1

In a presentation of the binomial distribution on this website, the probability of the success of a trial is presented as $\pi=g^{-1}(\eta)$, with $g$ the inverse link function. $\eta=\alpha+\boldsymbol{\mathrm{x}^\mathsf{T}\beta}$ is described as a "linear predictor."

Is $^\mathsf{T}$ here the transpose symbol? If so, what does it mean in the context of a linear model? If not, what is this symbol?

Namenlos
  • 375
  • 2
  • 8
  • 7
    It's a transpose symbol. $x$ is a column vector, so $x^t$ is a row vector. $x^t \beta$ is a row vector times a column vector, hence a scalar. Thus, $x^t \beta$ is just a fancy way to write a dot product. – Matthew Drury Jul 03 '18 at 04:37

1 Answers1

3

What the equation is describing is not a "binomial distribution", but a generalized linear model. Such model is described in terms of linear predictor:

$$ \eta = \alpha+\boldsymbol{\mathrm{x}^\mathsf{T}\beta} $$

link function

$$ \pi = g^{-1}(\eta) $$

and likelihood function

$$ Y \sim \mathcal{Bernoulli}(\pi) $$

as described in multiple places on this site. When using Bernoulli distribution as a likelihood function, we are talking about logistic regression.

As about the $^\mathsf{T}$ symbol, it stands for transpose (alternatively you could see $^{'}$), so it is a shortcut to saying

$$ \mathrm{a}^\mathsf{T} \mathrm{b} = a_1 b_1 + a_2 b_2 + \dots + a_k b_k $$

where, depending on context, it could be matrix times vector, or vector times vector.

Tim
  • 108,699
  • 20
  • 212
  • 390