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Chapter 11: Problem 17

A study used 1496 patients suffering from low levels of anxiety. The study randomly assigned each subject to a cognitive behavioral therapy (CBT) treatment or a placebo treatment. In this study, increased anxiety levels were observed for 45 of the 729 subjects taking a placebo and for 29 of the 767 subjects taking CBT. a. Report the data in the form of a \(2 \times 2\) contingency table. b. Show how to carry out all five steps of the null hypothesis that having an anxiety attack is not associated with whether one is taking a placebo or CBT. (You should get a chi-squared statistic equal to \(4.5 .\) ) Interpret.

Short Answer

Expert verified
Construct a 2x2 table, calculate chi-square as 4.5, interprets treatment and anxiety levels are likely associated.

Step by step solution

01

Understanding the Structure

To address part (a), we need a contingency table. A contingency table displays the frequency distribution of variables and helps in assessing the relationship between them. We need to create a table with treatments (CBT and Placebo) versus outcomes (Anxiety Increase and No Anxiety Increase).
02

Creating the Contingency Table

Construct a 2x2 table where rows represent the type of treatment (CBT and Placebo) and columns represent anxiety outcomes (Increased Anxiety and No Increased Anxiety). Fill the table with the given data: | Treatment | Increased Anxiety | No Increased Anxiety | Total | |------------|-------------------|----------------------|--------| | Placebo | 45 | 684 | 729 | | CBT | 29 | 738 | 767 | | Total | 74 | 1422 | 1496 |
03

Setting the Hypotheses

Define the null and alternative hypotheses for part (b): - Null Hypothesis ( H_0 ): There is no association between the treatment type (CBT vs. Placebo) and increased anxiety levels. - Alternative Hypothesis ( H_1 ): There is an association between the treatment type and increased anxiety levels.
04

Calculating Expected Frequencies

Using the formula for expected frequency, E_{ij} = rac{(row ext{ } total imes column ext{ } total)}{grand ext{ } total} calculate expected frequencies for each cell: - Placebo, Increased Anxiety: rac{729 imes 74}{1496} = 36.06 - Placebo, No Increased Anxiety: rac{729 imes 1422}{1496} = 692.94 - CBT, Increased Anxiety: rac{767 imes 74}{1496} = 37.94 - CBT, No Increased Anxiety: rac{767 imes 1422}{1496} = 729.06 .
05

Computing Chi-Square Statistic

Apply the chi-square formula: ext{Chi-square} = rac{(O_{ij}-E_{ij})^2}{E_{ij}} for each cell where O_{ij} is observed frequency and E_{ij} is expected frequency: - For (Placebo, Increased Anxiety): rac{(45-36.06)^2}{36.06} = 2.21 - For (Placebo, No Increased Anxiety): rac{(684-692.94)^2}{692.94} = 0.12 - For (CBT, Increased Anxiety): rac{(29-37.94)^2}{37.94} = 2.11 - For (CBT, No Increased Anxiety): rac{(738-729.06)^2}{729.06} = 0.11 The total chi-square statistic is the sum: 4.5 .
06

Interpretation

The calculated chi-square statistic is 4.5 . We compare it with the critical value from the chi-square distribution table at a certain significance level (e.g., alpha = 0.05 ) and degrees of freedom (df = 1 in this case, since df = (rows-1)(columns-1)). If the chi-square statistic is greater than the critical value, the null hypothesis is rejected, indicating a significant association.

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Key Concepts

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

Chi-Square Test
The Chi-Square Test is a statistical method used to determine if there is a significant association between two categorical variables. This test compares the observed frequencies in a contingency table to the frequencies we would expect to occur if the variables were independent.
  • It is commonly used in research when the data is in the form of counts or frequencies.
  • By calculating a chi-square statistic, researchers can assess whether the observed spread of frequencies is due to chance or if there's an underlying relationship.

In our example, the test helps us determine if anxiety level outcomes are independent of whether patients received Cognitive Behavioral Therapy (CBT) or a placebo. The calculated chi-square statistic tells us how far the observed results are from the expected if there was no association between the treatment and increased anxiety levels.
Null Hypothesis
In statistical tests like the Chi-Square Test, much of the analysis revolves around the concept of the null hypothesis. The null hypothesis is a statement suggesting that there is no relationship or effect between the variables under investigation.
  • It serves as a starting point for statistical testing.
  • The aim is to challenge or reject the null hypothesis in favor of an alternative hypothesis, which suggests a specific effect or association.

For instance, in the provided study, the null hypothesis states that there is no association between treatment type (CBT or placebo) and whether anxiety levels increase. Through statistical testing, such as calculating the chi-square statistic, researchers can decide whether to reject this hypothesis.
Cognitive Behavioral Therapy
Cognitive Behavioral Therapy (CBT) is a widely used psychological treatment, effective for a variety of mental health issues, including anxiety. It focuses on modifying dysfunctional emotions, behaviors, and thoughts through a structured, goal-oriented approach.
  • CBT helps individuals identify and change negative thinking patterns and behaviors that contribute to their anxiety.
  • It is typically conducted through sessions that provide coping strategies, problem-solving techniques, and skills training.

This therapy was compared with a placebo in the study to observe its impact on anxiety levels, providing insight into how effective CBT is relative to non-specific treatments or no treatment at all.
Anxiety Levels
Anxiety levels refer to the degree of anxiety experienced by an individual, ranging from mild to severe. In research settings, it's important to quantify these levels to assess the effectiveness of interventions like CBT.
  • Increased anxiety levels might manifest through symptoms such as restlessness, excessive worry, or difficulty concentrating.
  • Defining and measuring anxiety levels help researchers objectively evaluate the impact of different treatments.

In the study mentioned, increased anxiety levels were tracked to compare their occurrence between subjects undergoing CBT and those receiving a placebo. This kind of assessment helps determine if CBT can significantly reduce anxiety compared to not receiving the specific therapy.

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Most popular questions from this chapter

Babies and gray hair A young child wonders what causes women to have babies. For each woman who lives on her block, she observes whether her hair is gray and whether she has young children, with the results shown in the table that follows. a. Construct the \(2 \times 2\) contingency table that cross-tabulates gray hair (yes, no) with has young children (yes, no) for these nine women. b. Treating has young children as the response variable, obtain the conditional distributions for those women who have gray hair and for those who do not. Does there seem to be an association? c. Noticing this association, the child concludes that not having gray hair is what causes women to have children. Use this example to explain why association does not necessarily imply causation. \begin{tabular}{lcc} \hline Woman & Gray Hair & Young Children \\ \hline Andrea & No & Yes \\ Mary & Yes & No \\ Linda & No & Yes \\ Jane & No & Yes \\ Maureen & Yes & No \\ Judy & Yes & No \\ Margo & No & Yes \\ Carol & Yes & No \\ Donna & No & Yes \\ \hline \end{tabular}

In the GSS, subjects who were married were asked about the happiness of their marriage, the variable coded as HAPMAR. a. Go to the GSS website sda.berkeley.edu/GSS/, click GSS with no weight as the default, and construct a contingency table for 2012 relating family income (measured as in Table 11.1 ) to marital happiness: Enter FINRELA(r:4;3;2) as the row variable (where \(4=\) "above average," \(3=\) "average, \("\) and \(2=\) "below average") and HAPMAR(r:3;2;1) as the column variable \((3=\) not \(t o 0,2=\) pretty \(,\) and \(1=\) very happy \()\) As the selection filter, enter YEAR(2012). Under Output Options, put a check in the row box (instead of in the column box) for the Percentaging option and put a check in the Summary Statistics box further below. Click on Run the Table. b. Construct a table or graph that shows the conditional distributions of marital happiness, given family income. How would you describe the association? c. Compare the conditional distributions to those in Table \(11.2 .\) For a given family income, what tends to be higher, general happiness or marital happiness for those who are married? Explain.

In a study conducted by a pharmaceutical company, 605 out of 790 smokers and 122 out of 434 nonsmokers were diagnosed with lung cancer. a. Construct a \(2 \times 2\) contingency table relating smoking (SMOKING, categories smoker and nonsmoker) as the rows to lung cancer (LUNGCANCER, categories present and absent) as the columns. b. Find the four expected cell counts when assuming independence. Compare them to the observed cell counts, identifying cells having more observations than expected. c. For this data, \(X^{2}=272.89 .\) Verify this value by plugging into the formula for \(X^{2}\) and computing the sum.

Another predictor of happiness? Go to sda.berkeley.edu/ GSS and find a variable that is associated with happiness, other than variables used in this chapter. Use methods from this chapter to describe and make inferences about the association, in a one-page report.

Aspirin and heart attacks for women A study in the New England Journal of Medicine compared cardiovascular events for treatments of low-dose aspirin or placebo among 39,876 healthy female health eare providers for anaverage duration of about 10 years. Results indicated that women receiving aspirin and those receiving placebo did not differ for rates of a first major cardiovascular event, death from cardiovascular causes, or fatal or nonfatal heart attacks. However, women receiving aspirin had lower rates of stroke than those receiving placebo (data from N. Engl.J.Med., vol. \(352,2005, \mathrm{pp} .1293-1304) .\) \begin{tabular}{lccc} \hline \multicolumn{3}{c} { Women's Aspirin Study Data } \\ \hline Group & Mini-Stroke & Stroke & No Strokes \\ \hline Placebo & 240 & 259 & 19443 \\ Aspirin & 185 & 219 & 19530 \\ \hline \end{tabular} a. Use software to test independence. Show (i) assump- tions, (ii) hypotheses, (iii) test statistic, (iv) P-value, (v) conclusion in the context of this study. b. Describe the association by finding and interpreting the relative risk for the stroke category.
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