Question

A study was conducted to determine whether a particular drug injection reduced the harmful effects of a chemotherapy treatment on the survival time for rats. Two randomly selected groups of 12 rats received the toxic drug in a dose large enough to cause death, but in addition, one group received the antitoxin to reduce the toxic effect of the chemotherapy on normal cells. The test was terminated at the end of 20 days, or 480 hours. The survival times for the two groups of rats, to the nearest 4 hours, are shown in the table. Do the data provide sufficient evidence to indicate that rats receiving the antitoxin tend to survive longer after chemotherapy than those not receiving the antitoxin? Use the Wilcoxon rank sum test with \(\alpha=.05 .\) $$ \begin{array}{rc} \hline \text { Chemotherapy Only } & \text { Chemotherapy Plus Drug } \\ \hline 84 & 140 \\ 128 & 184 \\ 168 & 368 \\ 92 & 96 \\ 184 & 480 \\ 92 & 188 \\ 76 & 480 \\ 104 & 244 \\ 72 & 440 \\ 180 & 380 \\ 144 & 480 \\ 120 & 196 \end{array} $$

Short answer

Use a significance level of α = 0.05.

Step-by-step solution

Rank the survival times

Combine the survival times for both groups and rank them from the smallest to the largest. Assign equal ranks for identical survival times and average the ranks if there is a tie.

Calculate the Rank Sum for each group

Find the rank sum for each group by summating the ranks corresponding to the survival times. Let's call the rank sum for chemotherapy-only group as \(W_1\) and the rank sum for chemotherapy plus drug group as \(W_2\).

Calculate the expected rank sum and standard deviation

Compute the expected rank sum and standard deviation for group 1 (chemotherapy-only) under the null hypothesis using the following formulas: Expected rank sum, \(E(W_1) = \frac{n_1 (n_1 + n_2 + 1)}{2}\), where \(n_1\) is the number of rats in the chemotherapy-only group and \(n_2\) is the number of rats in the chemotherapy plus drug group. Standard deviation, \(SD(W_1) = \sqrt{\frac{n_1 n_2 (n_1 + n_2 + 1)}{12}}\)

Calculate the test statistic

Use the observed rank sum \(W_1\), expected rank sum \(E(W_1)\), and standard deviation \(SD(W_1)\) to compute the test statistic, \(z\): \(z = \frac{W_1 - E(W_1)}{SD(W_1)}\)

Compare the test statistic to the critical value

Using the significance level \(\alpha = 0.05\) and a one-tailed Z-distribution table, we can find the critical value for the test statistic \(z\). If the calculated test statistic is greater than the critical value, we can reject the null hypothesis and conclude that rats receiving the antitoxin tend to survive longer after chemotherapy than those not receiving the antitoxin. In other words, if \(z > z_\alpha\), then there is sufficient evidence to conclude that the antitoxin drug has a significant effect on rats' survival time after chemotherapy.

Key concepts

Survival Time Analysis

Survival time analysis is a branch of statistics that deals with analyzing the time until a certain event of interest occurs, such as death, failure, or recovery. In the context of medical research, such as the study described in the exercise, it's used to evaluate the effectiveness of treatments and to compare the survival rates of different treatment groups.

Understanding the survival times of patients or subjects under study can be critical for developing new treatments or understanding the impact of existing ones. In the exercise, we compare the survival times of two groups of rats, one receiving chemotherapy only and the other receiving chemotherapy plus an additional drug. Here, the question is whether the additional drug (antitoxin) increases the survival time post-chemotherapy, which could have significant implications for chemotherapy research.

Applying non-parametric statistical methods, such as the Wilcoxon rank sum test in this case, is essential when the survival time data doesn't follow a normal distribution, which is common in medical and biological studies. These methods are robust and can be used even with small sample sizes, as seen with the 12 rats in each group. Applying such analyses correctly can ultimately guide researchers towards more effective treatments and a better understanding of a drug's impact on survival.

Non-Parametric Statistical Methods

Non-parametric statistical methods are a set of techniques used when the data being analyzed doesn't fit the assumptions needed to apply parametric statistics, such as a normal distribution or equal variances. These methods are quite valuable when dealing with ordinal data, ranks, or when the data's underlying distribution is unknown or non-normal, as is often the case in survival time analysis.

In the exercise, we use one such method, the Wilcoxon rank sum test, which is a non-parametric alternative to the two-sample t-test. It compares the median values of two groups by looking at the ranks of data. This is particularly useful when handling skewed data or outliers, which can heavily influence the results of more parametric tests. In the context of the study with rats, using the Wilcoxon rank sum test allows the researchers to avoid the assumptions of normality and still draw meaningful conclusions about the differences in survival times between the two groups.

Moreover, the step-by-step solution of the exercise utilizes ranks to statistically infer whether the antitoxin has a significant impact on survival time without relying on means or standard deviations. This approach emphasizes both the practicality and importance of non-parametric methods in conducting research and analyzing data when conventional statistical methods aren't applicable.

Chemotherapy Research

Chemotherapy research is a critical area in the fight against cancer and involves the study and development of drugs designed to destroy or slow the growth of cancer cells. A significant aspect of this research is assessing how different substances interact with chemotherapy drugs and their effects on treatment outcomes such as survival time.

The exercise explores a common research question in chemotherapy, aiming to determine if adding an antitoxin can reduce the harmful effects of chemotherapy on the survival of rats. This reflects real-world scenarios where researchers look for ways to improve the efficacy of chemotherapy and mitigate its side effects on normal cells. The Wilcoxon rank sum test, as performed in the exercise, becomes an essential tool for researchers to statistically evaluate if an additional treatment has a tangible benefit.

Improvements in chemotherapy come from robust research where statistical evidence is used to highlight significant differences in treatment outcomes. Having a statistically significant result, like the one sought in the exercise, could contribute to shaping the direction of future chemotherapy research and improving treatment protocols. By using detailed and carefully selected statistical methods, researchers can present compelling evidence to support the advancement of cancer treatments, ultimately leading to better patient care and survival outcomes.