I'm interested in logistic regression for modeling a classification problem. I tried to study logistic regression with two books, "Discrete Choice Methods with Simulation" (Train) and "Applied Logistic Regression" (Hosmer and Lemeshow). To clarify my question, I picked two logit model examples from those books.
Household's choice between a gas and an electric heating system (binary logit)
Let the subscripts $g$ and $e$ denote gas and electric, $PP$ and $OC$ are the purchase price and operating cost, $\beta_1$ and $\beta_2$ are scalar parameters. Then the probability that the household choose gas heating is
$$P_g = \frac{e^{\beta_1PP_g+\beta_2OC_g}}{e^{\beta_1PP_g+\beta_2OC_g}+e^{\beta_1PP_e+\beta_2OC_e}}$$Relationship between age and the presence or absence of CHD (binary logit)
Let $x$ be the age, the subscripts $0$ and $1$ denote absence and presence of the CHD and $\beta_1^0$, $\beta_2^0$, $\beta_1^1$ and $\beta_2^1$ are scalar parameters. Then the probability that a $x$ years old person has a CHD is:
$$P_1 = \frac{e^{\beta_1^1+\beta_2^1x}}{e^{\beta_1^1+\beta_2^1x}+e^{\beta_1^0+\beta_2^0x}}$$
(Although the model needs two scalar parameters to be estimated, I wrote the formula in this form to clarify the question.)
I am familiar with the logistic regression showed in the second example. The independent variable (age in this example) is included in both choices (or classes), and of course, the value of independent variable is same for different classes. And there are coefficients for each class and each independent variable which has to be estimated, and those coefficients are different for different classes. However, the model of the first example has variables which are different for each choices ($PP_e$, $PP_g$, $OC_e$ and $OC_g$), and the coefficients are the same between choices.
Roughly, I can understand two models separately, but I cannot build the unified concept of the logistic regression. They seems different to me. For example, many textbooks and materials provide the (multinomial) logistic regression formula, which contains independent variables and the coefficients (which are different for each classes). With this formula, I can understand the second example, but can this formula incorporate the situation showed in the first example? In other words, does this formula have the variables which are different for each class?
