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Chapter 27: Problem 27

As part of an ANOVA that compares three treatments, you carry out Tukey pairwise tests at the overall \(5 \%\) significance level. The Tukey tests find that \(\boldsymbol{\mu}_{1}\) is significantly different from \(\boldsymbol{\mu}_{2}\) but that the other two comparisons are not significant. You can be \(95 \%\) confident that (a) \(\boldsymbol{\mu}_{1} \neq \boldsymbol{\mu}_{2}\) and \(\boldsymbol{\mu}_{1}=\boldsymbol{\mu}_{3}\) and \(\boldsymbol{\mu}_{2}=\boldsymbol{\mu}_{3}\). (b) just \(\boldsymbol{\mu}_{1} \neq \boldsymbol{\mu}_{2}\); there is not enough evidence to draw conclusions about the other pairs of means. (c) \(\boldsymbol{\mu}_{1}=\boldsymbol{\mu}_{3}\) and \(\boldsymbol{\mu}_{2}=\boldsymbol{\mu}_{3}\), and this implies that it must also be true that \(\boldsymbol{\mu}_{1}=\boldsymbol{\mu}_{2}\).

Short Answer

Expert verified
Option (b) is correct; \( \mu_1 \neq \mu_2 \) with no further conclusions possible for the other pairs.

Step by step solution

01

Understand the ANOVA Outcome

In the context of ANOVA, we are comparing the means of three treatments: \( \mu_1, \mu_2, \mu_3 \). The Tukey post-hoc tests indicate that \( \mu_1 \) is significantly different from \( \mu_2 \), while \( \mu_1 \) and \( \mu_3 \), as well as \( \mu_2 \) and \( \mu_3 \), do not have significant differences at the 5% significance level.
02

Analyze Each Statement

Let's break down each option:- **Option (a):** Proposes \( \mu_1 eq \mu_2 \), \( \mu_1 = \mu_3 \), and \( \mu_2 = \mu_3 \). The first part (\( \mu_1 eq \mu_2 \)) is correct, but there isn't enough evidence from the Tukey test results to definitively state \( \mu_1 = \mu_3 \) and \( \mu_2 = \mu_3 \).- **Option (b):** States just \( \mu_1 eq \mu_2 \). This aligns with the Tukey test results and accurately reflects the data.- **Option (c):** Suggests \( \mu_1 = \mu_3 \), \( \mu_2 = \mu_3 \), and implies \( \mu_1 = \mu_2 \). This conflicts with the Tukey test conclusion that \( \mu_1 eq \mu_2 \).
03

Select the Correct Answer

From analyzing the statements, it is clear that option (b) correctly reflects the results of the Tukey test. The test indicated significant differences only between \( \mu_1 \) and \( \mu_2 \), with insufficient evidence to draw conclusions about the other pairs.

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Key Concepts

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

Tukey test
The Tukey test, also known as Tukey's Honestly Significant Difference (HSD) test, is a statistical tool used in post-hoc analysis after performing an ANOVA. It's designed to identify which specific group means are different from each other. This test helps in making multiple pairwise comparisons between group means. The beauty of the Tukey test is that it controls the overall Type I error rate, which minimizes the chances of incorrectly identifying a difference when none exists. Thus, it ensures that the likelihood of finding a significant difference is truly due to the variance between the group means and not random chance.

The Tukey test operates by calculating a 'critical value', which is then compared against the differences between each pair of means. If the difference exceeds this critical value, the test deems it significant. This straightforward mechanism makes it a popular method among researchers who deal with numerous datasets and require a reliable method for comparison.
Pairwise comparisons
In statistical analysis, pairwise comparisons are integral when you want to determine which specific group means differ in a dataset. This process involves comparing each possible pair of group means to see if there is a statistically significant difference between them. For example, if you have three treatment groups with means \( \mu_1, \mu_2, \mu_3 \), pairwise comparisons assist you in assessing each pair: \( \mu_1 \) versus \( \mu_2 \), \( \mu_1 \) versus \( \mu_3 \), and \( \mu_2 \) versus \( \mu_3 \).

This type of analysis is crucial after ANOVA confirms there are significant differences somewhere among the group means but doesn't specify where. Pairwise comparisons narrow down the options, providing clearer insights into specific distinct relationships within the data. By examining these, researchers and statisticians can hone in on meaningful interpretations and conclusions from their data.
Significance level
The significance level in statistics is a threshold that determines whether a result from a test is statistically significant. It's often denoted by the Greek letter alpha (\( \alpha \)) and represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Commonly, a significance level of \(5\%\) or \(0.05\) is used.

In the context of the Tukey test and ANOVA, the significance level plays a vital role. It dictates the cutoff point for considering whether a pairwise comparison between means is significant. For instance, if the test results of comparing \( \mu_1 \) and \( \mu_2 \) fall within this threshold, then you can conclude there's a statistically significant difference between these treatments. Lower levels of significance, like \(1\%\), require stronger evidence to support a difference. By setting an appropriate significance level, researchers balance between being too lenient and too strict, thus avoiding incorrect conclusions.
Post-hoc analysis
Post-hoc analysis is an important step that follows an ANOVA test when the initial analysis reveals significant differences among group means. While ANOVA tells you if any differences exist, it doesn't specify where they occur. This is where post-hoc tests, like the Tukey test, come in. They delve deeper, helping to identify which specific pairs of means differ significantly.

With post-hoc analysis, each pair of groups is analyzed to discover notable differences in a controlled manner. This step is crucial because it guides researchers in making detailed conclusions about their datasets without increasing the risk of Type I errors. Often, multiple comparisons can lead to accidental false positives, but using post-hoc procedures like the Tukey test controls for this, ensuring the results are reliable and meaningful.

Therefore, post-hoc analyses are indispensable for interpreting complex data accurately when multiple comparisons are at play. They transform broad analyses into actionable insights, aiding in data-driven decision-making.

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Most popular questions from this chapter

To test whether a hearing aid is right for a patient, audiologists play a tape on which words are pronounced at low volume. The patient tries to repeat the words. There are several different lists of words that are supposed to be equally difficult. Are the lists equally difficult when there is background noise? To find out, an experimenter had subjects with normal hearing listen to four lists with a noisy background. The response variable was the percent of the 50 words in a list that the subject repeated correctly. The data set contains 96 responses. \({ }^{17}\) Here are two study designs that could produce these data: Design A. The experimenter assigns 96 subjects to four groups at random. Each group of 24 subjects listens to one of the lists. All individuals listen and respond separately. Design B. The experimenter has 24 subjects. Each subject listens to all four lists in random order. All individuals listen and respond separately. Does Design A allow use of one-way ANOVA to compare the lists? Does Design B allow use of one-way ANOVA to compare the lists? Briefly explain your answers.

Can the introduction of pleasant sensory stimuli lead to a more pleasant exercise environment and decrease perceived exertion during a four-minute stepping task? Forty-three students from a Southeastern university were assigned at random to three conditions: "taste," in which participants inserted a lemon-flavored mouth guard during the task; "placebo," in which participants inserted a non-flavored mouth guard; and "control," in which no mouth guard was used. Twelve students were assigned to the taste group, 15 to the placebo group, and 16 to the control group. Ratings of perceived exertion (RPE) scores were measured on standard 15-point scale ranging from 6 (very, very light) to 20 (exhausted). \(.^{13}\)

Exercise \(27.49\) describes one of the experiments done by researchers at the University of Minne-sota to demonstrate that even being reminded of money makes people more self-sufficient and less involved with other people. Here are three more of these experiments. For each experiment, which statistical test from Chapters \(21,23,25\), and 27 would you use, and why? (a) Randomly assign student subjects to money and control groups. The control group unscrambles neutral phrases, and the money group unscrambles money- oriented phrases, as described in Exercise 27.49. Then ask the subjects to volunteer to help the experimenter by coding data sheets, about five minutes per sheet. Subjects said how many sheets they would volunteer to code. Participants in the money condition volunteered to help code fewer data sheets than did participants in the control condition. (b) Randomly assign student subjects to high-money, low-money, and control groups. After playing Monopoly for a short time, the high-money group is left with \(\$ 4000\) in Monopoly money, the low-money group with \(\$ 200\), and the control group with no money. Each subject is asked to imagine a future with lots of money (high-money group), a little money (low-money group), or just their future plans (control group). Another student walks in and spills a box of 27 pencils. How many pencils does the subject pick up? "Participants in the high-money condition gathered fewer pencils" than subjects in the other two groups. (c) Randomly assign student subjects to three groups. All do paperwork while a computer on the desk shows a screensaver of currency floating underwater (Group 1), a screensaver of fish swimming underwater (Group 2), or a blank screen (Group 3). Each subject must now develop an advertisement and can choose whether to work alone or with a partner. Count how many in each group make each choice. Choosing to perform the task with a coworker was reduced among money condition participants.

Although consumers often want to touch products before purchasing them, they generally prefer that others have not touched products they would like to buy. Can another person touching a product create a positive reaction? Subjects were given instructions to contact a sales associate at a university bookstore who would provide them with a shirt to try on. When meeting the sales associate, subjects were told there was only one shirt left and it was being tried on by another "customer." The other customer trying on the shirt was a confederate of the experimenter and was either an attractive, well-dressed professional female model or an average-looking female college student wearing jeans and a T-shirt. Subjects, who were either males or females, saw the confederate leaving the dressing room where the shirt was left for them to try on. There was also a control group of subjects who were handed the shirt directly off the rack by the sales associate. Thus, there were five treatments: male subjects seeing a model, female subjects seeing a model, male subjects seeing a college student, female subjects seeing a college student, and the control group. Subjects evaluated the product on five dimensions, each dimension on a seven-point scale, with the five scores then averaged to give the subject's evaluation measure, with higher numbers indicating a more positive evaluation. Here are the sample sizes, means, and standard deviations for the five groups: \({ }^{15}\) $$ \begin{array}{lccc} \hline \text { Treatment Group } & n & \mathrm{x}^{-} \bar{x} & \boldsymbol{s} \\\ \hline \text { Males seeing a model } & 22 & 5.34 & 0.87 \\ \hline \text { Males seeing a student } & 23 & 3.32 & 1.21 \\ \hline \text { Females seeing a model } & 24 & 4.10 & 1.32 \\ \hline \text { Females seeing a student } & 23 & 3.50 & 1.43 \\ \hline \text { Controls } & 27 & 4.17 & 1.50 \\ \hline \end{array} $$ (a) Verify that the sample standard deviations allow the use of ANOVA to compare the population means. What do the means suggest about the effect of the subject's sex and the attractiveness of the confederate on the evaluation of the product? (b) The paper reports the ANOVA \(F=8.30\). What are the degrees of freedom for the ANOVA \(F\) statistic and the \(P\)-value? State your conclusions.

Comparisons. Exercise \(27.42\) examines the relationship between previous video game experience and a surgeon's ability to acquire skills for laparoscopic surgery. Software gives these Tukey \(95 \%\) simultaneous confidence intervals: (a) How confident are you that all three of these intervals capture the true differences between pairs of population means? (b) Write a short summary of the results of the ANOVA, including the multiple comparisons.
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