/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 For a \(2 \times 2\) contingency... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 3

For a \(2 \times 2\) contingency table with probabilities $$ \begin{array}{cc} \pi_{00} & \pi_{01} \\ \pi_{10} & \pi_{11} \end{array} $$ the maximal log-linear model may be written as $$ \begin{array}{ll} \eta_{00}=\alpha+\beta+\gamma+(\beta \gamma), & \eta_{01}=\alpha+\beta-\gamma-(\beta \gamma) \\ \eta_{10}=\alpha-\beta+\gamma-(\beta \gamma), & \eta_{11}=\alpha-\beta-\gamma+(\beta \gamma) \end{array} $$ where \(\eta_{j k}=\log \mathrm{E}\left(Y_{j k}\right)=\log \left(m \pi_{j k}\right)\) and \(m=\sum_{j, k} y_{j k} .\) Show that the 'interaction'term \((\beta \gamma)\) may be written \((\beta \gamma)=\frac{1}{4} \log \Delta\), where \(\Delta\) is the odds ratio \(\left(\pi_{00} \pi_{11}\right) /\left(\pi_{01} \pi_{10}\right)\), so that \((\beta \gamma)=0\) is equivalent to \(\Delta=1\)

Short Answer

Expert verified
\((\beta \gamma) = \frac{1}{4} \log \Delta\) shows interaction equals zero if \(\Delta=1\).

Step by step solution

01

Define the Odds Ratio

The odds ratio \( \Delta \) for a \( 2 \times 2 \) table is defined as \( \Delta = \frac{\pi_{00} \cdot \pi_{11}}{\pi_{01} \cdot \pi_{10}} \). This ratio compares the odds of one event occurring to the odds of another event.
02

Define Eta Terms

The maximal log-linear model is represented by \[ \begin{align*} \eta_{00} &= \alpha + \beta + \gamma + (\beta \gamma), \ \eta_{01} &= \alpha + \beta - \gamma - (\beta \gamma), \ \eta_{10} &= \alpha - \beta + \gamma - (\beta \gamma), \ \eta_{11} &= \alpha - \beta - \gamma + (\beta \gamma). \end{align*} \] These equations relate the log of expected counts to various effects and interactions.
03

Establish Relationships Between Eta and Delta

Using the expression for \( \eta \), establish the relationship in terms of the odds ratio. Substitute \( \eta_{jk} = \log(m \pi_{jk}) \) into each equation and express the ratios \( \eta_{00} + \eta_{11} \) and \( \eta_{01} + \eta_{10} \).
04

Express the Interaction Term in terms of Delta

Calculate \[ \begin{align*} \eta_{00} + \eta_{11} &= 2\alpha + 2(\beta \gamma), \ \eta_{01} + \eta_{10} &= 2\alpha - 2(\beta \gamma). \end{align*} \] With the assumption that \( \Delta = \frac{\pi_{00} \pi_{11}}{\pi_{01} \pi_{10}} \), take the ratio: \[ \eta_{00} + \eta_{11} - \eta_{01} - \eta_{10} = 4(\beta \gamma) = \log \Delta. \] Thus, \( (\beta \gamma) = \frac{1}{4} \log \Delta \).
05

Verify the Equivalence of the Interaction to Independence

When \( (\beta \gamma) = 0 \), then \( \frac{1}{4} \log \Delta = 0 \), which implies \( \log \Delta = 0 \). This simplifies to \( \Delta = 1 \), showing the relationship that interaction being zero is equivalent to the odds ratio being 1, indicating independence.

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Key Concepts

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

Odds Ratio
In statistics, the odds ratio is a measure of association between two binary variables, often used to summarize the relationship in a contingency table. For a basic understanding, consider a scenario like a clinical trial where we compare the odds of an outcome in two different groups. The odds ratio provides a way to compare whether the probability of a certain event is the same or different between these two groups.

In a simple \(2 \times 2\) contingency table with cell probabilities \( \pi_{00}, \pi_{01}, \pi_{10}, \pi_{11} \), the odds ratio \( \Delta \) is computed as:
  • \( \Delta = \frac{\pi_{00} \cdot \pi_{11}}{\pi_{01} \cdot \pi_{10}} \)
This equation calculates the odds of the outcome for one condition over another based on the table鈥檚 layout.

The importance of the odds ratio is highlighted when \( \Delta = 1 \). This particular result indicates that the odds of the event occurring is the same across groups, implying no effect or association between the variables.
Log-linear Models
Log-linear models are statistical tools used to examine the relationships between categorical variables by analyzing the cell frequencies in contingency tables. They are especially powerful because they allow for the modeling of all possible interactions between variables.

For a given \(2 \times 2\) table, the fitted counts are expressed using logarithms for transformation:
  • \( \eta_{jk} = \log(\mathrm{E}(Y_{jk})) = \log(m \pi_{jk}) \)
By using log-linear models, we can understand complex relationships as they factor in various effects and interactions among variables such as main effects of each variable and their interactions, denoted as \(\alpha, \beta, \gamma, \) and \((\beta \gamma)\).

These parameters collectively define how the expected counts in a contingency table are determined. When the interaction term \((\beta \gamma)\) is zero, it implies that the variables are statistically independent, which is a key point of analysis.
Statistical Independence
In the context of statistical analysis, two categorical variables are considered statistically independent if the occurrence of one does not affect the probability of occurrence of the other. This is a fundamental concept when analyzing contingency tables.

In a \(2 \times 2\) contingency table, statistical independence is reflected in the odds ratio, \(\Delta\), being equal to 1. This implies that the ratio of the cell probabilities maintains a perfect balance such that:
  • \( \Delta = 1 \Longrightarrow \frac{\pi_{00} \cdot \pi_{11}}{\pi_{01} \cdot \pi_{10}} = 1 \)
  • Which leads to \((\beta \gamma) = 0\)
The idea is simple but profound: if the interaction term, \((\beta \gamma)\), is zero, it confirms that there is no additional interaction effect beyond what is expected under independence. Therefore, statistically speaking, changes in one variable do not predict changes in the other, which can be crucial for drawing valid conclusions from data.

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Most popular questions from this chapter

Let \(Y_{1}, \ldots, Y_{n}\) be independent exponential variables with hazards \(\lambda_{j}=\exp \left(\beta^{\mathrm{T}} x_{j}\right)\). (a) Show that the expected information for \(\beta\) is \(X^{\mathrm{T}} X\), in the usual notation. (b) Now suppose that \(Y_{j}\) is subject to uninformative right censoring at time \(c_{j}\), so that \(y_{j}\) is a censoring time or a failure time as the case may be. Show that the log likelihood is $$ \ell_{U}(\beta)=\sum_{f} \beta^{\mathrm{T}} x_{j}-\sum_{j=1}^{n} \exp \left(\beta^{\mathrm{T}} x_{j}\right) y_{j} $$ where \(\sum_{f}\) denotes a sum over observations seen to fail. If the \(j\) th censoring-time is exponentially distributed with rate \(\kappa_{j}\), show that the expected information for \(\beta\) is \(X^{\mathrm{T}} X-\) \(X^{\mathrm{T}} C X\), where \(C=\operatorname{diag}\left\\{c_{1}, \ldots, c_{n}\right\\}\), and \(c_{j}=\kappa_{j} /\left(\kappa_{j}+\lambda_{j}\right)\) is the probability that the \(j\) th observation is censored. What is the implication for estimation of \(\beta\) if the \(c_{j}\) are constant? (c) Sometimes a variable \(W_{j}\) has been measured which can act as a surrogate response variable for censored individuals. We formulate this as \(W_{j}=Z_{j} / U_{j}\), where \(Z_{j}\) is the unobserved remaining life-time of the \(j\) th individual from the moment of censoring, and \(U_{j}\) is a noise component which has a fixed distribution independent of the censoring time and of \(x_{j}\). Owing to the exponential assumption, the excess life \(Z_{j}\) is independent of \(Y_{j}\) if censoring occurred. If \(U_{j}\) has gamma density $$ \alpha^{K} u^{\kappa-1} \exp (-\alpha u) / \Gamma(\kappa), \quad \alpha, \kappa>0, u>0 $$ show that \(W_{j}\) has density $$ \lambda_{j} \kappa \alpha^{\kappa} /\left(\alpha+\lambda_{j} w\right)^{\kappa+1}, \quad w>0 $$ Show that the log likelihood for the data, including the additional information in the \(W_{j}\), is $$ \ell(\beta)=L_{U}(\beta)+\sum_{c}\left\\{\beta^{\mathrm{T}} x_{j}+\log \kappa+\kappa \log \alpha-(\kappa+1) \log \left(\alpha+e^{\beta^{\mathrm{T}}} x_{j} w_{j}\right)\right\\} $$ where \(\sum_{c}\) denotes a sum over censored individuals, and we have assumed that \(\alpha\) and \(\kappa\) are known. Show that the expected information for \(\beta\) is $$ X^{\mathrm{T}} X-2 /(\kappa+2) X^{\mathrm{T}} C X $$ and compare this with (b). Explain qualitatively in terms of the variability of the distribution of \(U\) why the loss of information decreases as \(\kappa\) increases. \((\operatorname{Cox}, 1983)\)

Suppose that \(y\) is the number of events in a Poisson process of rate \(\lambda\) observed for a period of length \(T\). Show that \(y\) has a generalized linear model density and give \(\theta, b(\theta), \phi\) and \(c(y ; \phi)\)

Let \(Y\) be binomial with probability \(\pi=e^{\lambda} /\left(1+e^{\lambda}\right)\) and denominator \(m\). (a) Show that \(m-Y\) is binomial with \(\lambda^{\prime}=-\lambda\). Consider $$ \tilde{\lambda}=\log \left(\frac{Y+c_{1}}{m-Y+c_{2}}\right) $$ as an estimator of \(\lambda\). Show that in order to achieve consistency under the transformation \(Y \rightarrow m-Y\), we must have \(c_{1}=c_{2}\) (b) Write \(Y=m \pi+\sqrt{m \pi(1-\pi)} Z\), where \(Z=O_{p}(1)\) for large \(m\). Show that $$ \mathrm{E}\\{\log (Y+c)\\}=\log (m \pi)+\frac{c}{m \pi}-\frac{1-\pi}{2 m \pi}+O\left(m^{-3 / 2}\right) $$ Find the corresponding expansion for \(\mathrm{E}\\{\log (m-Y+c)\\}\), and with \(c_{1}=c_{2}=c\) find the value of \(c\) for which \(\tilde{\lambda}\) is unbiased for \(\lambda\) to order \(m^{-1}\). What is the connection to the empirical logistic transform? (Cox, 1970, Section 3.2)

One standard model for over-dispersed binomial data assumes that \(R\) is binomial with denominator \(m\) and probability \(\pi\), where \(\pi\) has the beta density $$ f(\pi ; a, b)=\frac{\Gamma(a+b)}{\Gamma(a) \Gamma(b)} \pi^{a-1}(1-\pi)^{b-1}, \quad 0<\pi<1, a, b>0 $$ (a) Show that this yields the beta-binomial density $$ \operatorname{Pr}(R=r ; a, b)=\frac{\Gamma(m+1) \Gamma(r+a) \Gamma(m-r+b) \Gamma(a+b)}{\Gamma(r+1) \Gamma(m-r+1) \Gamma(a) \Gamma(b) \Gamma(m+a+b)}, \quad r=0, \ldots, m $$ (b) Let \(\mu\) and \(\sigma^{2}\) denote the mean and variance of \(\pi .\) Show that in general, $$ \mathrm{E}(R)=m \mu, \quad \operatorname{var}(R)=m \mu(1-\mu)+m(m-1) \sigma^{2} $$ and that the beta density has \(\mu=a /(a+b)\) and \(s^{2}=a b /\\{(a+b)(a+b+1)\\} .\) Deduce that the beta-binomial density has mean and variance $$ \mathrm{E}(R)=m a /(a+b), \quad \operatorname{var}(R)=m \mu(1-\mu)\\{1+(m-1) \delta\\}, \quad \delta=(a+b+1)^{-1} $$ Hence re-express \(\operatorname{Pr}(R=r ; a, b)\) as a function of \(\mu\) and \(\delta .\) What is the condition for uniform overdispersion?

Show that if \(Y\) is continuous with cumulative hazard function \(H(y)\), then \(H(Y)\) has the unit exponential distribution. Hence establish that \(\mathrm{E}\\{H(Y) \mid Y>c\\}=1+H(c)\), and explain the reasoning behind (10.55).
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