Question

Suppose that \(Y\) has a density with generalized linear model form $$ f(y ; \theta, \phi)=\exp \left\\{\frac{y \theta-b(\theta)}{a(\phi)}+c(y ; \phi)\right\\} $$ where \(\theta=\theta(\eta)\) and \(\eta=\beta^{\mathrm{T}} x\). (a) Show that the weight for iterative weighted least squares based on expected information is $$ w=b^{\prime \prime}(\theta)(d \theta / d \eta)^{2} / a(\phi) $$ and deduce that \(w^{-1}=V(\mu) a(\phi)\\{d g(\mu) / d \mu\\}^{2}\), where \(V(\mu)\) is the variance function, and that the adjusted dependent variable is \(\eta+(y-\mu) d g(\mu) / d \mu\). Note that initial values are not required for \(\beta\), since \(w\) and \(z\) can be determined in terms of \(\eta\) and \(\mu\); initial values can be found from \(y\) as \(\mu^{1}=y\) and \(\eta^{1}=g(y)\). (b) Give explicit formulae for the weight and adjusted dependent variable when \(R=m Y\) is binomial with denominator \(m\) and probability \(\pi=e^{\eta} /\left(1+e^{\eta}\right)\).

Short answer

The weight \(w\) for IWLS is derived from the variance function and transformations between \(\theta\) and \(\eta\). For a binomial case, \(w = \pi (1 - \pi) m\) and the adjusted variable \(z = \eta + (y - \mu) \frac{1}{\mu(1 - \mu)}\).

Step-by-step solution

Understanding the given density function

The given density function is expressed in a generalized linear model form: \[f(y ; \theta, \phi) = \exp \left\{\frac{y \theta - b(\theta)}{a(\phi)} + c(y; \phi)\right\}\]This form highlights the roles of \(\theta\) and \(\phi\), where \(\theta\) is linked to the linear predictor \(\eta\) through \(\theta = \theta(\eta)\) and \(\eta = \beta^{\mathrm{T}} x\).

Calculating the weight for IWLS

The weight \(w\) in iterative weighted least squares (IWLS) is based on expected information and is derived from the second derivative of the log-likelihood with respect to \(\theta\). The formula is:\[w = b''(\theta)\left(\frac{d\theta}{d\eta}\right)^2 / a(\phi)\]Here, \(b''(\theta)\) is the second derivative of \(b(\theta)\) with respect to \(\theta\), and \(\left(\frac{d\theta}{d\eta}\right)^2\) captures the transformation from \(\eta\) to \(\theta\).

Deduction for inverse weight function

The inverse of the weight function is given by:\[w^{-1} = V(\mu) a(\phi) \left\{\frac{d g(\mu)}{d \mu}\right\}^2\]\(V(\mu)\) is the variance function for the distribution of \(Y\), and \(\frac{d g(\mu)}{d \mu}\) represents the derivative of the link function \(g(\mu)\) with respect to the mean \(\mu\).

Determining the adjusted dependent variable

The adjusted dependent variable, \(z\), used in weighted least squares is given by:\[z = \eta + (y - \mu) \frac{d g(\mu)}{d \mu}\]This expression adjusts \(\eta\) by the scaled difference between the observed data \(y\) and its expected value \(\mu\).

Identifying formulas for binomial distribution setup

For a binomial distribution \(R = mY\), with \(\pi = \frac{e^{\eta}}{1+e^{\eta}}\), the weight \(w\) and adjusted dependent variable \(z\) take the explicit forms:- Weight:\[w = \pi (1 - \pi) m\]- Adjusted dependent variable:\[z = \eta + (y - \mu) \frac{1}{\mu(1 - \mu)}\]

Initial values based on data

Initial values for iterative processes can be set without requiring \(\beta\). For initial calculations, use :\(\mu^1 = y\) and \(\eta^1 = g(y)\), where \(g(\cdot)\) is the link function corresponding to the GLM family.

Key concepts

Iterative Weighted Least Squares

Iterative Weighted Least Squares (IWLS) is an important algorithm used to find maximum likelihood estimates in Generalized Linear Models (GLMs). This method is particularly helpful because GLMs can be complex due to different variance and link functions. The goal of IWLS is to iteratively find estimates for the parameters that improve the fit of the model to the data with each iteration.

The key aspect of IWLS is the use of weights. These weights are derived from the second derivative of the log-likelihood function with respect to the model parameters, which provides a measure of how much the likelihood is expected to change. It's calculated as:

• Respectively, the weight formula in a generalized linear model is given by \(w = b''(\theta)\left(\frac{d\theta}{d\eta}\right)^2 / a(\phi)\).
• This expression effectively combines information about the variance and the link function.

This equation plays a crucial role in ensuring that the GLM estimates are optimal at each step, making the overall model robust to variations in the data.

During each iteration, the weights are recalculated, and the model parameters are updated until convergence is achieved. The iterative process allows for adjustment to a variety of data distributions, showcasing the versatility of IWLS.

Variance Function

The variance function, denoted as \(V(\mu)\), in a Generalized Linear Model (GLM) represents how the variance of observations changes as a function of the mean, \(\mu\). It's a critical element because it captures how much the expected variability in the data differs due to intrinsic factors.

Within the framework of GLMs:

• The variance function is specific to the type of distribution being modeled. For example, in the case of a Poisson distribution, the variance function is equal to the mean, i.e., \(V(\mu) = \mu\). For a binomial distribution, \(V(\mu) = \mu (1 - \mu)\).
• This function demonstrates how variability is expected to scale with the expected outcomes, which is crucial for accurately assigning weights in the IWLS method.

In practical terms, understanding the variance function aids in setting realistic models that can effectively predict outcomes based on their unique data characteristics.

This is why it is directly incorporated into the inverse weight calculations in the IWLS procedure. The inverse weight \(w^{-1}\) is dependent on \(V(\mu)\), allowing it to reflect the influence of variance directly in the estimation process.

Link Function

The link function in a Generalized Linear Model (GLM) allows for the fitting of diverse types of data by relating the linear predictor to the mean of the distribution function. It is essentially a transformation function that connects the expected value of the response variable to the linear predictors.

In most GLMs:

• Commonly used link functions include the identity link, log link, and logit link. For example, in a binomial distribution, the logit function \(g(\mu) = \log\left(\frac{\mu}{1-\mu}\right)\) is typically used, linking the probability of success to the predictor.
• The choice of link function can significantly affect the interpretation of the Regression coefficients, providing linear formats where necessary for analytical clarity.

A critical relationship involving the link function is its derivative \(\frac{d g(\mu)}{d \mu}\) as used in weight and adjusted dependent variable calculations. This derivative helps convert differences between observed and mean values into the linear predictor's scale, adjusting the fit of the model.
Consequently, the flexibility offered by link functions enables GLMs to accommodate a wide range of statistical data distributions.

Binomial Distribution

In the context of Generalized Linear Models (GLMs), understanding the binomial distribution is pivotal when dealing with binary or proportion data. The binomial distribution is often applied in scenarios where each observation represents a number of successes over several trials.

Important aspects include:

• A binomial distribution takes two parameters: \(n\) (number of trials) and \(\pi\) (probability of success per trial).
• In GLMs, particularly when using a logit link, the probability \(\pi\) is modeled as \(\pi = \frac{e^{\eta}}{1+e^{\eta}}\), where \(\eta\) is the linear predictor.

For instance, when considering a binomial setup such as \(R = mY\), where \(m\) is a known number of trials, the weight can be explicitly formulated as \(w = \pi (1-\pi) m\). This weight is critical for the IWLS refinement process.
The binomial distribution's variance function, \(V(\mu) = \mu (1-\mu)\), is integral to understanding the data's spread around its mean.By knowing how the binomial distribution properties interface with GLM frameworks, students can better model response variables constrained by two outcomes and achieve more accurate predictions.