Question

Men with prostate cancer were randomly assigned to undergo surgery \((n=347)\) or "watchful waiting" (no surgery, \(n=348\) ). Over the next several years there were 83 deaths in the first group and 106 deaths in the second group. The results are given in the table. \({ }^{18}\). (a) Let \(D\) and \(A\) represent died and alive, respectively, and let \(S\) and \(W W\) represent surgery and watchful waiting. Calculate \(\operatorname{Pr}\\{\mathrm{D} \mid \mathrm{S}\\}\) and \(\operatorname{Pr}\\{\mathrm{D} \mid \mathrm{WW}\\}\). (b) The value of the contingency-table chi-square statistic for these data is \(\chi_{s}^{2}=3.75 .\) Test for a relationship between the treatment and survival. Use a nondirectional alternative and let \(\alpha=0.05 .\)

Short answer

The probabilities are \(\operatorname{Pr}\{D \mid S\} = \frac{83}{347}\) and \(\operatorname{Pr}\{D \mid WW\} = \frac{106}{348}\). The chi-square test with \(\chi_{s}^{2} = 3.75\) does not lead to a rejection of the null hypothesis at the \(0.05\) level, thus failing to show a significant relationship between treatment and survival.

Step-by-step solution

Title - Calculate Probability of Death Given Surgery

To calculate the probability of death given surgery, denoted as \(\operatorname{Pr}\{D \mid S\}\), divide the number of deaths in the surgery group by the total number of patients in the surgery group: \[\operatorname{Pr}\{D \mid S\} = \frac{\text{Number of deaths in surgery group}}{\text{Total number of patients in surgery group}}\] Therefore: \[\operatorname{Pr}\{D \mid S\} = \frac{83}{347}\]

Title - Calculate Probability of Death Given Watchful Waiting

Similarly, to calculate the probability of death given watchful waiting, denoted as \(\operatorname{Pr}\{D \mid WW\}\), divide the number of deaths in the watchful waiting group by the total number of patients in the watchful waiting group: \[\operatorname{Pr}\{D \mid WW\} = \frac{\text{Number of deaths in watchful waiting group}}{\text{Total number of patients in watchful waiting group}}\] Therefore: \[\operatorname{Pr}\{D \mid WW\} = \frac{106}{348}\]

Title - State the Hypotheses

To test for a relationship between the treatment and survival, we will set up a chi-square test with the following hypotheses: \[H_0: \text{There is no relationship between treatment and survival (independence).}\] \[H_a: \text{There is a relationship between treatment and survival (dependence).}\] The null hypothesis assumes that the treatment does not affect the survival, while the alternative hypothesis assumes that there is an effect.

Title - Compare Chi-Square Test Statistic to Critical Value

Using a critical chi-square value table for the given degrees of freedom \((df = 1)\) and the significance level \(\alpha = 0.05\), we find the critical value for a nondirectional (two-tailed) test. If our test statistic \(\chi_{s}^{2} = 3.75\) is greater than the critical value, we will reject the null hypothesis. The critical value from the chi-square distribution table with \(df = 1\) and \(\alpha = 0.05\) is approximately 3.841. Since \(3.75 < 3.841\), we do not reject the null hypothesis.

Key concepts

Probability of Death Given Treatment

Understanding the likelihood of an event occurring under a specific condition is a core aspect of statistics often analyzed in medical research. One such example is determining the probability of death given a particular medical treatment. In the given exercise, we analyze this probability in the context of prostate cancer treatments: either undergoing surgery or opting for 'watchful waiting.' The method involves taking the total number of deaths within each treatment group and dividing it by the total number of patients receiving that treatment.

This calculation yields valuable insights into the risks associated with each treatment option. For instance, a higher probability of death under one treatment could imply a riskier approach, whereas a lower probability might suggest a safer or more effective treatment. However, it is crucial to remember that these probabilities do not confirm causation and should be interpreted carefully within the broader context of the research.

Testing for Independence

In statistics, testing for independence is a critical technique used to determine if there is a relationship between two categorical variables. In our context, we're interested in understanding whether the type of treatment for prostate cancer (surgery vs. watchful waiting) is independent of the eventual outcome (death or survival).

The null hypothesis typically posits that the two variables are independent, meaning the choice of treatment does not significantly affect the mortality rate. If, after performing a statistical test like the Chi-Square Test, we find evidence to reject the null hypothesis, it would suggest that there may be a dependence or association between the treatment method and the patient's survival. This information is crucial as it can drive medical recommendations and influence patient choices.

Contingency Table Analysis

A contingency table, also known as a cross-tabulation or two-way table, is a powerful tool in statistical analysis used to examine the relationship between two categorical variables. It displays the frequency distribution of variables and allows researchers to observe the interaction between them.

In the prostate cancer treatment analysis, a contingency table helps organize the data into four categories: deaths and survivals for both the surgery group and the watchful waiting group. The table then facilitates the computation of the Chi-Square statistic, which measures how expected counts compare to observed counts. A large discrepancy between these counts indicates an association between the treatment method and survival outcomes. The analysis of such tables is integral to making informed, data-driven decisions in the medical field.

Hypothesis Testing

Hypothesis testing is the backbone of statistical inference, allowing us to make decisions about populations based on sample data. It involves setting up two opposing hypotheses: the null hypothesis, which represents the status quo or no effect, and the alternative hypothesis, which suggests a new effect or relationship.

In the context of our exercise, we used a Chi-Square Test to decide whether treatment type affects the survival of prostate cancer patients. The calculated Chi-Square statistic was compared against a critical value, determining whether the observed data significantly diverged from what would be expected under the null hypothesis. If the test statistic exceeds the critical value, the null hypothesis is rejected, suggesting a significant relationship between treatment and survival. This process aids in validating research findings and ensuring that medical practices are grounded in statistically significant data.