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Controlling asthma A study of 13 children suffering from asthma (Clinical and Experimental Allergy, vol. 20 , pp. \(429-432,1990\) ) compared single inhaled doses of formoterol (F) and salbutamol (S). Each child was evaluated using both medications. The outcome measured was the child's peak expiratory flow (PEF) eight hours following treament. Is there a difference in the PEF level for the two medications? The data on PEF follow: $$ \begin{array}{ccc} \hline \text { Child } & \text { F } & \text { S } \\ \hline 1 & 310 & 270 \\ 2 & 385 & 370 \\ 3 & 400 & 310 \\ 4 & 310 & 260 \\ 5 & 410 & 380 \\ 6 & 370 & 300 \\ 7 & 410 & 390 \\ 8 & 320 & 290 \\ 9 & 330 & 365 \\ 10 & 250 & 210 \\ 11 & 380 & 350 \\ 12 & 340 & 260 \\ 13 & 220 & 90 \\ \hline \end{array} $$ a. Construct plots to compare formoterol and salbutamol. Write a short summary comparing the two distributions of the peak expiratory flow. b. Consider the distribution of differences between the PEF levels of the two medications. Find the 13 differences and construct and interpret a plot of the differences. If on the average there is no difference between the PEF level for the two brands, where would you expect the differences to be centered?

Step by step solution

01

Organize the Data for Each Drug

We have 13 pairs of PEF measurements for the formoterol (F) and salbutamol (S) treatments. Begin by listing all the PEF values for each treatment side by side:\[\begin{aligned}F &: 310, 385, 400, 310, 410, 370, 410, 320, 330, 250, 380, 340, 220 \S &: 270, 370, 310, 260, 380, 300, 390, 290, 365, 210, 350, 260, 90\end{aligned}\]

02

Create Comparative Box Plots

Using the data from Step 1, create side-by-side box plots for the PEF values of formoterol and salbutamol. This visual representation helps compare the distributions in terms of spread, central tendency, and potential outliers. Look for differences in medians, ranges, and shapes to summarize their distributions.

03

Calculate Differences for Each Child

For each child, calculate the difference in PEF values as \( D_i = F_i - S_i \). Compute the differences:\[\begin{aligned}D_1 &= 310 - 270 = 40, & D_2 &= 385 - 370 = 15, \D_3 &= 400 - 310 = 90, & D_4 &= 310 - 260 = 50, \D_5 &= 410 - 380 = 30, & D_6 &= 370 - 300 = 70, \D_7 &= 410 - 390 = 20, & D_8 &= 320 - 290 = 30, \D_9 &= 330 - 365 = -35, & D_{10} &= 250 - 210 = 40, \D_{11} &= 380 - 350 = 30, & D_{12} &= 340 - 260 = 80, \D_{13} &= 220 - 90 = 130\end{aligned}\]

04

Create a Plot of the Differences

Construct a plot such as a dot plot or a histogram for the differences calculated in Step 3. This will help visualize the distribution of differences between the two medications. Pay attention to whether the plot is centered around zero, as this would suggest no average difference between the two medications.

05

Interpret the Differences Plot

By examining the plot, determine the central tendency, usually the mean or median, of the differences. If there is no average difference, the plot of differences should be approximately centered at zero. Analyze whether the data support a conclusion that one treatment is generally more effective based on the center and spread of differences.

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Key Concepts

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

Asthma Treatment Analysis

Asthma is a chronic respiratory condition that affects millions of individuals, notably children. The objective in treating asthma is to alleviate symptoms and enhance quality of life by utilizing compounds like bronchodilators.
In comparative studies such as the one at hand, different treatments are evaluated to identify which one provides better outcomes.

• Formoterol (F) and salbutamol (S) are common bronchodilators used to treat asthma.
• Such studies help determine which drug is more effective in terms of peak expiratory flow (PEF), an important measurement in respiratory medicine.

By analyzing these medications, researchers can conclude which treatment provides more respiratory relief.
This analysis helps to make informed medical decisions, ensuring optimal patient care.

Peak Expiratory Flow

Peak Expiratory Flow (PEF) is a critical measure in respiratory health, particularly when assessing asthma treatment effectiveness. PEF measures how quickly a person can exhale air, providing insights into lung function.
In asthma, the PEF can indicate how the airways are responding to treatment.

• A higher PEF suggests better airway clearance and less bronchoconstriction.
• Conversely, a lower PEF may indicate that the airways are still constricted, suggesting less effective treatment.

Regular monitoring of PEF helps track how well asthma treatments are working and aids in making necessary adjustments to treatment plans.
This information is extremely valuable in managing long-term asthma effectively.

Box Plot Analysis

A box plot, or box-and-whisker plot, is a powerful visual tool in statistics. It is used to display the spread and skewness of data in asthma treatment studies, such as comparing the effectiveness of formoterol and salbutamol.
The box plot helps visualize several key aspects:

• Central tendency through the median line within the box.
• Spread or variability, indicated by the range between the lower and upper quartiles (the box itself).
• Potential outliers, shown as individual points outside the plot's whiskers.

In this context, a box plot can help determine if one medication leads to consistently higher PEF readings than another.
Such visual aids simplify complex data, allowing for an intuitive comparison and understanding of distributions.

Difference Distribution Visualization

Visualizing the differences in peak expiratory flow (PEF) between two asthma treatments can offer profound insights into their effectiveness. This involves calculating the difference between the PEF values for the formoterol and salbutamol treatments for each child.

• A plot such as a dot plot or a histogram can effectively display these differences.
• If the plot center aligns around zero, it suggests no significant average difference between the two treatments.

Such visualization shows the distribution of effectiveness, indicating which treatment may offer better respiratory function improvement.
This method provides a straightforward display of data, supporting easier interpretation and aiding in decision-making processes regarding treatment efficacy.