Problem 5 - An Introduction to Generalized Linear Models

Question

Let \(Y_{i}\) be the number of successes in \(n_{i}\) trials with $$Y_{i} \sim \operatorname{Bin}\left(n_{i}, \pi_{i}\right)$$ where the probabilities \(\pi_{i}\) have a Beta distribution $$\pi_{i} \sim \operatorname{Be}(\alpha, \beta)$$ The probability density function for the beta distribution is \(f(x ; \alpha, \beta)=\) \(x^{\alpha-1}(1-x)^{(\beta-1)} / B(\alpha, \beta)\) for \(x\) in \([0,1], \alpha > 0, \beta > 0\) and the beta function \(B(\alpha, \beta)\) defining the normalizing constant required to ensure that \(\int_{0}^{1} f(x ; \alpha, \beta) d x=1 .\) It can be shown that \(\mathrm{E}(X)=\alpha /(\alpha+\beta)\) and \(\operatorname{var}(X)=\) \(\alpha \beta /\left[(\alpha+\beta)^{2}(\alpha+\beta+1)\right] .\) Let \(\theta=\alpha /(\alpha+\beta),\) and hence, show that (a) \(\mathrm{E}\left(\pi_{i}\right)=\theta\) (b) \(\operatorname{var}\left(\pi_{i}\right)=\theta(1-\theta) /(\alpha+\beta+1)=\phi \theta(1-\theta)\) \((\mathrm{c}) \mathrm{E}\left(Y_{i}\right)=n_{i} \theta\) (d) \(\operatorname{var}\left(Y_{i}\right)=n_{i} \theta(1-\theta)\left[1+\left(n_{i}-1\right) \phi\right]\) so that \(\operatorname{var}\left(Y_{i}\right)\) is larger than the Binomial variance (unless \(n_{i}=1\) or \(\phi=0\) ).

Step-by-Step Solution

Answer

(a) \(\mathbb{E}(\pi_i) = \theta\), (b) \(\text{Var}(\pi_i) = \phi \theta (1 - \theta)\), (c) \(\mathbb{E}(Y_i) = n_i \theta\), (d) \(\text{Var}(Y_i) = n_i \theta (1 - \theta)[1 + (n_i - 1)\phi]\).

Step 1: Understand the Beta Distribution

The Beta distribution for \(\pi_i\) is given by \(\pi_i \sim \text{Be}(\alpha, \beta)\). The expected value of a Beta distribution is \(\mathbb{E}(X) = \frac{\alpha}{\alpha + \beta}\). Identify that \(\theta = \frac{\alpha}{\alpha + \beta}\) and hence, \(\mathbb{E}(\pi_i) = \theta\). This proves part (a).

Step 2: Compute Variance of Beta Distribution

The variance of a Beta distribution is given by \(\text{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\). Substituting \(\theta = \frac{\alpha}{\alpha + \beta}\), we have \(\text{Var}(\pi_i) = \theta(1-\theta)/(\alpha+\beta+1)\). Define \(\phi = \frac{1}{\alpha+\beta+1}\), thus \(\text{Var}(\pi_i) = \phi \theta (1 - \theta)\). This completes part (b).

Step 3: Determine Expectation of Binomial \(Y_i\)

Given \(Y_i \sim \text{Bin}(n_i, \pi_i)\), the expected value is \(\mathbb{E}(Y_i) = n_i \mathbb{E}(\pi_i)\). Since \(\mathbb{E}(\pi_i) = \theta\), we get \(\mathbb{E}(Y_i) = n_i \theta\). Thus, part (c) is shown.

Step 4: Derive Variance of \(Y_i\)

For the variance of \(Y_i\), use \(\text{Var}(Y_i) = n_i \theta (1-\theta) + n_i^2 \text{Var}(\pi_i)\). From part (b), substitute \(\text{Var}(\pi_i) = \phi \theta (1 - \theta)\), leading to \(\text{Var}(Y_i) = n_i \theta (1 - \theta)[1 + (n_i - 1)\phi]\). This result shows that \(\text{Var}(Y_i)\) is typically larger than the variance of a binomial random variable, unless \(n_i = 1\) or \(\phi = 0\), proving part (d).

Key Concepts

Beta Distribution

The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. It is parameterized by two positive shape parameters, \(\alpha\) and \(\beta\). The distribution is very flexible because it can take on different shapes depending on these parameters. The probability density function (PDF) for the Beta distribution is given by:

• \(f(x; \alpha, \beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)}\)

Here, \(B(\alpha, \beta)\) is the Beta function, which acts as a normalizing constant to ensure that the total area under the PDF is 1.

The Beta distribution is often used as a prior distribution in Bayesian statistics, especially for variables such as probabilities or proportions because they naturally lie in the range [0, 1].

It is crucial in various fields like Bayesian inference due to its ability to model distribution of probabilities.

Binomial Distribution

The Binomial distribution is a discrete probability distribution of the number of successes in a sequence of \(n\) independent experiments, each asking a yes-no question, and each with its own Boolean-valued outcome. The parameters of the Binomial distribution are:

• \(n\) – total number of trials
• \(\pi\) – probability of success on each trial

In mathematical form, if a random variable \(Y_i\) follows a Binomial distribution, it is written as \(Y_i \sim \text{Bin}(n_i, \pi_i)\).

The Binomial distribution is fundamental when dealing with experiments having binary outcomes, making it extremely useful in fields like quality control, survey analysis, and general statistics.

Binomial random variables help us understand scenarios where different outcomes might depend on repeated and independent occurrences of the same event.

Variance

Variance measures how far a set of numbers are spread out from their average value. In probability and statistics, it represents how much a random variable can differ from its expected value, offering insight into how spread out the outcomes of a probability distribution are.

For the Beta distribution, the variance is given by:

• \(\text{Var}(X) = \frac{\alpha \beta}{(\alpha+\beta)^2 (\alpha+\beta+1)}\)

Using \(\theta = \frac{\alpha}{\alpha + \beta}\), we can express the variance of \(\pi_i\) as:

• \(\text{Var}(\pi_i) = \theta (1 - \theta) / (\alpha + \beta + 1)\)

Variance provides important indicators of variability within datasets. It can show whether data values are always close to the mean or widely spread out.

Expected Value

Expected value, or mean, is the anticipated value of a random variable. In simplest terms, it provides the average outcome of a random variable if the experiment is conducted a large number of times. For any probability distribution, it is calculated as the sum of all possible values each multiplied by their respective probabilities.

For the Beta distribution's random variable \(\pi_i\), the expected value is given by:

• \(\text{E}(\pi_i) = \theta = \frac{\alpha}{\alpha + \beta}\)

In the case of a Binomial distribution, the expected value can be

\(\text{E}(Y_i) = n_i \theta\)

Expected values provide a central or typical value for a probability distribution and are a key concept in fields like decision theory and economics, where they help in calculating the 'average' anticipated outcome.