Question

Consider a study to investigate a certain suspected disease-causing agent. One thousand people are to be chosen at random from the population; each individual is to be classified as diseased or not diseased and as exposed or not exposed to the agent. The results are to be cast in the following contingency table: Let EY and EN denote exposure and nonexposure and let DY and DN denote presence and absence of the disease. Express each of the following statements in terms of conditional probabilities. (Note that "a majority" means "more than half.") (a) The disease is more common among exposed than among nonexposed people. (b) Exposure is more common among diseased people than among nondiseased people. (c) Exposure is more common among diseased people than is nonexposure. (d) A majority of diseased people are exposed. (e) A majority of exposed people are diseased. (f) Exposed people are more likely to be diseased than are nonexposed people. (g) Exposed people are more likely to be diseased than to be nondiseased

Short answer

Conditional probabilities corresponding to the statements: (a) \( P(DY | EY) > P(DY | EN) \), (b) \( P(EY | DY) > P(EY | DN) \), (c) \( P(EY | DY) > P(EN | DY) \), (d) \( P(EY | DY) > 0.5 \), (e) \( P(DY | EY) > 0.5 \), (f) \( P(DY | EY) > P(DY | EN) \), (g) \( P(DY | EY) > P(DN | EY) \).

Step-by-step solution

Restating the problem using conditional probabilities

To express the statements using conditional probabilities, we need to use the correct notation. Let's define the events: EY for exposure to the agent, EN for nonexposure to the agent, DY for having the disease, and DN for not having the disease.

Statement (a) Using Conditional Probabilities

The statement 'The disease is more common among exposed than among nonexposed people' can be expressed as the conditional probability that an individual has the disease given that they are exposed to the agent being greater than the probability that an individual has the disease given that they are not exposed: \( P(DY | EY) > P(DY | EN) \).

Statement (b) Using Conditional Probabilities

The statement 'Exposure is more common among diseased people than among nondiseased people' can be transformed into the conditional probability that an individual is exposed given that they have the disease being greater than the probability that an individual is exposed given that they do not have the disease: \( P(EY | DY) > P(EY | DN) \).

Statement (c) Using Conditional Probabilities

Expressing 'Exposure is more common among diseased people than is nonexposure', in terms of conditional probability, means the probability of exposure given disease is greater than the probability of nonexposure given disease: \( P(EY | DY) > P(EN | DY) \).

Statement (d) Using Conditional Probabilities

To say 'A majority of diseased people are exposed' can be stated as the conditional probability that more than half of the diseased people are exposed: \( P(EY | DY) > 0.5 \).

Statement (e) Using Conditional Probabilities

The statement 'A majority of exposed people are diseased' would be restated as the conditional probability that more than half of the exposed people have the disease: \( P(DY | EY) > 0.5 \).

Statement (f) Using Conditional Probabilities

Expressing 'Exposed people are more likely to be diseased than are nonexposed people' with conditional probability, we get that the probability of disease given exposure is greater than the probability of disease given nonexposure: \( P(DY | EY) > P(DY | EN) \).

Statement (g) Using Conditional Probabilities

Finally, the statement 'Exposed people are more likely to be diseased than to be nondiseased' can be restated as the conditional probability of being diseased given exposure is greater than being nondiseased given exposure: \( P(DY | EY) > P(DN | EY) \).

Key concepts

Contingency Table

In probability theory and statistics, a contingency table is a type of table that displays the frequency distribution of certain variables. It provides a way of displaying multivariate categorical data in a grid format that can be used to study the relationships between different variables. For example, in an epidemiological study, it might show the presence (or absence) of a disease in relation to exposure (or non-exposure) to a potential risk factor.

In the case of the exercise, the contingency table is used to categorize people based on two variables: their disease status (diseased or not diseased) and their exposure status to a certain agent (exposed or not exposed). This table becomes essential for calculating various conditional probabilities, enabling researchers to determine if there is a significant association between exposure and the disease.

Probability Theory

Probability theory is the branch of mathematics that deals with quantifying the likelihood of events. It forms the foundation for statistical inference, allowing us to make predictions and decisions even when faced with uncertainty. Conditional probability, a key concept in probability theory, measures the probability of an event occurring given that another event has already occurred.

This concept is central to analyzing the textbook exercise. Conditional probabilities such as P(DY | EY), P(EY | DY), and others compare the likelihoods of disease presence based on exposure status or vice versa. Understanding these probabilities is fundamental to interpreting results from the contingency table and making inferences about the relationship between disease and exposure.

Epidemiological Study Design

Epidemiological study design refers to the methods and strategies used in research to investigate the patterns, causes, and effects of health and disease conditions in defined populations. The study mentioned in the exercise likely follows a cross-sectional design, which observes a population at a single point in time or over a short period.

Designing the study to systematically classify participants based on disease and exposure ensures that the resulting data can provide insights into potential relationships. Conditional probabilities derived from the data help to clarify if exposure is associated with increased disease occurrence, thus informing epidemiological study design decisions in future research.

Statistical Inference

Lastly, statistical inference enables us to make conclusions about a population based on sample data. This process uses techniques from probability theory, such as calculating conditional probabilities, to draw conclusions and make predictions. In the exercise's context, statistical inference may involve using the calculated conditional probabilities to decide whether there is evidence to support the suspicion that the agent in question causes the disease.

By comparing the probabilities (for example, P(DY | EY) greater than P(DY | EN)), researchers can infer whether disease is indeed more common among those exposed to the agent. Inferences should always be made carefully, taking into account possible confounding variables and the inherent limitations of sample data.