Generalized Linear Model/Examples/Bernoulli Variable/Application
Jump to navigation
Jump to search
Example of Generalized Linear Model
Let $Y$ be a Bernoulli variable with expectation $\mu = p$.
Let the link function $\map g \mu$ of the generalized linear model for $Y$ be given by:
- $\map g \mu = \map \ln {\dfrac p {1 - p} }$
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