Question

To compare the effects of three toxic chemicals, \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C},\) on the skin of rats, 2 -centimeter-side squares of skin were treated with the chemicals and then scored from 0 to 10 depending on the degree of irritation. Three adjacent 2-centimeter-side squares were marked on the backs of eight rats, and each of the three chemicals was applied to each rat. Thus, the experiment was blocked on rats to eliminate the variation in skin sensitivity from rat to rat. a. Do the data provide sufficient evidence to indicate a difference in the toxic effects of the three chemicals? Test using the Friedman \(F_{r}\) -test with \(\alpha=.05 .\) b. Find the approximate \(p\) -value for the test and interpret it.

Short answer

Answer: The conclusion depends on the comparison of the calculated F_r value with either the critical value of the chi-square distribution or the significance level of 0.05 against the p-value derived from the F_r statistic. If either the F_r value is greater than the critical value or the p-value is less than the significance level, then we can conclude that the toxic effects of the chemicals are different.

Step-by-step solution

State the null and alternative hypotheses

We need to check if there is enough evidence to show the difference in the toxic effects of the three chemicals. Let \(\mu_A\), \(\mu_B\), and \(\mu_C\) be the average irritation scores of chemicals A, B, and C. Null hypothesis \(H_0\): The three chemicals have the same toxicity, i.e. \(\mu_A=\mu_B=\mu_C\). Alternative hypothesis \(H_a\): The three chemicals have different toxicity, i.e. at least one of \(\mu_A\), \(\mu_B\), and \(\mu_C\) is different from the others.

Calculate the rank sum for each chemical

To perform the Friedman F_r-test, we need to rank the degree of irritation scores for each rat first. If there are tied scores in a row, the average rank is assigned. After calculating the ranks, calculate the rank sum R_j for each chemical j: \(R_j = \sum_{i=1}^8 rank(\text{irritation score for chemical } j \text{ on rat } i)\) Where j is the chemicals (A, B, or C) and i is the rat number.

Calculate the Friedman test statistic F_r

The Friedman test statistic \(F_r\) is calculated with the following formula: \(F_r = \frac{12}{k(n-1)} \sum_{j=1}^k \frac{R_j^2}{n} - 3(n+1)\) Where \(k\) is the number of treatments (3 chemicals) and \(n\) is the number of blocks (8 rats).

Find the critical value and make a decision

We proceed to find the critical value of \(\chi^2\) distribution with \(k-1=3-1=2\) degrees of freedom at a significance level of \(\alpha=0.05\). Compare the calculated \(F_r\) value with the critical value of the \(\chi^2\) distribution. If \(F_r\) is greater than the critical value, reject the null hypothesis, which means that there is a significant difference between the toxic effects of the chemicals.

Calculate the approximated p-value

We can also test the null hypothesis using the p-value (which is asked in part b). To do this, we need to find the p-value corresponding to our \(F_r\) statistic from the chi-square distribution with \(k-1\) degrees of freedom. Compare this p-value with the significance level of 0.05. If the p-value is less than the significance level (0.05), we reject the null hypothesis in favor of the alternative hypothesis, which means that there is enough evidence to claim that the toxic effects of the three chemicals are different.

Conclusion

By performing the Friedman \(F_r\)-test, we tested if there was enough evidence to show that the toxic effects of the three chemicals (A, B, and C) are different. Additionally, we found the approximate p-value and compared it to the significance level of 0.05. The results of the test will give us either enough evidence or not to claim that the toxic effects of the chemicals are different.

Key concepts

Understanding the Null Hypothesis

The null hypothesis is a crucial starting point for statistical testing. It acts as a default statement that there is no effect or no difference present. In layman's terms, it's like assuming everyone has the same skill level in a game until proven otherwise. When scientists or researchers make claims, they use the null hypothesis to test whether their claim has enough evidence to show a significant difference.

In the context of the Friedman test, the null hypothesis suggests that the three chemicals, A, B, and C, all have the same level of toxicity on rat skin. Formally, it's expressed as \(\mu_A = \mu_B = \mu_C\). This hypothesis is tested against the alternative, which posits that at least one chemical has a different effect. Drawing a conclusion in the study often relies on whether or not we can reject the null hypothesis based on the data collected. To reject the null, we need strong enough evidence that shows noticeable differences among the chemicals' effects.

Calculating the Friedman F_r Statistic

The Friedman F_r statistic serves as a beacon, guiding researchers to see if their data can confidently suggest differences in grouped scenarios. Think of it as a tool that scrutinizes rankings of different treatments across multiple subjects or blocks. It's particularly useful when normality can't be assumed, or when dealing with non-parametric data. The F_r statistic uses ranks of data rather than the actual data values, which can be beneficial when the data is not evenly distributed.

To compute the F_r statistic, we rank the irritation scores for each rat's reaction to the chemicals. Sum each chemical's ranks across all rats to get their rank sum R_j. The Friedman F_r statistic is then calculated using the formula \( F_r = \frac{12}{k(n-1)} \sum_{j=1}^k \frac{R_j^2}{n} - 3(n+1) \), where \( j \) represents each chemical and \( i \) each rat. With this value in hand, we can proceed to compare it to a critical value from the chi-square distribution to assess the evidence against our null hypothesis.

Interpreting the p-value

The p-value is statistical shorthand for the probability that the data could occur under the assumption that the null hypothesis is true. It gives us a numerical way to decide whether to reject this null hypothesis. Picture the p-value as a measure of surprise; a lower p-value means your data is more surprising under the assumption that the null hypothesis is correct, indicating that your hypothesis might need to be rejected.

Upon calculating the Friedman F_r statistic, we then determine how unusual this statistic would be if the null hypothesis were true by finding the corresponding p-value. If this p-value is below a chosen significance level (commonly 0.05), we conclude the data is sufficiently surprising to reject the null hypothesis. In the context of our exercise, we compare the p-value to \(\alpha = 0.05\). If our p-value is less, we have strong evidence to suggest that the toxic effects of the three chemicals are significantly different.