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Short question,

When specifying a logistic regression model as below, does one also include B0?

In other words, does the bottom part of the equation look like:

A. 1 + exp(-(B1X1 + B2X2)

B. 1 + exp(-(B0 + B1X1 + B2X2)

And how about and error term?

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It's always option B. So in the notation of your source:

$$ t = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k $$

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