Question

7.10.3 In a pharmacological study, researchers measured the concentration of the brain chemical dopamine in six rats exposed to toluene and six control rats. (This is the same study described in Example \(7.2 .1 .\) ) The concentrations in the striatum region of the brain were as shown in the table. $$ \begin{array}{|cc|} \hline {\text { Dopamine (ng/gm) }} \\ \hline \text { Toluene } & \text { Control } \\ \hline 3,420 & 1,820 \\ 2,314 & 1,843 \\ 1,911 & 1,397 \\ 2,464 & 1,803 \\ 2,781 & 2,539 \\ 2,803 & 1,990 \\ \hline \end{array} $$ (a) Use a Wilcoxon-Mann-Whitney test to compare the treatments at \(\alpha=0.05 .\) Use a nondirectional alternative. (b) Proceed as in part (a), but let the alternative hypothesis be that toluene increases dopamine concentration.

Short answer

Part (a): Perform Wilcoxon-Mann-Whitney test and compare P-value to \(\alpha=0.05\) for nondirectional hypothesis. Part (b): Use one-tailed P-value approach for the hypothesis that toluene increases dopamine concentration.

Step-by-step solution

Understanding the Exercise

To perform the Wilcoxon-Mann-Whitney test, first arrange all observations into a single ranked list. Then sum the ranks for the observations from each group.

Ranking the Observations

List all dopamine concentrations from both Toluene and Control groups. Assign ranks starting with 1 for the lowest value, increasing by 1 for each value, and handling ties by assigning the average rank to each tied value.

Calculate Rank Sums

Calculate the sum of ranks for each group, this will give you the rank sums: Toluene rank sum, and Control rank sum.

Determine Test Statistic

The test statistic for the Wilcoxon-Mann-Whitney test is typically the smaller of the two rank sums.

Find Critical Value or P-value

Using statistical tables or software, find the critical value or P-value corresponding to the calculated test statistic and the sample sizes for the two groups.

Make Decision for Part (a) Hypothesis Test

Compare the P-value with the significance level \(\alpha=0.05\). If the P-value is greater than \(\alpha\), we fail to reject the null hypothesis; otherwise, we reject it in favor of the nondirectional alternative hypothesis.

Make Decision for Part (b) Hypothesis Test

For the one-tailed alternative hypothesis that toluene increases dopamine concentration, use the test statistic and P-value calculated previously but consider only the direction that supports the alternative hypothesis to make the decision.

Key concepts

Nonparametric Hypothesis Testing

When it comes to comparing two groups, as in the case with toluene and control rats' dopamine levels, nonparametric hypothesis testing is a robust analytical approach. This type of testing is not based on parameterized distributions, meaning it doesn't assume a normal distribution of the data, making it particularly useful for small samples or when the underlying distribution is unknown.

Nonparametric tests, like the Wilcoxon-Mann-Whitney test used in this exercise, are also known as distribution-free tests. They operate on ranks rather than data values, thus they are less sensitive to outliers. Such methods are highly practical in biomedical research, where data may not adhere to typical distribution patterns due to small sample sizes or skewed measurements.

Dopamine Concentration Comparison

Comparing dopamine concentrations between groups can reveal the effects of a substance, like toluene, on biological processes. In nonparametric hypothesis testing, we aren't as much interested in the actual levels of dopamine but rather in the relative differences between the two groups. We wish to understand if one group tends to have higher or lower concentrations than the other, which could imply a physiological effect of the substance being tested.

In the provided exercise, researchers want to determine if toluene exposure affects dopamine levels in rats. Through this comparative analysis, scientists can gain insights into toluene's neurochemical impact, potentially extrapolating such findings to understand its effects on human health.

Rank Sum Method

The rank sum method is the cornerstone of the Wilcoxon-Mann-Whitney test. Here's how it works in simple steps: first, all observations from both groups are combined and each value is replaced with its rank within the combined set. In the case of ties, average ranks are assigned.

Once ranked, the observations are split back into their original groups and the ranks in each group are summed up – these are the rank sums. The test statistic is usually the smaller rank sum, which is then used to assess the likelihood of observing such a rank sum by chance. This method is quite powerful as it makes no assumptions about the original data distributions, making it suitable in various research scenarios, including comparing dopamine levels as in our exercise.

Statistical Significance

Statistical significance is used to determine if the observed effect is likely due to something other than random chance. In hypothesis testing, we compare a calculated p-value against a predetermined significance level, often denoted as \( \alpha \).

Commonly, \( \alpha \)=0.05 is used, which implies a 5% risk of concluding that an effect exists when there is none (Type I error). If the p-value is less than \( \alpha \), the result is considered statistically significant, and we reject the null hypothesis. In the context of the exercise, this means we would conclude that toluene has a significant effect on dopamine concentration if our p-value falls below 0.05.