Question

Patients with coronary artery disease were randomly assigned to either receive angioplasty plus medical therapy \((n=1149)\) or medical therapy alone \((n=1138)\) in a clinical trial. Over the next several years 85 angioplasty and 95 medical therapy patients died, with cause of death categorized as cardiac, other, or unknown. The following table shows a cross classification of the data. \(^{39}\) Is there statistically significant evidence, at the 0.10 level, to conclude that there is an association between treatment group (angioplasty versus medical therapy) and outcome? (a) State the null and alternative hypotheses in context. (b) How many degrees of freedom are there for a chi-square test? (c) The \(P\) -value for the chi-square test is \(0.87 .\) If \(\alpha=0.10\), what is your conclusion regarding \(H_{0}\) ? $$ \begin{array}{|lccccc|} \hline & && {\text { Unknown }} \\ & {\text { Cardiac death }} & \text { Other death } & \text { cause of death } & \text { Alive } & \text { Total } \\ \hline \text { Angioplasty } & 23 & 45 & 17 & 1,064 & 1,149 \\ \text { Medical therapy } & 25 & 51 & 19 & 1,043 & 1,138 \\ \text { Total } & 48 & 96 & 36 & 2,107 & 2,287 \\ \hline \end{array} $$

Short answer

The null hypothesis is not rejected because the P-value of 0.87 exceeds the significance level of 0.10; thus, no association between treatment and outcome is established.

Step-by-step solution

State the Null and Alternative Hypotheses

The null hypothesis (\(H_0\)) asserts that there is no association between the type of treatment (angioplasty plus medical therapy or medical therapy alone) and the outcome (death classified as cardiac, other, or unknown, or alive). The alternative hypothesis (\(H_1\)) suggests that there is an association between treatment type and outcome.

Determine the Degrees of Freedom

The degrees of freedom (df) for a chi-square test is calculated by the formula: df = (number of rows - 1) * (number of columns - 1). In this case, there are two treatment groups and four outcome categories, so df = (2 - 1) * (4 - 1) = 1 * 3 = 3.

Draw a Conclusion from the P-Value

Since the P-value (\(0.87\)) is greater than the significance level \(\alpha=0.10\)), we fail to reject the null hypothesis. There is not enough evidence to conclude that there is an association between treatment group and outcome.

Key concepts

Null and Alternative Hypotheses

Understanding null and alternative hypotheses is crucial when conducting a statistical test, such as the chi-square test for association. In the context of investigating the treatment for coronary artery disease, the null hypothesis (\( H_0 \)) represents the baseline assumption that there is no association between the type of treatment (angioplasty plus medical therapy or medical therapy alone) and the patient outcomes.

The alternative hypothesis (\( H_1 \)), on the other hand, represents the researcher's suspicion or the theory being tested, suggesting that there is an association between treatment type and outcome. The purpose of the chi-square test is to determine whether the observed outcomes sufficiently contradict the null hypothesis to support the alternative hypothesis. A failure to reject the null hypothesis implies that the data do not show a statistically significant association, whereas rejecting it indicates evidence that such an association may exist.

Degrees of Freedom

The concept of degrees of freedom (\( df \)) in statistics is tied to the number of independent values that can vary in an analysis without breaking any constraints. In the context of the chi-square test, degrees of freedom are used to determine the critical value from the chi-square distribution, which is then compared to the test statistic to draw conclusions.

For a contingency table like the one used in assessing the treatment for coronary artery disease, the formula for calculating degrees of freedom is given by \( df = (r - 1) * (c - 1) \) where \(r\) is the number of levels for one categorical variable, and \(c\) is the number of levels for the other categorical variable. In this study, with two treatment types and four possible outcomes, there are \( df = (2-1) * (4-1) = 3 \) degrees of freedom. Knowing the degrees of freedom helps researchers determine the appropriate distribution to reference when interpreting their test statistic.

P-value Interpretation

The P-value is a critical component of hypothesis testing as it provides the probability of observing the data, or something more extreme, assuming the null hypothesis is true. In simple terms, it tells us how 'surprising' the data are, given that there is no real effect or association.

In the chi-square test for the association between treatment groups and outcomes in coronary artery disease patients, a P-value of \(0.87\) is notably higher than the significance level of \(\alpha = 0.10\). This means that the observed data are very likely to occur even if there is no real association between the treatments and outcomes. With a high P-value, we do not have evidence to reject the null hypothesis and must conclude that the data do not provide sufficient evidence of an association.

Coronary Artery Disease Treatment

Coronary artery disease (CAD) is a serious condition that can lead to heart attacks and death. The treatment of CAD often involves a combination of lifestyle changes, medication, and procedures like angioplasty. In the context of our chi-square test, the two groups compared were those receiving angioplasty plus medical therapy versus those receiving medical therapy alone.

Both these treatments aim to improve patient outcomes by reducing symptoms and preventing future cardiac events. However, determining the statistical association between the types of treatments and patient outcomes is crucial for understanding the effectiveness of these approaches. Even though the chi-square test in this study revealed no significant association, this does not necessarily indicate equal effectiveness; factors such as the nature and severity of CAD, patient health, and other medical conditions play a vital role in treatment success. Clinical decisions should always be based on a comprehensive understanding of the patient's condition and the best available evidence, which can sometimes extend beyond what is captured in a statistical test.