Generalized Linear Model/Examples/Bernoulli Variable

Example of Generalized Linear Model

Let $Y$ be a Bernoulli variable with expectation $\mu = p$.

Then the link function $\map g \mu$ of the generalized linear model for $Y$ is given by:

$\map g \mu = \map \ln {\dfrac p {1 - p} }$

Application

This can be applied to the study of the toxicity of insecticides.

Let $p_i$ be the probability that a particular insect will die when exposed to a dose $x_i$ of an insecticide.

Under a set of usual conditions:

$\map \ln {\dfrac {p_i} {1 - p_i} } = \beta_0 + \beta_1 x$

If batches of insects are subjected to doses $x_1, x_2, \ldots, x_i, \ldots, x_n$ of the insecticide, the proportion that dies in batch $i$ provides an estimate of $p_i$.

The parameters $\beta_0$ and $\beta_1$ can be obtained by maximum likelihood estimation.

Iterative procedures are usually needed, because the variance of $\map \ln {\dfrac p {1 - p} }$ is a function of $p$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generalized linear models
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generalized linear models