Question

A source at location \(x=0\) pollutes the environment. Are cases of a rare disease \(\mathcal{D}\) later observed at positions \(x_{1}, \ldots, x_{n}\) linked to the source? Cases of another rare disease \(\mathcal{D}^{\prime}\) known to be unrelated to the pollutant but with the same susceptible population as \(\mathcal{D}\) are observed at \(x_{1}^{\prime}, \ldots, x_{m}^{\prime} .\) If the probabilities of contracting \(\mathcal{D}\) and \(\mathcal{D}^{\prime}\) are respectively \(\psi(x)\) and \(\psi^{\prime}\), and the population of susceptible individuals has density \(\lambda(x)\), show that the probability of \(\mathcal{D}\) at \(x\), given that \(\mathcal{D}\) or \(\mathcal{D}^{\prime}\) occurs there, is $$ \pi(x)=\frac{\psi(x) \lambda(x)}{\psi(x) \lambda(x)+\psi^{\prime} \lambda(x)} $$ Deduce that the probability of the observed configuration of diseased persons, conditional on their positions, is $$ \prod_{j=1}^{n} \pi\left(x_{j}\right) \prod_{i=1}^{m}\left\\{1-\pi\left(x_{i}^{\prime}\right)\right\\} $$ The null hypothesis that \(\mathcal{D}\) is unrelated to the pollutant asserts that \(\psi(x)\) is independent of \(x\). Show that in this case the unknown parameters may be eliminated by conditioning on having observed \(n\) cases of \(\mathcal{D}\) out of a total \(n+m\) cases. Deduce that the null probability of the observed pattern is \(\left({ }_{n}^{n+m}\right)^{-1}\). If \(T\) is a statistic designed to detect decline of \(\psi(x)\) with \(x\), explain how permutation of case labels \(\mathcal{D}, \mathcal{D}^{\prime}\) may be used to obtain a significance level \(p_{\text {obs }}\). Such a test is typically only conducted after a suspicious pattern of cases of \(\mathcal{D}\) has been observed. How will this influence \(p_{\text {obs }}\) ?

Short answer

The null probability of the pattern is \( \left({ }_{n}^{n+m}\right)^{-1} \). Post-hoc analysis may lower \( p_{\text{obs}} \), indicating bias.

Step-by-step solution

Identify Components of the Problem

We need to find the probability of occurrence of disease \( \mathcal{D} \) at position \( x \), given that either \( \mathcal{D} \) or \( \mathcal{D}^{\prime} \) happens there. The density of people susceptible to these diseases is \( \lambda(x) \), and the probabilities of contracting \( \mathcal{D} \) and \( \mathcal{D}^{\prime} \) are \( \psi(x) \) and \( \psi^{\prime} \), respectively.

Calculate Probability \( \pi(x) \)

The probability of \( \mathcal{D} \) occurring, given that either \( \mathcal{D} \) or \( \mathcal{D}^{\prime} \) occurs, is the ratio of the probability of \( \mathcal{D} \) at \( x \) to the total probability of either disease at \( x \). This gives:\[\pi(x) = \frac{\psi(x) \lambda(x)}{\psi(x) \lambda(x) + \psi^{\prime} \lambda(x)} = \frac{\psi(x)}{\psi(x) + \psi^{\prime}}\]

Probability of the Observed Configuration

The probability of observing \( n \) cases of \( \mathcal{D} \) at positions \( x_1, ..., x_n \) and \( m \) cases of \( \mathcal{D}^{\prime} \) at positions \( x_1^{\prime}, ..., x_m^{\prime} \) is:\[\prod_{j=1}^{n} \pi(x_j) \prod_{i=1}^{m} \{ 1 - \pi(x_i^{\prime}) \}\]

Null Hypothesis and Conditional Probability

Under the null hypothesis, \( \psi(x) \) is independent of \( x \). Thus, the ratio \( \pi(x) \) becomes constant, and conditioning on exactly \( n \) out of \( n+m \) cases being \( \mathcal{D} \) leads to the binomial probability which simplifies to \( \left( \binom{n}{n+m} \right)^{-1} \), eliminating other parameters.

Permutation Test and Significance Level

Permuting labels \( \mathcal{D} \) and \( \mathcal{D}^{\prime} \) allows assessing whether observed cases are unusual under the null hypothesis, by comparing \( T \) (designed to detect \( \psi(x) \)'s decline with \( x \)) with its distribution. If \( \,p_{\text{obs}} \, \) represents a test conducted post-observation, it may be biased or lower as the pattern that instigated the test increases perceived evidence against the null.

Key concepts

Environmental Statistics

Environmental statistics involves collecting, analyzing, and interpreting data about environmental conditions and impacts. In the exercise above, the task is to determine whether cases of a rare disease, m\mathcal{D}, are linked to a pollution source at location \(x=0\). This requires understanding the distribution of disease cases across a geographical area.
Using environmental statistics, you can analyze whether any observed disease pattern indicates a potential environmental hazard, such as pollution. This might involve collecting data on disease incidences, assessing the spatial distribution of cases, and evaluating potential correlations with environmental factors like local pollution sources. By analyzing this information, conclusions can be made about whether environmental factors might be driving disease patterns.
The key objectives in environmental statistics involve risk assessment and decision-making to mitigate negative environmental impacts on human health. With statistical tools, scientists can make informed decisions and devise strategies to reduce the risks associated with environmental hazards.

Probability Distributions

Probability distributions are essential in statistical hypothesis testing, as they describe the likelihood of different outcomes in a random experiment. In the given scenario with diseases \(\mathcal{D}\) and \(\mathcal{D}'\), we encounter the probability distributions \(\psi(x)\) and \(\psi'(x)\) which represent the likelihood of disease occurrence at position \(x\).
A probability distribution provides a comprehensive picture of all possible outcomes and their associated probabilities. For example, \(\psi(x)\) indicates how likely it is that disease \(\mathcal{D}\) occurs at a specific location \(x\). Similarly, \(\lambda(x)\) informs about the density of susceptible individuals in that region. These distributions help quantify the risk of disease transmission across different locations.
Understanding these probability distributions is critical for deducing patterns and assessing whether the disease occurrence is truly random or influenced by an environmental factor. By comparing \(\psi(x)\) and \(\psi'(x)\), one can examine if exposure to a pollutant affects disease rates.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. In our example, to assess whether disease \(\mathcal{D}\) at position \(x\) is due to environmental factors, we examine the probability that \(\mathcal{D}\) occurs given that either \(\mathcal{D}\) or \(\mathcal{D}'\) occurs.
The formula for calculating conditional probability \(\pi(x)\) is given by:
\[ \pi(x) = \frac{\psi(x) \lambda(x)}{\psi(x) \lambda(x) + \psi'(x) \lambda(x)} \]
This equation shows how to calculate the likelihood of \(\mathcal{D}\) occurring at \(x\) when juxtaposed with both diseases occurring. By separating the probabilities of \(\mathcal{D}\) and \(\mathcal{D}'\), it reveals conditional relationships influenced by the presence of other diseases. Through such calculations, it's possible to discern whether disease incidences are independent of environmental factors or if they escalate due to localized pollution.
Conditional probability is a robust statistical method, allowing researchers to dissect complex dependencies and make informed assessments in hypothesis tests.

Permutation Tests

Permutation tests are non-parametric statistical tests used to determine the significance of the observed effects in a dataset. This approach is useful when trying to verify the null hypothesis that a disease is not related to a pollutant.
In this exercise, permutation tests help determine whether the pattern of diseases \(\mathcal{D}\) and \(\mathcal{D}'\) corresponds to randomness or some underlying influencing factor. By permuting the labels of the cases for \(\mathcal{D}\) and \(\mathcal{D}'\), the goal is to see if the observed pattern under the null hypothesis is likely to occur by chance.
If the calculated significance level \(p_{\text{obs}}\) from permutations is notably low, it suggests that the observed pattern is unlikely by chance, thus indicating potential environmental influence. However, since the test is often executed after noticing a suspicious pattern indicative of environmental impact, \(p_{\text{obs}}\) might be bias-prone, possibly yielding lower significance values due to its reactive nature. Permutation tests thereby offer a robust way to investigate statistical significance without heavy reliance on assumptions about underlying distributions.