomy and lumpectomy combined with radiation.

If the null hypothesis is like a status quo, the alternative hypothesis is the challenger. Proving this hypothesis typically means that there is sufficient evidence to suggest a variable change or a noticeable effect. In our example, a significant finding would involve determining that one surgical treatment leads to a better long-term survival outcome than the other.

Statistical Significance

Statistical significance is a term used to indicate whether the observed effect in a study justifies rejecting the null hypothesis. When researchers find a result statistically significant, it means the data suggests there is a low probability the observed effect is due to chance alone. Often, a significance level (denoted by \( \alpha\)) of 0.05 is used as a standard threshold. This signifies that there is a 5% probability that the results occurred by random chance rather than a true effect.

In the breast cancer study, the researchers concluded that there was "no statistically significant difference" in the survival rates between the two treatment groups. This indicates that the data did not provide strong enough evidence against the null hypothesis, hence the researchers failed to reject it. It doesn't prove that the null hypothesis is true, but rather that there isn't enough evidence to declare that it's false.

Survival Analysis

Survival analysis is a statistical method used to analyze data in which the outcome variable is the time until an event occurs. It's particularly useful when studying the effectiveness of treatments or interventions over time. This method accounts not only for whether a patient survived a treatment period, but also how long they survived after the treatment.

The breast cancer treatment study is a classic example of survival analysis, focusing on the long-term survival of patients after undergoing different surgical procedures. The key statistic of interest is the survival probability at a certain time point, in this case, 20 years. By using survival analysis, researchers could comprehensively compare the efficacy of the two treatment options over time, providing a more nuanced understanding of their impact on patient longevity.