Problem 1 - An Introduction to Generalized Linear Models

Question

The data in Table 4.5 show the numbers of cases of AIDS in Australia by date of diagnosis for successive 3 -months periods from 1984 to 1988 (Data from National Centre for HIV Epidemiology and Clinical Research \(1994 .\) In this early phase of the epidemic, the numbers of cases seemed to be increasing exponentially. (a) Plot the number of cases \(y_{i}\) against time period \(i(i=1, \ldots, 20)\) (b) A possible model is the Poisson distribution with parameter \(\lambda_{i}=i^{\theta}\) or equivalently \\[ \log \lambda_{i}=\theta \log i \\] Plot \(\log y_{i}\) against \(\log i\) to examine this model (c) Fit a generalized linear model to these data using the Poisson distribution, the log-link function and the equation \\[ g\left(\lambda_{i}\right)=\log \lambda_{i}=\beta_{1}+\beta_{2} x_{i} \\] $$\begin{array}{crrrr} \hline & {}{\underline{\phantom{xx}}} {\text { Quarter }} \\ \text { Year } & 1 & 2 & 3 & 4 \\ \hline 1984 & 1 & 6 & 16 & 23 \\ 1985 & 27 & 39 & 31 & 30 \\ 1986 & 43 & 51 & 63 & 70 \\ 1987 & 88 & 97 & 91 & 104 \\ 1988 & 110 & 113 & 149 & 159 \\ \hline \end{array}$$ where \(x_{i}=\log i .\) Firstly, do this from first principles, working out expressions for the weight matrix \(\mathbf{W}\) and other terms needed for the iterative equation \\[ \mathbf{X}^{T} \mathbf{W} \mathbf{X} \mathbf{b}^{(m)}=\mathbf{X}^{T} \mathbf{W} \mathbf{z} \\] and using software which can perform matrix operations to carry out the calculations. (d) Fit the model described in (c) using statistical software which can perform Poisson regression. Compare the results with those obtained in (c).

Step-by-Step Solution

Answer

Plot the data, apply a GLM using Poisson regression, and verify using software; compare results for consistency.

Step 1: Organize the Data

List the number of AIDS cases by quarter from 1984 to 1988. This will help in plotting and analyzing the data. The quarters correspond to 3-month periods, meaning each year has four entries.

Step 2: Plot Number of Cases vs Time Periods

Create a scatter plot with the time period (from 1984-Q1 to 1988-Q4) on the x-axis and the number of AIDS cases on the y-axis. There are 20 data points to represent consecutive quarters across the years provided.

Step 3: Logarithmic Transformation

Transform the number of cases and time periods using logarithms. Plot \(\log y_i\) against \(\log i\) to visually evaluate whether the data follows the expected model \(\log \lambda_{i}=\theta \log i\). This can help in preliminary assessment of the Poisson distribution.

Step 4: Generalized Linear Model Setup

Define the generalized linear model (GLM) with a Poisson distribution and a log-link function. Use the equation \(g\left(\lambda_{i}\right)=\log \lambda_{i}=\beta_{1}+\beta_{2} \log i\). Prepare matrices and vectors needed for GLM fitting.

Step 5: Construct Weight Matrix and Other Terms

Construct the weight matrix \(\mathbf{W}\), which is diagonal with elements equal to \(\lambda_i\). Prepare the model matrix \(\mathbf{X}\), observed data \(\mathbf{y}\), and initial estimates for \(\beta\). Compute the adjusted dependent variable \(\mathbf{z}\) using the link function and initial response estimates.

Step 6: Iterative Equation Solution

Solve the equation \(\mathbf{X}^{T} \mathbf{W} \mathbf{X} \mathbf{b}^{(m)}=\mathbf{X}^{T} \mathbf{W} \mathbf{z}\) iteratively using numerical methods or software for matrix operations. Update \(\beta\) until convergence is achieved.

Step 7: Model Fitting Using Software

Utilize statistical software capable of performing Poisson regression to fit the model based on the data. Use the log-link function and compare results with the first-principles calculations to verify consistency.

Step 8: Analyze and Compare Results

Compare the scaled deviance, coefficients, and goodness of fit from both methods to validate the model. Check whether the parameter estimates and residuals from the software align with manual calculations and if they confirm the assumed model structure.

Key Concepts

Poisson Distribution

The Poisson distribution is a statistical distribution used to model count data. It is particularly useful when the events are rare and occur independently. In the context of modeling AIDS cases, the Poisson distribution helps in understanding how frequently new cases were diagnosed over time.

The key parameter of the Poisson distribution is \(\lambda \), which represents the mean number of events (here, AIDS cases) in a fixed interval of time. Since the cases are expected to grow, \(\lambda_i = i^{\theta}\)suggests an increasing pattern, dependent on the time period \(i\).

Key points to remember about the Poisson distribution:

• The distribution is defined for positive integers, which naturally relate to count data such as the number of AIDS cases.
• It assumes that events are distributed independently.
• The variance of the distribution equals its mean.

By plotting the number of cases against time, we can visually assess how well the Poisson model fits the data.

Log-Link Function

In a Generalized Linear Model (GLM), the link function is used to transform the mean of the response variable to be modeled linearly. The log-link function is a popular choice when working with count data, as it ensures that predictions remain positive by linking the linear predictors to the logarithm of the mean of the distribution.

The log-link function can be defined as:\[g(\lambda_{i}) = \log(\lambda_{i})\]where \(\lambda_{i}\)is the expected count, and \(\beta_1 + \beta_2 x_i\)represents the linear predictor.

Benefits of using the log-link function in this model:

• Transforms multiplicative relationships in the data to additive ones.
• Maintains non-negativity of the predictions.
• Simplifies dealing with exponential growth, common in epidemic data.

When the transformed data follows a straight line in the log-log plot, it suggests that the model is appropriate.

Iterative Equation

When fitting a model using a GLM, an iterative approach is often employed to find the best estimates for the model parameters. This is necessary because closed-form solutions for the coefficients in models like the Poisson GLM with a log-link function do not usually exist.

The iterative equation used is:\[\mathbf{X}^{T} \mathbf{W} \mathbf{X} \mathbf{b}^{(m)} = \mathbf{X}^{T} \mathbf{W} \mathbf{z}\]here, \(\mathbf{W}\)is the weight matrix related to current estimates ofg(the predicted response), and \(\mathbf{z}\)is the adjusted dependent variable.

Steps involved:

• Build initial estimates of parameters \(\beta\).
• Calculate the weighted least-squares updating equation.
• Iterate: update parameter estimates using matrix operations, check for convergence.
• Use statistical software when manual calculations become complex and time-consuming.

Through iterations, this method refines \(\beta\) until convergence, ensuring a strong fit to the epidemic data.

AIDS Epidemic Data Analysis

Analyzing AIDS epidemic data requires handling historical and increasing trends effectively. The initial outbreak phase, like the one in Australia from 1984 to 1988, saw exponential growth in the number of cases.

To capture such trends:

• Organize data chronologically to track changes over time.
• Plot the number of cases against time periods to visualize trends.
• Use logarithmic transformations to stabilize variance and linearize growth patterns.
• Apply GLMs, particularly with a Poisson distribution and log-link function for count data.

Comparing manual and software-generated Poisson regression models is crucial. This ensures accuracy and that the model adequately reflects the data.

Understanding the methods and rationale behind these steps aids in modeling real-world epidemic patterns, enabling the development of informed public health strategies.