Question

The paper "Predictors of Complementary Therapy Use Among Asthma Patients: Results of a Primary Care Survey" (Health and Social Care in the Community [2008]: \(155-164)\) described a study in which each person in a large sample of asthma patients responded to two questions: Question 1: Do conventional asthma medications usually help your symptoms? Question 2: Do you use complementary therapies (such as herbs, acupuncture, aroma therapy) in the treatment of your asthma? Suppose that this sample is representative of asthma patients. Consider the following events: \(E=\) event that the patient uses complementary therapies \(F=\) event that the patient reports conventional medications usually help The data from the sample were used to estimate the following probabilities: $$P(E)=0.146 \quad P(F)=0.879 \quad P(E \cap F)=0.122$$ a. Use the given probability information to set up a "hypothetical 1000 " table with columns corresponding to \(E\) and \(n o t E\) and rows corresponding to \(F\) and not \(F\). b. Use the table from Part (a) to find the following probabilities: i. The probability that an asthma patient responds that conventional medications do not help and that patient uses complementary therapies. ii. The probability that an asthma patient responds that conventional medications do not help and that patient does not use complementary therapies. iii. The probability that an asthma patient responds that conventional medications usually help or the patient uses complementary therapies. c. Are the events \(E\) and \(F\) independent? Explain.

Short answer

The constructed table with probabilities is as follows: \[F \cap E : 0.122, F \cap not E : 0.757, not F \cap E : 0.024, not F \cap not E : 0.097\]. The event that the patient uses complementary therapies \(E\) and the event that conventional medications help \(F\) are not independent.

Step-by-step solution

Setup a Hypothetical 1000 Table

We are given three probabilities, we will scale them by a factor of 1000 to create the table. First, calculate the number of patients who use complementary therapies \(E\) as \(P(E) \times 1000 = 0.146 \times 1000 = 146\). Similarly the number of patients who report conventional medications usually help \(F\) as \(P(F) \times 1000 = 0.879 \times 1000 = 879\). And the number of patients for whom both events occur \(E \cap F\) as \(P(E \cap F) \times 1000 = 0.122 \times 1000 = 122\).

Find the required probabilities

Now we can calculate the rest of the values for the table. \( E \cap not F = E - E \cap F = 146 - 122 = 24\) (these patients use complementary therapies, but don’t get help from conventional treatments), \( F \cap not E = F - E \cap F = 879 - 122 = 757\) (conventional treatments are effective, and these patients don’t use complementary treatments), \( not E \cap not F = 1000 - (E + F - E \cap F) = 1000 - (146 + 879 - 122) = 97\) (these patients don’t use complementary nor find conventional treatments effective). To find the probabilities, divide each table entries by 1000.

Check for Independence of Events

Two events \(E\) and \(F\) are said to be independent if the occurrence of \(E\) has no effect on the occurrence of \(F\) and vice versa. This is mathematically represented as \(P(E \cap F) = P(E) \cdot P(F)\). Substitute the given probabilities into the equation: \(0.122 = 0.146 \times 0.879\). Compute the right side of the equation: \(0.146 \times 0.879 = 0.128314\), which is not equal to 0.122. Therefore, the events \(E\) and \(F\) are not independent.

Key concepts

Complementary Therapies in Asthma Treatment

Asthma is a chronic respiratory condition that affects the airways and can lead to difficulty breathing. The standard approach to asthma management typically involves the use of conventional medications, such as inhalers and corticosteroids, aimed at reducing inflammation and relieving symptoms. However, a number of patients may not find complete relief from conventional treatments alone and turn to complementary therapies.

Complementary therapies in asthma treatment encompass a variety of non-conventional approaches, including herbal remedies, acupuncture, yoga, massage, and aroma therapy, among others. These therapies aim to provide relief by reducing stress, decreasing airway inflammation, and enhancing overall well-being. While there is ongoing research into the efficacy of these therapies, many patients report subjective improvement in their symptoms when using them in conjunction with conventional medication.

It must be noted, however, that while complementary therapies are used by some, they should not replace prescribed medical treatments but rather serve to complement them. Patients should always consult with healthcare professionals before incorporating any new therapy into their treatment plan, to ensure it is safe and appropriate for their specific condition.

Probability of Events

Probability is a branch of mathematics concerned with the likelihood of an event occurring. It's a fundamental concept in statistics and is critical to various types of analyses. Probability values range from 0 to 1, where 0 indicates impossibility and 1 represents certainty. The probability of an event is calculated as the number of desired outcomes divided by the number of all possible outcomes.

When examining the probability of events, for instance, researchers may investigate the likelihood that a particular treatment will benefit a patient. In the context of the asthma study, the probabilities were given for patients using complementary therapies and for those who found conventional medications helpful. These probabilities help healthcare providers and patients understand the effectiveness and usage patterns of different treatment approaches, and they also allow researchers to explore potential correlations or trends within populations.

Independence of Events

The concept of independence in probability refers to the situation where the occurrence of one event does not influence the occurrence of another. In other words, two events are independent if the probability of one event occurring has no impact on the probability of the other event occurring.

This is mathematically represented by the formula: if events A and B are independent, then the probability of both events occurring together, denoted by \(P(A \cap B)\), is equal to the product of their individual probabilities: \(P(A) \times P(B)\). Independence is a crucial concept in probability theory because it simplifies the computation of the probability of multiple events occurring together by allowing us to multiply their individual probabilities.

However, in many real-world situations, such as in the study of asthma treatments, events may not be independent. For instance, the usage of complementary therapies may be related to patients' perceptions of the effectiveness of conventional medications. This type of relationship can influence the analysis of the data and the conclusions that can be drawn about the effectiveness of treatments.