91Ó°ÊÓ

The model for the data from a randomized block experiment for comparing \(I\) treatments was \(X_{i j}=\mu+\alpha_{i}+\beta_{j}+\varepsilon_{i j}\), where the \(\alpha\) 's are treatment effects, the \(\beta\) 's are block effects, and the \(\varepsilon\) 's were assumed normal with mean 0 and variance \(\sigma^{2}\). We now replace normality by the assumption that the \(\varepsilon\) 's have the same continuous distribution. A distribution-free test of the null hypothesis of no treatment effects, called Friedman's test, involves first ranking the observations in each block separately from 1 to \(I\). The rank average \(\bar{R}_{i}\) is then calculated for each of the \(I\) treatments. If \(H_{0}\) is true, the expected value of each rank average is \((I+1) / 2\). The test statistic is $$ F_{r}=\frac{12 J}{I(I+1)} \sum\left(\bar{R}_{i}-\frac{I+1}{2}\right)^{2} $$ For even moderate values of \(J\), the test statistic has approximately a chi- squared distribution with \(I-1 \mathrm{df}\) when \(H_{0}\) is true. The article "Physiological Effects During Hypnotically Requested Emotions" (Psychosomatic Med., 1963: 334-343) reports the following data \(\left(x_{i j}\right)\) on skin potential in millivolts when the emotions of fear, happiness, depression, and calmness were requested from each of eight subjects. $$ \begin{array}{lllrl} \text { Fear } & 23.1 & 57.6 & 10.5 & 23.6 \\ \text { Happiness } & 22.7 & 53.2 & 9.7 & 19.6 \\ \text { Depression } & 22.5 & 53.7 & 10.8 & 21.1 \\ \text { Calmness } & 22.6 & 53.1 & 8.3 & 21.6 \\ & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \cline { 2 - 5 } \text { Fear } & 11.9 & 54.6 & 21.0 & 20.3 \\ \text { Happiness } & 13.8 & 47.1 & 13.6 & 23.6 \\ \text { Depression } & 13.7 & 39.2 & 13.7 & 16.3 \\ \text { Calmness } & 13.3 & 37.0 & 14.8 & 14.8 \\ \hline \end{array} $$ Use Friedman's test to decide whether emotion has an effect on skin potential.

Step by step solution

01

Arrange Data by Subject

Start by organizing the data for each subject (block) separately, which will allow us to rank the observations for each subject. We have eight subjects, so make sure to list the observations for Fear, Happiness, Depression, and Calmness for each subject.

02

Rank Observations within Each Block

For each subject, rank the skin potential observations from 1 to 4 since there are four emotions (treatments) being tested. Assign the smallest value a rank of 1, and the largest value a rank of 4. If there are ties, each tied value gets the average of their rank positions.

03

Calculate Rank Averages

Compute the average rank \(\bar{R}_{i}\) for each emotion by summing the ranks across all subjects and dividing by the number of subjects, which is 8.

04

Compute Test Statistic

Use Friedman's test statistic formula: \[F_{r}=\frac{12 \times 8}{4(4+1)} \sum\left(\bar{R}_{i}-\frac{4+1}{2}\right)^{2}\] Calculate \(\sum\left(\bar{R}_{i}-2.5\right)^{2}\) for each emotion and then compute \(F_{r}\).

05

Determine Chi-Squared Critical Value

For \(I-1\) degrees of freedom (3, since \(I = 4\)), determine the critical chi-squared value from the chi-squared distribution table at a significance level (usually 0.05).

06

Decision Rule and Conclusion

Compare the computed \(F_{r}\) value to the critical chi-squared value. If \(F_{r}\) is greater than the critical value, reject the null hypothesis, indicating that emotion does have an effect on skin potential.

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Key Concepts

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

Randomized Block Experiment

A randomized block experiment is a type of experimental design used when subjects can be grouped into blocks based on a certain "blocking" factor. This factor is expected to affect the response variable, so controlling it helps improve the accuracy of the experiment. In this setup, the blocks are used to create groups of similar subjects or conditions, and then the treatments are randomly assigned within each block.

For example, in the provided problem about testing emotions' effects on skin potential, each subject (or individual) serves as a block. This means that the variability between subjects is controlled, allowing for a more focused investigation into how different emotional treatments affect skin potential without the added noise of inter-subject differences.

Key aspects of a randomized block design include:

• Within each block, the treatments are assigned randomly.
• The blocks are as homogenous as possible in terms of the blocking factor; in this case, individual subject characteristics.
• The main goal is to reduce the impact of confounding variables by isolating the variable of interest, which is the treatment effect.

By taking these steps, the experiment ensures a fair comparison of treatments by minimizing biases and variances not related to the treatment effects.

Distribution-Free Test

A distribution-free test, also known as a non-parametric test, does not assume a specific probability distribution in the population from which the samples are drawn. These tests are particularly useful when the assumptions necessary for parametric tests (like normality) cannot be met.

In the given scenario, Friedman's test is used as a distribution-free method to evaluate the null hypothesis that assumes there are no treatment (emotional) effects on skin potential across the blocks (or subjects).

This test is ideal when:

• The data do not meet the assumptions of parametric tests, such as normal distribution or equal variances.
• The sample sizes are small.
• The measurement scale is ordinal, that is, the data can be ranked.

Friedman's test works by converting the original observations into ranks and then analyzing the rank differences, which reduces dependency on data distribution assumptions and focuses on the median rather than means, making it robust to outliers and skewed distributions.

Chi-Squared Distribution

The chi-squared distribution is a probability distribution commonly used in hypothesis testing, especially in tests that involve variance. In the context of Friedman's test, the test statistic approximately follows a chi-squared distribution when the null hypothesis is true, making it an integral part of determining whether observed treatment effects are significant.

Characteristics of the chi-squared distribution include:

• It is always positively skewed, with the shape becoming more symmetrical as the degrees of freedom increase.
• It takes values ranging from zero to infinity.
• The chi-squared distribution is defined by its degrees of freedom, which are generally related to the number of independent pieces of information in your data.

For Friedman's test, the degrees of freedom are equal to the number of treatments minus one ( I-1 ), which tells us how the calculated Friedman test statistic should be compared against a chi-squared distribution table to decide if the null hypothesis should be rejected.

Statistical Hypothesis Testing

Statistical hypothesis testing is a crucial component of scientific research, providing a structured framework for making inferences about populations based on sample data. The process involves comparing a null hypothesis, which typically states that there is no effect or difference, against an alternative hypothesis, which suggests that there is an effect or difference.

In the context of Friedman's test, the null hypothesis ( H_0 ) posits that the treatment effects are negligible, meaning the emotions do not impact the skin potential readings. The test evaluates the likelihood that the observed data would occur if the null hypothesis were true.

Steps in hypothesis testing include:

• Formulating the null and alternative hypotheses.
• Choosing a significance level (usually 0.05).
• Calculating the test statistic based on the sample data.
• Comparing the test statistic to a critical value derived from a relevant probability distribution (such as chi-squared) to decide whether to reject or fail to reject the null hypothesis.

Ultimately, hypothesis testing helps researchers evaluate statistical evidence in their data, supporting or refuting the initial hypotheses with a quantifiable level of confidence.