Question

A sample of 15 patients was randomly split into two groups as part of a double-blind experiment to compare two pain relievers. \({ }^{19}\) The 7 patients in the first group were given Demerol and reported the following numbers of hours of pain relief: $$ 2,6,4,13,5,8,4 $$ The 8 patients in the second group were given an experimental drug and reported the following numbers of hours of pain relief. $$ 0,8,1,4,2,2,1,3 $$ How might these data be analyzed?

Short answer

The data can be analyzed by calculating measures of central tendency (mean, median, mode) and variation (range, variance, standard deviation) for each group and comparing them. Furthermore, hypothesis testing, such as an independent t-test, could be performed to determine if there are statistically significant differences between the groups' responses to the pain relievers.

Step-by-step solution

Understanding the problem

In this study, two groups of patients are given two different pain relievers and report the number of hours they experienced pain relief. To analyze the data, statistical methods can be used to compare the central tendency, variation, and overall effectiveness of the two drugs.

Organizing the data

Firstly, organize the data into two sets, one for each group. For Demerol (Group 1), the hours of pain relief are: 2, 6, 4, 13, 5, 8, 4. For the experimental drug (Group 2), the hours are: 0, 8, 1, 4, 2, 2, 1, 3.

Calculating measures of central tendency

Calculate the mean, median, and mode for each group to summarize the central tendency of the data.

Calculating measures of variation

Calculate the range, variance, and standard deviation for each group to assess the variation in the data.

Comparing the data

Compare the calculated statistical measures between the two groups to determine if there are significant differences in the effectiveness of the pain relievers.

Conducting hypothesis testing

Perform a hypothesis test such as an independent t-test to determine if any observed differences between the two groups are statistically significant.

Interpreting the results

Interpret the results of the statistical analysis to make an informed conclusion about the comparative effectiveness of the two pain relievers.

Key concepts

Central Tendency Measures

When analyzing data from experiments like the double-blind comparison of pain relievers, central tendency measures give us a snapshot of the typical values in our data set. We usually look at three key measures: the mean (average), the median (the middle value when the data is ordered), and the mode (the most frequently occurring value).

For the pain relief data, we might calculate the mean number of hours of relief for each group. Mean is sensitive to extreme values, known as outliers, like the 13 hours of relief reported by one patient for Demerol, which can skew the results. We would also find the median, a robust central tendency measure, relatively unaffected by outliers. If we have a mode, it can indicate the most common experience among patients, but there might not always be one clear mode.

Variation Measures

Measures of variation, including the range, variance, and standard deviation, tell us about the spread of the data. The range is a quick snapshot giving the difference between the highest and lowest values. For our pain relief data, this could highlight the maximum pain relief duration difference between Demerol and the experimental drug.

The variance and standard deviation provide more detail about how much the data varies around the mean. A high variance or standard deviation indicating that the hours of pain relief are spread out over a wide range suggests less consistency in the drug's effectiveness. Knowing these measures helps to understand the reliability of the pain relievers.

Hypothesis Testing

Hypothesis testing is a crucial statistical tool used to determine the likelihood that the difference observed between groups is due to chance. We start with a null hypothesis, which suggests that there is no difference between the experimental groups—in this case, that both pain relievers provide the same amount of relief.

An alternative hypothesis is that there is a difference. Using a test statistic, we assess the probability that the observed results are compatible with the null hypothesis. If this probability (the p-value) is very low (typically, below 0.05), we reject the null hypothesis, suggesting that the drugs indeed have different effects.

Independent t-test

The independent t-test is a statistical procedure used to test whether there is a significant difference between the means of two unrelated groups. In the context of our pain relief study, we're comparing the average hours of pain relief from Demerol to that of the experimental drug.

An independent t-test will consider both the difference in the means and the spread of the scores (the standard deviations) to calculate the test statistic (t-value). If the t-value is associated with a p-value below 0.05, we might claim that the difference in pain relief duration is statistically significant, giving us more confidence to prefer one drug over the other.

Double-blind Experiment

A double-blind experiment is a method where neither the participants nor the experimenters know who is receiving a particular treatment. This design prevents bias in the treatment's administration or subjective reporting of effects, leading to more reliable results.

In our example, neither the patients nor the doctors knew who received Demerol or the experimental drug, helping to ensure that the reported hours of pain relief are influenced solely by the drug's effectiveness. This method is essential in avoiding biases that could invalidate the experiment's conclusion.