5J (Statistical Modelling)

A generalized linear model has independent responses whose distributions belong to an exponential family:

f_{i} (y_{i}; θ_{i}, ϕ) = exp {\frac{y _{i} θ _{i} - b ( θ _{i} )}{a _{i} ( ϕ )} + c_{i} (y_{i}, ϕ)} .

μ_{i} = b^{'} (θ_{i})

and

Var (Y_{i}) = a_{i} (ϕ) b^{''} (θ_{i})

. A specified one-to-one link function

g

connects the mean to the linear predictor:

g (μ_{i}) = x_{i}^{T} β

g (μ_{i}) = θ_{i}

Y_{i} \sim Bin (m_{i}, p_{i})

independently, with known trial counts

m_{i}

and probabilities related to the predictor. The logistic regression and probit regression specifications are respectively

lo g \frac{p _{i}}{1 - p _{i}} = x_{i}^{T} β, p_{i} = \frac{e ^{x_{i}^{T} β}}{1 + e ^{x_{i}^{T} β}}; Φ^{- 1} (p_{i}) = x_{i}^{T} β, p_{i} = Φ (x_{i}^{T} β) .

Here

Φ

is the standard normal distribution function. The binomial log-density contains

y_{i} lo g (p_{i} / (1 - p_{i})) + m_{i} lo g (1 - p_{i})

, so its natural parameter is the log-odds: the logistic link is canonical. If the response mean is written as

μ_{i} = m_{i} p_{i}

, the canonical link is

g_{i} (μ_{i}) = lo g (μ_{i} / (m_{i} - μ_{i}))

 Ancestors (9)

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